# Regularity of the superstring supermeasure and the superperiod map

Giovanni Felder  Department of mathematics, ETH Zurich, 8092 Zurich,
Switzerland<EMAIL_ADDRESS>,  David Kazhdan  Einstein Institute of
Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
<EMAIL_ADDRESS>and  Alexander Polishchuk  Department of Mathematics,
University of Oregon, Eugene, OR 97403, USA; National Research University
Higher School of Economics; and Korea Institute for Advanced Study
<EMAIL_ADDRESS>
###### Abstract.

The supermeasure whose integral is the genus $g$ vacuum amplitude of
superstring theory is potentially singular on the locus in the moduli space of
supercurves where the corresponding even theta-characteristic has nontrivial
sections. We show that the supermeasure is actually regular for $g\leq 11$.
The result relies on the study of the superperiod map. We also show that the
minimal power of the classical Schottky ideal that annihilates the image of
the superperiod map is equal to $g$ if $g$ is odd and is equal to $g$ or $g-1$
if $g$ is even.

##  1\. Introduction

The moduli space of supercurves ${\mathcal{M}}$ (of given genus $g$) has been
studied in many papers, including [3, 4, 6, 11], motivated by superstring
perturbation theory, see [20] for a recent account with references to the
original literature. The space ${\mathcal{M}}$ is a superstack of complex
dimension $3g-3|2g-2$. The corresponding reduced stack ${\mathcal{M}}_{0}$ is
the moduli stack of pairs $(C,L)$, where $C$ is a usual smooth projective
curve of genus $g$ and $L$ is a theta-characteristic on $C$ (also known as
spin structure). The space ${\mathcal{M}}_{0}$ (and hence ${\mathcal{M}}$) has
two components, which we call even and odd, that are distinguished by the
condition that $L$ is even or odd.

In the first part of this paper we study the behavior of the superstring
supermeasure on the even component ${\mathcal{M}}^{+}$ of the moduli space of
supercurves ${\mathcal{M}}$ (the supermeasure vanishes on the odd component).
The supermeasure is a priori only defined on a Zariski open subset of the
moduli space where the corresponding theta-characteristic has no nontrivial
section. E. Witten recently revisited and extended the study of the
supermeasure in [21] and argued that the supermeasure actually extends
smoothly to the whole moduli space for low genus, indicating potential
difficulties starting at genus 11.

Our main result is the regularity of this supermeasure for $g\leq 11$ along
the locus where the corresponding theta-characteristic acquires a section (see
Theorem 5.1), confirming Witten’s expectation.

More precisely, the contribution of genus $g$ to the vacuum amplitude of type
II superstring theory is given as an integral of a supermeasure (section of
the Berezinian of the cotangent bundle) on a suitable cycle in
${\mathcal{M}}^{+}\times\overline{{\mathcal{M}}^{+}}$ (where the bar denotes
the complex conjugate space). This supermeasure can be expressed in
holomorphic terms in terms of the super analogue of the Mumford isomorphism,
constructed in [18], see [17]. The relevant special case of the Mumford
isomorphism is a canonical isomorphism

$\operatorname{Ber}^{5}(R\pi_{*}\omega_{X/{\mathcal{M}}})\cong\operatorname{Ber}(T^{*}_{\mathcal{M}})$

of the fifth tensor power of the Berezinian of the derived direct image of the
relative dualizing sheaf (Berezinian) $\omega_{X/{\mathcal{M}}}$ of the
universal curve $\pi\colon X\to{\mathcal{M}}$ with the Berezinian of the
holomorphic cotangent bundle.

The supermeasure is defined on a certain open subset of the product
${\mathcal{M}}^{+}\times\overline{{\mathcal{M}}^{+}}$. In the corresponding
reduced space ${\mathcal{M}}^{+}_{0}\times\overline{{\mathcal{M}}^{+}_{0}}$ we
have a natural locus of pairs of curves with theta-characteristics
$((C,L),(C^{\prime},L^{\prime}))$ such that $C=C^{\prime}$, which we call the
quasi-diagonal. The supermeasure can be defined as a meromorphic section of
the holomorphic Berezinian on a neighborhood of the quasi-diagonal in
${\mathcal{M}}^{+}\times\overline{{\mathcal{M}}^{+}}$. It is the image of the
canonical section of

$\operatorname{Ber}^{5}(R\pi_{*}\omega_{X/{\mathcal{M}}})\boxtimes\operatorname{Ber}^{5}(R\pi_{*}\omega_{\overline{X}/\overline{\mathcal{M}}})$

induced by the embedding of $\pi_{*}\omega_{X/{\mathcal{M}}}$ into the local
system $R^{1}\pi_{*}\pi^{-1}\mathcal{O}_{{\mathcal{M}}}$, which has a
canonical pairing. The possible problem with the regularity occurs when the
higher direct images $R^{i}\pi_{*}\omega_{X/{\mathcal{M}}}$ are not locally
free. This happens near points $(C,L)$ such that $H^{0}(C,L)\neq 0$. We show
in Theorem 5.1 that the supermeasure extends to a regular section for $g\leq
11$.

Note that if we view ${\mathcal{M}}^{+}\times\overline{{\mathcal{M}}^{+}}$ as
a real supermanifold then a section of the holomorphic Berezinian can be
viewed as a codimension $6g-6|0$ closed integral form on it. Thus, we have to
extend the quasi-diagonal to a sub-supermanifold $\Delta$ in
${\mathcal{M}}^{+}\times\overline{{\mathcal{M}}^{+}}$ of real dimension
$6g-6|4g-4$. We can integrate over $\Delta$ the restriction of the holomorphic
Berezinian provided the supermeasure is regular in its neighborhood. The
integral does not depend on a choice of $\Delta$ since the integral form is
closed (there are convergence issues at infinity, which we do not consider in
this paper).

In the second part of the paper we study the superperiod map at the generic
point of the component ${\mathcal{M}}^{+}$. The superperiod map gives rise to
a natural super-Schottky ideal ${\mathcal{I}}_{s-Sch}$ on the period domain,
which is containd in the usual Schottky ideal ${\mathcal{I}}_{Sch}$ describing
the image of the classical period map. The problem of studying
${\mathcal{I}}_{s-Sch}$ was raised in [2, Rem. 6.10]. Our second main result
concerns the minimal power $d$ such that

${\mathcal{I}}_{Sch}^{d}\subset{\mathcal{I}}_{s-Sch}.$

It is easy to see that $d\leq g$. We prove that for $g$ odd, one has $d=g$,
while for $g$ even, $d\geq g-1$.

The paper is organized as follows: in Section 2 we review some standard facts
about supercurves and their moduli. In Section 3 we study the direct image
$\pi_{*}\omega_{X/S}$ and the higher direct image $R^{1}\pi_{*}\omega_{X/S}$
of the relative Berezinian of the universal supercurve and show that they are
locally free on the open set where the even theta-characteristic has no
nontrivial sections and are not locally free away from it, as well as on the
component of moduli space with odd theta-characteristic. In the following
Section 5 we study the behaviour of the period matrix as we approach the
divisor of even theta-characteristics with nontrivial sections. The key idea
is to realize the symplectic local system $R^{1}\pi_{*}\mathbb{C}_{X/S}$, for
a family of supercurves $\pi\colon X\to S$, where
$\mathbb{C}_{X/S}=\pi^{-1}\mathcal{O}_{S}$, as a symplectic reduction of a
bigger sumplectic bundle ${\mathcal{V}}$ with a Lagrangian subbundle
$L_{\mathrm{can}}$ whose reduction is $\pi_{*}\omega_{X/S}$. Away from the
divisor of even theta-characteristics with nontrivial sections, the Lagrangian
subbundle $\pi_{*}\omega_{X/S}$ of $R^{1}\pi_{*}\mathbb{C}_{X/S}$ is locally
the graph of the period matrix, a map $\Omega\colon\Lambda^{\prime}\to\Lambda$
for some Lagrangian splitting
$R^{1}\pi_{*}\mathbb{C}_{X/S}=\Lambda\oplus\Lambda^{\prime}$. The
singularities of $\Omega$ are governed by the map
$L_{\Lambda}\to\mathcal{V}/L_{\mathrm{can}}$ where the Lagrangian subbundle
$L_{\Lambda}\subset\mathcal{V}$ projects to $\Lambda$. The period matrix has a
pole on the locus where this map fails to be an isomorphism. We show that this
locus is locally the zero set of a Pfaffian $f$ and that $f\Omega$ is regular.
This result allows us to compute the behavior of the supermeasure as we
approach the locus, leading to our regularity result. In the remaining
sections we study the second variation of the period matrix in the odd
directions at a generic point of the moduli space of supercurves with even
theta characteristic. An analytic formula for the second variation was given
by D’Hoker and Phong in [5] in terms of the Szegö kernel. We present in
Section 6 an alternative approach by expressing the second variation as a
Massey product. We study this Massey product in Section 7 giving explicit
formulas, in particular for hyperelliptic curves. This formula is then used in
Section 8 to prove our result on the super-Schottky ideal.

### Conventions

We work with either complex algebraic or analytic superspaces. Whenever we
work with the sheaf $\operatorname{\mathbb{C}}_{X/S}$, we use classical
topology and analytic framework. For a superscheme $S$ we denote by $|S|$ the
underlying topological space. By a curve we always mean a connected smooth
projective curve over $\operatorname{\mathbb{C}}$. When considering moduli
spaces of curves we always assume that the genus $g$ of a curve is $\geq 2$.
By a supercurve we always mean a compact complex supermanifold (or a proper
smooth algebraic supervariety) of dimension $1|1$ with a superconformal
structure (see Section 2.1 for details). These are often called super Riemann
surfaces. For a family of smooth curves $C/S$ (resp., supercurves $X/S$) we
denote by $\omega_{C/S}$ (resp., $\omega_{X/S})$ the relative dualizing sheaf
(given by the relative Berezinian in the supercase). Since we work in the
super-context throughout the paper, we drop the prefix “super” in supervector
bundles and supervector spaces.

### Acknowledgements

G.F. is partially supported by the National Centre of Competence in Research
“SwissMAP — The Mathematics of Physics” of the Swiss National Science
Foundation. He thanks the Hebrew University of Jerusalem, where part of this
work was done, for hospitality. D.K. is partially supported by the ERC under
grant agreement 669655. A.P. is partially supported by the NSF grant
DMS-1700642, by the National Center of Competence in Research “SwissMAP — The
Mathematics of Physics” of the Swiss National Science Foundation, and by the
Russian Academic Excellence Project ‘5-100’. While working on this project,
A.P. was visiting ETH Zurich and Hebrew University of Jerusalem. He would like
to thank these institutions for hospitality and excellent working conditions.

##  2\. Supercurves

In this section we review some standard material on supercurves (also called
super Riemann surfaces).

###  2.1. Berezinian of a supercurve

Let $\pi\colon X\to S$ be a smooth proper supercurve over a superscheme $S$.
By definition the tangent sheaf $T_{X/S}$ has a locally free
$\mathcal{O}_{X}$-submodule ${\mathcal{D}}$ of rank $(0,1)$ such that the
composition of the canonical projection with the Lie bracket
${\mathcal{D}}\otimes{\mathcal{D}}\to T_{X/S}/{\mathcal{D}}$, which is a map
of $\mathcal{O}_{X}$-modules, is an isomorphism. Thus, we have an exact
sequence

$0\to{\mathcal{D}}\to T_{X/S}\to{\mathcal{D}}^{\otimes 2}\to 0$ (2.1)

The dual sequence is

$0\to({\mathcal{D}}^{\vee})^{\otimes
2}\to\Omega_{X/S}^{1}\to{\mathcal{D}}^{\vee}\to 0$ (2.2)

and taking Berezinians shows that
${\mathcal{D}}^{\vee}\cong\omega_{X/S}:=\operatorname{Ber}(\Omega_{X/S}^{1})$.

Below we work with sheaves in the classical topology.

The map $\Omega_{X/S}^{1}\to{\mathcal{D}}^{\vee}$ defines an
$\mathcal{O}_{S}$-linear derivation

$\delta\colon\mathcal{O}_{X}\to\omega_{X/S}.$

The map $\delta$ is surjective and its kernel is the sheaf
$\mathbb{C}_{X/S}=\pi^{-1}\mathcal{O}_{S}$ of functions that are locally
constant along the fibers.

Locally there exist fiber coordinates $z,\theta$ such that $\mathcal{D}$ is
spanned by $D=\partial_{\theta}+\theta\partial_{z}$. In these coordinates the
map $\delta$ is $f\mapsto[dz|d\theta]D(f)$ where $[dz|d\theta]$ is the local
section of the Berezinian determined by the basis $dz,d\theta$ of
$\Omega^{1}_{X/S}$.

The long exact sequence associated with

$0\to\mathbb{C}_{X/S}\to\mathcal{O}_{X}\rTo{\delta}\omega_{X/S}\to 0$ (2.3)

includes

$\pi_{*}\omega_{X/S}\to R^{1}\pi_{*}\mathbb{C}_{X/S}\to
R^{1}\pi_{*}\mathcal{O}_{X}\to R^{1}\pi_{*}\omega_{X/S}\to
R^{2}\pi_{*}\mathbb{C}_{X/S}\to 0,$ (2.4)

(since $R^{2}\pi_{*}\mathcal{O}_{X}=0$). Locally we have
$R^{i}\pi_{*}\mathbb{C}_{X/S}\cong
H^{i}(|X_{0}|,\mathbb{C})\otimes\mathcal{O}_{S}$, where $|X_{0}|$ is the
underlying topological space of a fibre $X_{0}$. Thus,
$R^{1}\pi_{*}\mathbb{C}_{X/S}\cong\mathcal{O}_{S}^{\oplus 2g}$ and
$R^{2}\pi_{*}\mathbb{C}_{X/S}\cong\mathcal{O}_{S}$. In fact, the latter
isomorphism is global, and we have in particular a surjective trace map

$\tau\colon R^{1}\pi_{*}\omega_{X/S}\to\mathcal{O}_{S}$ (2.5)

that plays a role in duality theory (see Sec. 2.8 below).

###  2.2. Supercurves over an even base

Assume that $S$ is even. Then the decomposition of ${\mathcal{O}}_{X}$ into
the even and odd components gives a splitting of $X$:

$\mathcal{O}_{X}\cong{\bigwedge}^{\bullet}_{\mathcal{O}_{C}}L=\mathcal{O}_{C}\oplus
L$

where $L$ is a relative theta-characteristic on the underlying usual family of
curves $C\to S$, with a fixed isomorphism

$L\otimes L\to\omega_{C/S}.$ (2.6)

The dualizing sheaf $\omega_{X/S}$ is $\omega_{C}\oplus L$, with the
$\mathcal{O}_{X}$-module structure such that $L$ acts by zero on $\omega_{C}$
and its action on $L$ is given by the structure map (2.6). The derivation
$\delta$ is $(d,\mathrm{id})$. The distribution ${\mathcal{D}}$ is the
orthogonal complement of $\mathrm{Ker}(\bar{\delta})$ where
$\bar{\delta}\colon\Omega^{1}_{X/S}\to\omega_{X/S}$ is the map of
$\mathcal{O}_{X}$-modules associated with $\delta$.

######  Lemma 2.1.

Let $\theta$ be a local nowhere vanishing section of $L$, and let
$\alpha=\theta^{2}\in\omega_{C/S}$ be the image of $\theta\otimes\theta$ under
the structure map (2.6). Then

${\mathcal{D}}=\mathrm{Ker}(\alpha-\theta d\theta),$

where we use the natural embedding of $\omega_{C/S}=\Omega_{C/S}$ into
$\Omega_{X/S}$.

###### Proof.

We need to check that $\alpha-\theta d\theta$ is a nonzero section of
$\mathrm{Ker}(\bar{\delta})$. By definition $\bar{\delta}$ is the map of
$\mathcal{O}_{X}$ modules sending an exact form $d(f\oplus\psi)$ to
$df\oplus\psi$. Thus, $\bar{\delta}(\theta d\theta)=\theta^{2}$ and, since
$\alpha\in\mathcal{O}_{C}d\mathcal{O}_{C}$, $\delta\alpha=\alpha$. ∎

###  2.3. Local automorphisms

Let $X_{0}\to S_{0}$ be a smooth supercurve over an even superscheme $S_{0}$.
Let $S_{0}\hookrightarrow S$ be a closed embedding into a superscheme
corresponding to the nilpotent ideal $I_{S}$ of $\mathcal{O}_{S}$ (so that
$S_{0}=S_{\mathrm{red}}$). Let $\pi\colon X\to S$ be a smooth supercurve such
that $X|_{S_{0}}=X\times_{S}S_{0}\cong X_{0}$. The $S$-automorphisms of the
supercurve $X$ that induce the identity on $X_{0}$ act as the identity on the
underlying topological space of $X$. They are thus sections of a sheaf
$\mathrm{Aut}^{0}_{X/S}$ of unipotent groups. The Lie algebra
$\mathfrak{g}_{X/S}$ of $\mathrm{Aut}^{0}_{X/S}$ consists of even
superconformal vector fields vanishing on $X_{0}$. It is a sheaf of nilpotent
graded Lie algebras on $X$ over $\pi^{-1}\mathcal{O}_{S}$.

###  2.4. Extensions along odd moduli

Suppose again that a smooth supercurve $X_{0}\to S_{0}$ over an even
superscheme is given. Up to isomorphism this is the same as giving an ordinary
curve over $S_{0}$ with a theta-characteristic. For each $S$ with
$S_{\mathrm{red}}=S_{0}$ we ask about smooth supercurves $X\to S$ restricting
to $X_{0}$ over $S_{0}$.

######  Definition 2.2.

An extension of $X_{0}\to S_{0}$ along $S$ is a supercurve $X$ over $S$
equipped with an isomorphism $X\times_{S_{0}}S\cong X_{0}$ as supercurves over
$S_{0}$. Two extensions $X,X^{\prime}$ along $S$ are isomorphic if there is an
isomorphism $X\to X^{\prime}$ of supercurves over $S$ inducing the identity on
$X_{0}$.

Working locally on $S$, we may assume that $S=(|S_{0}|,\mathcal{O}_{S})$ with
$\mathcal{O}_{S}=\mathcal{O}_{S_{0}}\otimes_{\mathbb{C}}\bigwedge W$ for some
finite dimensional vector space $W$. Then we get an extension $X\to S$, called
the trivial extension, by setting $X=(|X_{0}|,\mathcal{O}_{X})$ and
$\mathcal{O}_{X}=\mathcal{O}_{X_{0}}\otimes\bigwedge W$. Any other extension
along $S$ is obtained from the trivial extension by twisting by a $1$-cocycle.

######  Lemma 2.3.

(cf. [11, Lem. 2.3]) Let $X$ be an extension of $X_{0}$ along $S$. Then the
isomorphism classes of extensions of $X_{0}$ along $S$ are in natural one to
one correspondence with the first non-abelian cohomology
$H^{1}(X,\operatorname{Aut}^{0}_{X/S})$.

###### Proof.

All extensions are locally isomorphic. Thus every point $x\in|X_{0}|$ has a
neighbourhood $U\subset|X_{0}|$ so that any extension of
$(U,\mathcal{O}_{X_{0}}|_{U})$ along $S$ is isomorphic to
$(U,\mathcal{O}_{X}|_{U})$. Thus for any extension $X^{\prime}$ we can find an
open covering $\mathcal{U}=(|U_{i}|)$ of $|X_{0}|$ with corresponding
supercurves $U_{i}=(|U_{i}|,\mathcal{O}_{X}|_{|U_{i}|})$, and isomorphisms
$\varphi_{i}\colon U_{i}\to U_{i}^{\prime}$. On non-empty intersections
$|U_{i}|\cap|U_{j}|$ we get automorphisms
$\varphi_{ij}=\varphi_{j}^{-1}\circ\varphi_{i}\in\mathrm{Aut}^{0}_{X/S}(U_{i}\cap
U_{j})$ defining a $1$-cocycle with values in $\mathrm{Aut}^{0}_{X/S}$. If
$\varphi_{i}^{\prime}$ are a different choice of local isomorphisms then
$\varphi_{i}^{\prime}=\varphi_{i}\circ f_{i}$ with
$f_{i}\in\mathrm{Aut}_{X/S}^{0}$ and $\varphi_{ij}$ is replaced by the
equivalent cocycle $f_{j}^{-1}\circ\varphi_{ij}\circ f_{i}$. Thus to every
extension $X^{\prime}$ we associate a well-defined characteristic class
$[\varphi_{ij}]\in H^{1}(X,\operatorname{Aut}_{X/S}^{0})$. Conversely, given a
class in $H^{1}(X,\operatorname{Aut}_{X/S}^{0})$ represented by a Čech cocycle
$(\varphi_{ij})$ we obtain an extension $X^{\prime}$ by gluing $\sqcup U_{i}$
over $|U_{i}|\cap|U_{j}|$ by the isomorphisms $\varphi_{ij}$. By construction
the characteristic class of $X^{\prime}$ is $[\varphi_{ij}]$. ∎

###  2.5. Local description of $\operatorname{Aut}^{0}_{X/S}$

We give a local description of $\operatorname{Aut}^{0}_{X/S}$ (cf. [3] and
[20, Section 2.1.1]). We can choose local fibre coordinates $z,\theta$ on an
open set $U\subset X$ so that the distribution ${\mathcal{D}}$ defining the
supercurve is spanned by $\partial_{\theta}+\theta\partial_{z}$ or,
equivalently, is the kernel of $dz-\theta d\theta$.

We are going to construct elements of $\Gamma(U,\operatorname{Aut}^{0}_{X/S})$
of two types. For $f=f(z)\in\mathcal{O}_{X}^{+}(U)$, an even function such
that $\partial_{\theta}f=0$ and $f\equiv z\mod I_{S}$, we define a
superconformal map $S_{f}$ by

$z\mapsto S_{f}^{*}z=f(z),\quad\theta\mapsto
S_{f}^{*}\theta=\sqrt{f^{\prime}(z)}\,\theta,$ (2.7)

where $f^{\prime}=\partial_{z}f$. Note that the square root is well-defined
since $f^{\prime}(z)=1+$nilpotent. On the other hand, for
$\varphi=\varphi(z)\in\mathcal{O}_{X}^{-}(U)$, an odd function such that
$\partial_{\theta}\varphi=0$, we define $T_{\varphi}$ by

$z\mapsto T_{\varphi}^{*}z=z+\theta\varphi(z),\quad\theta\mapsto
T_{\varphi}^{*}\theta=\theta+\varphi(z)+\textstyle{\frac{1}{2}}\,\theta\,\varphi(z)\,\varphi^{\prime}(z).$
(2.8)

######  Lemma 2.4.

The maps $S_{f},T_{\varphi}$ are in $\Gamma(U,\operatorname{Aut}^{0}_{X/S})$
and every element of this group can be uniquely written in the form
$S_{f}\circ T_{\varphi}$.

###### Proof.

A superconformal transformation is an automorphism such that $g^{*}(dz-\theta
d\theta)=\lambda(dz-\theta d\theta)$ for some function $\lambda$. Let
$g^{*}z=g_{0}(z)+\theta g_{1}(z)$ and $g^{*}\theta=h_{0}(z)+\theta h_{1}(z)$,
with $g_{0},h_{1}$ even and $g_{1},h_{0}$ odd. The condition that $g$ reduces
to the identity on $X_{0}$ is $g_{0}(z)=z+$nilpotent and
$h_{1}(z)=1+$nilpotent. Thus $g$ is superconformal if and only if

$h_{0}(z)=\frac{g_{1}(z)}{h_{1}(z)},\quad
h_{1}(z)^{2}=g^{\prime}_{0}(z)+\frac{g_{1}(z)g_{1}^{\prime}(z)}{g_{0}^{\prime}(z)}.$

In this case $\lambda=g_{0}^{\prime}(z)+h_{0}(z)h_{0}^{\prime}(z)+2\theta
h_{0}(z)h_{1}^{\prime}(z)$. Since $g_{0}^{\prime}(z)=1+$nilpotent, we can take
the square root:

$h_{1}(z)=\pm\sqrt{g_{0}^{\prime}(z)}\left(1+\frac{g_{1}(z)g_{1}^{\prime}(z)}{2g^{\prime}_{0}(z)^{2}}\right)$

The condition that $g$ restricts to the identity on $X_{0}$ selects the
positive sign. This gives

$g^{*}z=g_{0}(z)+\theta g_{1}(z),\quad
g^{*}\theta=\sqrt{g_{0}^{\prime}(z)}\left(\frac{g_{1}(z)}{g_{0}^{\prime}(z)}+\theta\left(1+\frac{g_{1}(z)g_{1}^{\prime}(z)}{2g^{\prime}_{0}(z)^{2}}\right)\right).$

If we set $g_{0}(z)=f(z),g_{1}(z)=f^{\prime}(z)\varphi(z)$, the formulas
simplify slightly:

$g^{*}z=f(z)+\theta f^{\prime}(z)\varphi(z)\quad
g^{*}\theta=\sqrt{f^{\prime}(z)}\left(\varphi(z)+\theta(1+\textstyle{\frac{1}{2}}\varphi(z)\varphi^{\prime}(z))\right).$

This reduces to $S_{f}^{*}$ for $\varphi=0$ and to $T^{*}_{\varphi}$ for
$f(z)=z$. For general $f,\varphi$ we get $g^{*}=T_{\varphi}^{*}\circ
S_{f}^{*}$. ∎

######  Remark 2.5.

The automorphism $T_{\varphi}$ is the exponential of an odd superconformal
vector field (an odd vector field such that
$[v,\mathcal{D}]\subset\mathcal{D}$]). Namely $T_{\varphi}=\exp(v(\varphi))$
with

$v(\varphi)=\theta\varphi\partial_{z}+\varphi\partial_{\theta}.$

###  2.6. Coordinate free formulation

The local generators of the sheaf of groups $\mathrm{Aut}^{0}_{X/S}$ of fibre
automorphisms restricting to the identity on $X_{\mathrm{red}}$ are described
above using local coordinates, but we only need a local splitting of $X$ to
define them. Let $X|_{U}\cong(U,\mathcal{O}_{C}\oplus L)$ be a splitting of
$X\to S$ on some open set $U$. Here $C$ is an ordinary curve over $S$ with
theta characteristic $L$ and a section $f\oplus\psi$ with $\psi=\theta g$ for
a local basis $\theta$ of the $\mathcal{O}_{C}$ module $L$ corresponds to
$f(z)+\theta g(z)$ in the local coordinate description. Then it is clear how
to interpret (2.7), (2.8) except possibly for the term
$\varphi(z)\varphi^{\prime}(z)$ in (2.8):

  * •

$f$ is a local automorphism of $C$ which is the identity modulo the nilpotent
ideal $I_{S}$ of $S$ and $S^{*}_{f}$ is its natural action on
$\mathcal{O}_{C}\oplus L$. It preserves the splitting.

  * •

$\varphi$ is an odd section of $L^{-1}$ and $T^{*}_{\varphi}$ acts on
$\mathcal{O}_{X}\oplus L$ as

$g\oplus\psi\to(g+\varphi\,\psi)\oplus(\psi+\varphi\,dg+\textstyle{\frac{1}{2}Q}(\varphi)\psi),$

for a quadratic form $Q\colon L^{-1}\to\mathcal{O}_{C}$ with local coordinate
expression $\varphi\varphi^{\prime}$ to be described below. The product is in
the algebra $\oplus_{j\in\mathbb{Z}}L^{j}$: for example $\varphi\,dg$ is the
product of the section $\varphi$ of $L^{-1}$ and $dg$ of $\omega_{C}\cong
L^{2}$; the result is a section of $L$.

It remains to describe the quadratic form $Q$.

######  Lemma 2.6.

Let $L$ be a theta-characteristic on $C\to S$. There is a unique map of
sheaves of vector spaces

$w\colon L^{-1}\otimes_{\mathbb{C}}L^{-1}\to\mathcal{O}_{C}$

such that

  1. (1)

$w(a,b)=-(-1)^{p(a)\,p(b)}w(a,b)$,

  2. (2)

$w(fa,b)=fw(a,b)+\langle d_{C/S}f,a\otimes b\rangle$,

for all local sections $a,b\in L^{-1}$, of parity $p(a),p(b)$ and
$f\in\mathcal{O}_{C}$. Here $\langle\;,\;\rangle$ is the canonical pairing
$\omega_{C}\otimes(L^{-1}\otimes L^{-1})\to\mathcal{O}_{C}$ induced by the
structure map $L^{2}\to\omega_{C/S}$.

###### Proof.

Let $e$ be a local basis of $L$. Then, for any $f,g\in\mathcal{O}_{C}$,
$w(fe,ge)=\langle(df\,g-fdg),e\otimes e\rangle$. This shows uniqueness. It
remains to show that this formula defines a well-defined map $w$, independent
of the choice of $e$. This follows from the identity

$d(hf)\,hg-hg\,d(hf)=h^{2}(fdg-gdf),\quad f,g,h\in\mathcal{O}_{C}.$

∎

######  Remark 2.7.

Locally,

$w(f(z)dz^{-1/2},g(z)dz^{-1/2})=f^{\prime}(z)g(z)-f(z)g^{\prime}(z)$

and Lemma 2.6 implies that this expression is well-defined, independently of
the choice of local coordinate $z$.

The quadratic $Q\colon L^{-1}\to\mathcal{O}_{C}$ appearing in $T_{\varphi}$ is
the quadratic form associated to this bilinear form:

$Q(a)=\frac{1}{2}w(a,a).$

###  2.7. Gluing construction of versal families in odd directions

Let us start with any family $C\to S_{0}$ of curves with a relative theta-
characteristic $L$ and a section $p:S_{0}\to C$. This family defines a split
supercurve $X_{0}=(C,\mathcal{O}_{C}\oplus L)$ over $S_{0}$. We want to extend
$X_{0}$ to a supercurve $X\to S$ where
$\mathcal{O}_{S}=\mathcal{O}_{S_{0}}\otimes\bigwedge W$ with
$W=H^{1}(C_{s},L_{s}^{-1})^{\vee}$ for some $s\in S_{0}$ as in Section 2.4.

For this we assume in addition that $S_{0}$ is affine,
$h^{0}(C_{s},L_{s}^{-1})=0$ for all $s\in S_{0}$, and the locally free
${\mathcal{O}}(S_{0})$-module $H^{1}(C,L^{-1})$ is trivialized:

$H^{1}(C,L^{-1})\simeq W^{\vee}\otimes{\mathcal{O}}_{S_{0}}.$

The required supercurve over $S$ is obtained from the trivial extension
$(|X_{0}|,\mathcal{O}_{X_{0}}\otimes\bigwedge W)$ by twisting by a class in
$H^{1}(X_{0},\mathrm{Aut}^{0}_{X_{0}/S})$. To define this class we cover $C$
by two open sets: an affine neighbourhood $U_{0}$ of $p(S_{0})\subset C$, and
$U_{1}=C\smallsetminus p(S_{0})$. Then a class in
$H^{1}(X_{0},\mathrm{Aut}^{0}_{X_{0}/S})$ is represented by a cocycle in
$\Gamma(U_{0}\cap U_{1},\operatorname{Aut}^{0}_{X_{0}/S})$ which we take to be

$T_{\varphi}=\exp(\varphi),\quad\varphi=\sum\eta_{i}b_{i},$

with $b_{i}\in\Gamma(U_{0}\cap U_{1},L^{-1})$ representing a basis of
$W^{\vee}\subset H^{1}(C,L^{-1})$ and the odd coordinates $(\eta_{i})$ forming
the dual basis of $W$. Then the class $[b_{i}]\in H^{1}(C,L^{-1})$ is the
Kodaira-Spencer class corresponding to the vector field
$\partial/\partial\eta_{i}$.

###  2.8. Hodge complex and duality

Recall that the Hodge bundle of an ordinary smooth (proper, flat) family of
curves $C\to S$ of genus $g$ is $\pi_{*}\omega_{C/S}$. It is locally free of
rank $g$. The situation is more subtle for supercurves: consider for example a
supercurve over a point with theta-characteristic $L$ such that
$\operatorname{dim}\,H^{0}(L)=m>0$. Then
$\pi_{*}\omega_{X}=H^{0}(\omega_{X})=H^{0}(\omega_{C})\oplus H^{0}(L)$ has
rank $g|m$, which varies from one supercurve to another. A more adequate
object to work with is the Hodge complex of a supercurve $X\to S$ defined as
the derived direct image $R\pi_{*}\omega_{X/S}$ of its relative dualizing
sheaf.

Below we will show that over the locus where the underlying theta-
characterstic has no nonzero sections, the sheaf $\pi_{*}\omega_{X/S}$ is
locally free of rank $g|0$. On the other hand, we will show that for the
universal family, the sheaves $\pi_{*}\omega_{X/S}$ and
$R^{1}\pi_{*}\omega_{X/S}$ are not locally free near every point where
$h^{0}(L)\neq 0$ (see Theorem 3.3).

Recall that for a family of supercurves $\pi:X\to S$ one has a canonical map

$\tau:R^{1}\pi_{*}\omega_{X/S}\to{\mathcal{O}}_{S}$

(see (2.5)). Since $R\pi_{*}\omega_{X/S}$ has cohomology in degrees $0$ and
$1$ we can view this map as a morphism in the derived category

$R\pi_{*}\omega_{X/S}\to{\mathcal{O}}_{S}[-1].$

Assume that $S$ is affine and the corresponding reduced family $C\to S_{0}$
has a marked point $\overline{p}:S_{0}\to C$, and let $U_{0},U_{1}$ be an open
covering of $C$, such that $U_{0}$ is an affine neighborhood of
$\overline{p}(S_{0})$ and $U_{1}$ is the complement to $\overline{p}(S_{0})$.
Then we can calculate $R^{1}\pi_{*}\omega_{X/S}(S)$ as

$\operatorname{coker}(\omega_{X/S}(U_{0})\oplus\omega_{X/S}(U_{1})\to\omega_{X/S}(U_{01})).$

Thus, we get a canonical (even) residue map

$\operatorname{Res}_{\overline{p}}:\omega_{X/S}(U_{01})\to{\mathcal{O}}_{S},$

which vanishes on the image of $\omega_{X/S}(U_{0})$. Furthermore, since
$\tau$ vanishes on the image of $\delta:R^{1}\pi_{*}\omega_{X/S}$, it follows
that $\operatorname{Res}_{\overline{p}}$ vanishes on
$\delta(\omega_{X/S}(U_{01}))$.

We can also replace $U_{01}$ be a formal punctured disk around $\overline{p}$.

######  Remark 2.8.

Let us consider a (split) formal superdisk $D$ and punctured superdisk
$D^{\prime}$ over $S$, given by the algebras

${\mathcal{O}}_{D}={\mathcal{O}}_{S}[\\![z]\\!]\oplus{\mathcal{O}}_{S}[\\![z]\\!]\theta,$
${\mathcal{O}}_{D^{\prime}}={\mathcal{O}}_{S}(\\!(z)\\!)\oplus{\mathcal{O}}_{S}(\\!(z)\\!)\theta,$

where we think of $\theta$ as a formal square root $dz^{1/2}$. We have

$\omega_{D}={\mathcal{O}}_{D}[dz|d\theta],\ \
\omega_{D^{\prime}}={\mathcal{O}}_{D^{\prime}}[dz|d\theta],$

and the derivation $\delta$ is given by

$\delta:{\mathcal{O}}_{D}\to\omega_{D},{\mathcal{O}}_{D^{\prime}}\to\omega_{D^{\prime}}:f(z)+g(z)\theta\mapsto(g(z)+f^{\prime}(z)\theta)[dz|d\theta].$

Then the map

$\operatorname{Res}:{\mathcal{O}}_{D^{\prime}}[dz|d\theta]\to{\mathcal{O}}_{S}$

is the composition of the projection to
${\mathcal{O}}_{S}(\\!(z)\\!)\theta[dz|d\theta]$ with the usual residue map on
${\mathcal{O}}_{S}(\\!(z)\\!)$. Note that the fact that $\operatorname{Res}$
vanishes on ${\mathcal{O}}_{S}(\\!(z)\\!)[dz|d\theta]$ is a consequence of the
fact that it vanishes on the image of $\delta$.

There is an analogue of Grothendieck-Serre duality for families of supercurves
in which $\omega_{X/S}$ plays a role of a relative dualizing sheaf, see [19,
Sec. 2] (see also [16], where the case of Serre duality on supercurves is
worked out in detail). This duality gives an isomorphism in the derived
category of $S$,

$R\pi_{*}(V)^{\vee}\simeq R\pi_{*}(V^{\vee}\otimes\omega_{X/S})[1],$

for any perfect complex $V$ over $X$. A part of this theory is the trace map

$\tau^{\prime}:R\pi_{*}\omega_{X/S}\to{\mathcal{O}}_{S}[-1].$

We claim that $\tau^{\prime}$ coincides with $\tau$ up to a sign. Indeed, this
follows from the compatibility of $\tau^{\prime}$ with the residue map: if
$X/S$ is equipped with a relative divisor $p\subset X$, where $p\to S$ is of
relative dimension $0|1$, then we have an exact triangle

$R\pi_{*}\omega_{X/S}\to
R\pi_{*}\omega_{X/S}(p)\to\omega_{X/S}(p)|_{p}\rTo{\alpha}R\pi_{*}\omega_{X/S}[1]$

such that the composition of $\alpha$ with $\tau^{\prime}$ is the residue map
$\omega_{X/S}(p)|_{p}\to{\mathcal{O}}_{S}$.

##  3\. Local freeness and non-freeness

###  3.1. Base change formalism

Let $f:X\to S$ be a projective map of supervarieties, and let ${\mathcal{F}}$
be a coherent sheaf on $X$, flat over $S$.

Assume also that $S$ is affine, $S=\operatorname{Spec}(A)$, where $A$ is a
supercommutative ring (finitely generated over a field). Then exactly as in
the purely even case (see e.g., [9, Prop. III.12.2]) one proves that there
exists a bounded above complex of projective $A$-modules $Q^{\bullet}$, such
that for every finitely generated $A$-module $M$, one has a functorial
isomorphism

$H^{i}(X,{\mathcal{F}}\otimes_{A}M)\simeq H^{i}(Q^{\bullet}\otimes_{A}M)$

Denoting these cohomology functors as $M\mapsto T^{i}(M)$, one has the
following results (proved in the same way as in the purely even case). For
simplicity we assume $A$ to be a local ring with maximal ideal $\mathfrak{m}$.

Note that for every $A$-module $M$ there is a natural map

$\phi^{i}(M):T^{i}(A)\otimes M\to T^{i}(M).$

Let $s$ be the closed point of $S$ with residue field $k(s)=A/\mathfrak{m}$.

######  Theorem 3.1.

(see [9, Sec. III.12]) (a) If $\phi^{i}(k(s))$ is surjective then
$\phi^{i}(M)$ is an isomorphism for any $M$.

(b) Assume that $\phi^{i}(k(s))$ is surjective. Then $\phi^{i-1}(k(s))$ is
surjective if and only if $T^{i}(A)$ is a free $A$-module.

Note that $T^{i}(A)\simeq H^{0}(S,R^{i}f_{*}{\mathcal{F}})$, so we can use the
above Theorem to check local freeness of $R^{i}f_{*}{\mathcal{F}}$.

###  3.2. Local freeness for a family of supercurves

Now assume that the fiber dimension of the map between reduced (usual) schemes
induced by $f$ is $1$. Then $T^{>1}=0$, so by Theorem 3.1 applied to $i=2$,
$\phi^{1}(k(s))$ is an isomoprhism. Thus, by Theorem 3.1(b) applied to $i=1$,
if $\phi^{0}(k(s))$ is surjective then $R^{1}f_{*}({\mathcal{F}})$ is locally
free. Since $\phi^{-1}(k(s))$ is always surjective, under the same assumption
$f_{*}{\mathcal{F}}$ is also locally free.

######  Proposition 3.2.

Let $\pi:X\to S$ be a family of supercurves, and let $\overline{\pi}:C\to
S_{0}$ be the corresponding usual family of curves with a relative theta-
characteristic $L$. Assume that $R\overline{\pi}_{*}L=0$ (or equivalently,
$H^{*}(L_{s})=0$ on every fiber). Then $R^{1}\pi_{*}{\mathcal{O}}_{X}$,
$\pi_{*}\omega_{X/S}$ and $R^{1}\pi_{*}\omega_{X/S}$ are locally free, and the
natural map

${\mathcal{O}}_{S}\to\pi_{*}{\mathcal{O}}_{X}$

is an isomorphism.

###### Proof.

First, let us apply Theorem 3.1 to the sheaf ${\mathcal{O}}_{X}$. The
assumption on $L$ implies that for every point $s\in S$ the map

$\phi^{0}(k(s)):\pi_{*}{\mathcal{O}}_{X}\otimes k(s)\to
H^{0}(C_{s},{\mathcal{O}})$

is surjective. Hence, $\pi_{*}{\mathcal{O}}_{X}$ and
$R^{1}\pi_{*}{\mathcal{O}}_{X}$ are locally free, and the map
${\mathcal{O}}_{S}\to\pi_{*}{\mathcal{O}}_{X}$ is an isomorphism. This implies
that locally we have a splitting
$R\pi_{*}{\mathcal{O}}_{X}\simeq\pi_{*}{\mathcal{O}}_{X}\oplus
R^{1}\pi_{*}{\mathcal{O}}_{X}[-1]$.

By Grothendieck duality, we have

$R\pi_{*}\omega_{X/S}\simeq(R\pi_{*}{\mathcal{O}}_{X})^{\vee}[-1]\simeq(\pi_{*}{\mathcal{O}}_{X}\oplus
R^{1}\pi_{*}{\mathcal{O}}_{X}[-1])^{\vee}[-1].$

It follows that
$R^{1}\pi_{*}\omega_{X/S}\simeq(\pi_{*}{\mathcal{O}}_{C})^{\vee}$ and
$\pi_{*}\omega_{X/S}\simeq(R^{1}\pi_{*}{\mathcal{O}}_{C})^{\vee}$ are locally
free. ∎

###  3.3. Local non-freeness

The following result implies that $R^{i}\pi_{*}\omega_{X/S}$ is not locally
free in the vicinity of any point of the moduli space where the corresponding
theta-characteristic has non-trivial sections. In particular, for the
universal family $\pi:X\to{\mathcal{M}}^{-}$ over the component corresponding
to odd theta-characteristics, the sheaves $\pi_{*}\omega_{X/S}$ and
$R^{1}\pi_{*}\omega_{X/S}$ are not locally free.

######  Theorem 3.3.

Let $\pi:X\to S$ be the universal family of supercurves over the moduli stack
of supercurves. Assume that the corresponding theta-characteristic $L_{s}$ on
$C_{s}$ has $h^{0}(C_{s},L_{s})\neq 0$ for some point $s\in S$. Then the
sheaves $\pi_{*}\omega_{X/S}$ and $R^{1}\pi_{*}\omega_{X/S}$ are not locally
free near $s$.

We will use the following general obstacle to local freeness.

######  Lemma 3.4.

Let $Q^{\bullet}=[Q^{0}\rTo{f}Q^{1}]$ be a two-term complex of vector bundles
over a superscheme $S$. For every point $s\in S$ let us define the map

$\kappa(s)=\kappa_{Q^{\bullet}}(s):T_{s}S\otimes H^{0}(Q^{\bullet}|_{s})\to
H^{1}(Q^{\bullet}|_{s}),$

where $T_{s}S$ is the Zariski tangent space to $S$ at $s$ as follows. Let
${\mathfrak{m}}$ be the maximal ideal in the local ring of $s$. Then we have

$H^{0}(Q^{\bullet}|_{s})=\ker(Q^{0}/{\mathfrak{m}}Q^{0}\to
Q^{1}/{\mathfrak{m}}Q^{1}),$

so we have a natural map

$H^{0}(Q^{\bullet}|_{s})\to{\mathfrak{m}}Q^{1}/({\mathfrak{m}}f(Q^{0})+{\mathfrak{m}}^{2}Q^{1})\simeq{\mathfrak{m}}/{\mathfrak{m}}^{2}\otimes(Q^{1}/f(Q^{0})),$
(3.1)

from which $\kappa$ is obtained by dualization.

(i) The map $\kappa(s)$ depends only on a quasi-isomorphism class of
$Q^{\bullet}$.

(ii) Locally near $s$ there exists a complex

$H^{0}(Q^{\bullet}|_{s})\otimes{\mathcal{O}}_{S}\rTo{g}H^{1}(Q^{\bullet}|_{s})\otimes{\mathcal{O}}_{S},$

quasi-isomorphic to $Q^{\bullet}$, such that the entries of $g$ are in
${\mathfrak{m}}$ and $g\mod{\mathfrak{m}}^{2}$ corresponds to $\kappa(s)$.

(iii) If $\kappa(s)$ is nontrivial then $\underline{H}^{1}Q^{\bullet}$ is not
locally free near $s$.

###### Proof.

(i) This follows from the naturality of the map (3.1) with respect to chain
maps between two-term complexes.

(ii) We can find bases of $Q^{0}|_{s}$ and $Q^{1}|_{s}$ such that the matrix
of $f\mod{\mathfrak{m}}$ has form

$f\mod{\mathfrak{m}}=\left(\begin{matrix}1_{r}&0\\\ 0&0\end{matrix}\right).$

Lifting these bases to local bases of $Q^{0}$ and $Q^{1}$ and then adjusting
them appropriately, we can arrange the matrix of $f$ to be of the form

$f=\left(\begin{matrix}1_{r}&0\\\ 0&M\end{matrix}\right)$

where $M$ has entries in ${\mathfrak{m}}$. But this means that there exists a
quasi-isomorphic two-term complex with the differential given by $M$, and the
assertion follows.

(iii) If $\underline{H^{1}}Q^{\bullet}$ is locally free near $s$ then in a
neighborhood of $s$ there exists a two-term complex quasi-isomorphic to
$Q^{\bullet}$ with zero differential. Hence, by (i), $\kappa(s)=0$. ∎

The crucial part of the proof of Theorem 3.3 is the following calculation of a
component of the obstruction map.

######  Lemma 3.5.

Let $[Q^{0}\to Q^{1}]$ be a resolution of $R\pi_{*}\omega_{X/S}$. Then for
every point $s$, we have decompositions into even and odd components,

$H^{i}(\omega_{X_{s}})=H^{i}(\omega_{C_{s}})\oplus H^{i}(L_{s})$

and the component

$H^{1}(L_{s}^{-1})\otimes H^{0}(\omega_{C_{s}})\to H^{1}(L_{s})$ (3.2)

of $\kappa(s)$ is given by the cup-product. Furthermore, the map (3.2) is
surjective.

###### Proof.

It is enough to consider the family $\pi:X\to S$ over
$S=\operatorname{Spec}(\operatorname{\mathbb{C}}[\epsilon]/\epsilon^{2})$,
restricting to $X_{0}$, with ${\mathcal{O}}_{X_{0}}={\mathcal{O}}_{C}\oplus L$
over $\operatorname{Spec}(\operatorname{\mathbb{C}})$, where $C=C_{s}$,
$L=L_{s}$, and associated with an odd tangent vector $v\in H^{1}(C,L^{-1})$.

Let $C=U_{0}\cup U_{1}$ be an open affine covering of $C$. Then
$R\pi_{*}\omega_{X/S}$ is represented by the complex

$[\omega_{X/S}(U_{1})\to\omega_{X/S}(U_{01})/\omega_{X/S}(U_{0})].$

Furthermore, we have an identification of ${\mathcal{O}}_{X}(U_{i})$ with
${\mathcal{O}}_{X_{0}}(U_{i})[\epsilon]$, so that the two identifications of
${\mathcal{O}}_{X}(U_{01})$ with ${\mathcal{O}}_{X_{0}}(U_{01})[\epsilon]$
differ by an automorphism $T$ given by $\exp(\epsilon\widetilde{v})$, where
$\widetilde{v}$ is an odd vector field over $U_{01}$ representing the class
$v\in H^{1}(C,L^{-1})$. Thus, we can represent $R\pi_{*}\omega_{X/S}$ by the
complex

$[\omega_{X_{0}/S}(U_{1})[\epsilon]\rTo{\alpha\mapsto
T(\alpha|_{U_{01}})}\omega_{X_{0}/S}(U_{01})/\omega_{X_{0}/S}(U_{0})].$ (3.3)

We claim that the restriction of the induced action of $T$ on the Berezinian
of ${\mathcal{O}}_{X_{0}}(U_{01})[\epsilon]$ over
$\operatorname{\mathbb{C}}[\epsilon]/\epsilon^{2}$ to $\omega_{C}(U_{01})$ is
given by

$\omega_{C}(U_{01})\to\omega_{C}(U_{01})[\epsilon]\oplus
L(U_{01})[\epsilon]:\alpha\mapsto\alpha+\epsilon\langle\widetilde{v},\alpha\rangle.$

Indeed, in general this action is given
$\alpha\mapsto\alpha+L_{\widetilde{v}}(\alpha)$, where $L_{\widetilde{v}}$ is
the Lie derivative. Since $\widetilde{v}$ is an odd vector field, our claim
follows easily from the description of the differential $\delta$ in terms of
the splitting ${\mathcal{O}}_{X_{0}}={\mathcal{O}}_{C}\oplus L$ (see Sec.
2.2).

Thus, if $\alpha$ is a global section of $\omega_{C}$ then the differential of
the complex (3.3) applied to $\alpha|_{U_{1}}$ gives
$\epsilon\langle\widetilde{v},\alpha\rangle$, which leads to the required
formula for $\kappa(s)$.

To check the surjectivity statement we observe that by Serre duality, the map
(3.2) is dual to the map

$H^{0}(L_{s})\to\operatorname{Hom}(H^{0}(\omega_{C_{s}}),H^{0}(\omega_{C_{s}}\otimes
L_{s})):s\mapsto(\alpha\mapsto\alpha s),$

which is clearly injective: for a nonzero $s\in H^{0}(L_{s})$, the
corresponding map $H^{0}(\omega_{C_{s}})\to H^{0}(\omega_{C_{s}}\otimes
L_{s})$ is injective, hence nonzero. ∎

We also need the following algebraic result about free modules over an
exterior algebra.

######  Lemma 3.6.

Let $R$ be a commutative ring, and let
${\mathcal{A}}=\bigwedge^{*}_{R}(\bigoplus_{i=1}^{n}R\theta_{i})$ be the
exterior algebra with $n$ generators $\theta_{1},\ldots,\theta_{n}$ over $R$.
Let us consider the ideal ${\mathcal{N}}=\bigwedge^{\geq
1}_{R}(R^{n})\subset{\mathcal{A}}$. For an ${\mathcal{A}}$-module $M$, we
denote by $M_{\mathcal{N}}$ the annihilator of ${\mathcal{N}}$ in $M$. Suppose
we have a morphism $\phi:M\to M^{\prime}$ of free ${\mathcal{A}}$-modules of
finite rank, such that the induced morphism $M_{\mathcal{N}}\to
M^{\prime}_{\mathcal{N}}$ is an isomorphism. Then $\phi$ is an isomorphism.

###### Proof.

For a free ${\mathcal{A}}$-module of finite rank $M$, the map induced by the
${\mathcal{A}}$-action,

$\theta_{1}\cdot\ldots\cdot\theta_{n}\cdot:M/{\mathcal{N}}M\to
M_{\mathcal{N}}$

is an isomorphism. Since $\phi$ is compatible with the ${\mathcal{A}}$-action,
we deduce that the morphism

$M/{\mathcal{N}}M\to M^{\prime}/{\mathcal{N}}M^{\prime}$

is an isomorphism. Since ${\mathcal{N}}$ is nilpotent, this implies that
$\phi$ is surjective. Hence, $\phi$ is a projection onto a direct summand.
Thus, if $K=\ker(\phi)$, we deduce that $K/{\mathcal{N}}K=0$, so $K=0$. ∎

###### Proof of Theorem 3.3.

Set $C=C_{s}$, $L=L_{s}$. By Serre duality, the cup-product map
$H^{1}(C,L^{-1})\otimes H^{0}(C,\omega_{C})\to H^{1}(C,L)$ corresponds to the
cup-product map

$H^{0}(C,\omega_{C})\otimes H^{0}(C,L)\to H^{0}(C,\omega\otimes L).$ (3.4)

The latter map is always nonzero whenever $H^{0}(C,L)\neq 0$. Thus, using
Lemma 3.5, we see that the obstruction $\kappa(s)$ associated with a
resolution of $R\pi_{*}\omega_{X/S}$ is nonzero, so by Lemma 3.4, the sheaf
$R^{1}\pi_{*}\omega_{X/S}$ is not locally free near $s$.

Next, we will deal with the sheaf $\pi_{*}\omega_{X/S}$, replacing $S$ by an
affine neighborhood of $s$. The trace map (2.5) can be viewed as a morphism in
derived category

$R\pi_{*}\omega_{X/S}\to{\mathcal{O}}_{S}[-1].$

More precisely, if $R\pi_{*}\omega_{C/S}$ is represented by a complex of
vector bundles $[A\rTo{f}B]$ then $\tau$ corresponds to a map
$B/f(A)\to{\mathcal{O}}_{S}$, so we can view it as a chain map $[A\to
B]\to{\mathcal{O}}_{S}$. Furthermore, since $\tau$ is surjective, there exists
a splitting ${\mathcal{O}}_{S}\to B$ of the projection
$B\to{\mathcal{O}}_{S}$, which we can view as a chain map
${\mathcal{O}}_{S}\to[A\to B]$. Thus, we get a splitting in derived category,

$R\pi_{*}\omega_{X/S}\simeq P^{\bullet}\oplus{\mathcal{O}}_{S}[-1].$

Next, by Lemma 3.4(ii), locally near $s$, the complex $P^{\bullet}$ is quasi-
isomorphic to a two-term complex $Q^{\bullet}$ of the form

$H^{0}(\omega_{C})\otimes{\mathcal{O}}_{S}\oplus
H^{0}(L)\otimes{\mathcal{O}}_{S}\rTo{(f^{-},f^{+})}H^{1}(L)\otimes{\mathcal{O}}_{S}$

(here the space $H^{0}(\omega_{C})$ is even, while the spaces $H^{i}(L)$ are
odd), where the linear terms of $f^{-}$ are given by the map (3.2).

Let $S_{0}\subset S$ be the reduced subscheme. Then the restriction of
$Q^{\bullet}$ to $S_{0}$ splits into the direct sum of
$H^{0}(\omega_{C})\otimes{\mathcal{O}}_{S_{0}}$ and the complex

$H^{0}(L)\otimes{\mathcal{O}}_{S_{0}}\rTo{f^{+}|_{S_{0}}}H^{1}(L)\otimes{\mathcal{O}}_{S_{0}}$
(3.5)

computing the direct image of the universal theta-characteristic.

Set ${\mathcal{F}}=\underline{H}^{0}Q^{\bullet}=\ker(f^{-},f^{+})$. We want to
check that ${\mathcal{F}}$ is not locally free. We distinguish two cases.

Case $h^{0}(L)$ is even. Then the complex (3.5) is generically acyclic, so
$f^{+}|_{S_{0}}$ is generically an isomorphism. In particular, $f^{+}$ is
injective and the sheaf $\operatorname{coker}(f^{+})$ has rank $0$. It follows
that ${\mathcal{F}}$ can be identified with the kernel of the morphism

$H^{0}(\omega_{C})\otimes{\mathcal{O}}_{S}\to\operatorname{coker}(f^{+})$

induced by $f^{-}$. In particular, ${\mathcal{F}}$ is an
${\mathcal{O}}_{S}$-submodule of $H^{0}(\omega_{C})\otimes{\mathcal{O}}_{S}$.

Locally we have a splitting
${\mathcal{O}}_{S}\simeq{\bigwedge}(W)\otimes{\mathcal{O}}_{S_{0}}$, with
${\mathcal{N}}={\bigwedge}^{\geq 1}(W)\otimes{\mathcal{O}}_{S_{0}}$. Let us
consider the submodule ${\mathcal{F}}_{\mathcal{N}}$ consisting of sections
annihilated by ${\mathcal{N}}$. Since $f^{-}$ has entries in ${\mathcal{N}}$,
we get that

${\mathcal{F}}_{\mathcal{N}}=H^{0}(\omega_{C})\otimes{\bigwedge}^{2g-2}(W)\otimes{\mathcal{O}}_{S_{0}}.$

Hence, by Lemma 3.6,
${\mathcal{F}}=H^{0}(\omega_{C})\otimes{\mathcal{O}}_{S}$. In other words, the
image of $f^{-}:H^{0}(\omega_{C})\otimes{\mathcal{O}}_{S}\to
H^{1}(L)\otimes{\mathcal{O}}_{S}$ has to be contained in the image of
$f^{+}:H^{0}(L)\otimes{\mathcal{O}}_{S}\to H^{1}(L)\otimes{\mathcal{O}}_{S}$.
Let consider the following component of $f^{-}$:

$f^{-}_{1}:H^{0}(\omega_{C})\otimes{\bigwedge}^{2g-3}(W)\otimes{\mathcal{O}}_{S_{0}}\to
H^{1}(L)\otimes{\bigwedge}^{2g-2}(W){\mathcal{O}}_{S_{0}}.$

Modulo the maximal ideal we get the linear map

$H^{0}(\omega_{C})\otimes W^{\vee}\simeq H^{0}(\omega_{C})\otimes
H^{1}(L^{-1})\to H^{1}(L)$

which is given by the cup-product and is surjective (see Lemma 3.5). Hence,
$H^{1}(L)\otimes{\bigwedge}^{2g-2}(W){\mathcal{O}}_{S_{0}}$
$=(H^{1}(L)\otimes{\mathcal{O}}_{S})_{{\mathcal{N}}}$ has to be in the image
of $f^{+}$. Since $f^{+}$ is injective, it can only be in the image of
$(H^{0}(L)\otimes{\mathcal{O}}_{S})_{{\mathcal{N}}}$. But the map

$H^{0}(L)\otimes{\mathcal{O}}_{S_{0}}\simeq(H^{0}(L)\otimes{\mathcal{O}}_{S})_{{\mathcal{N}}}\to(H^{1}(L)\otimes{\mathcal{O}}_{S})_{{\mathcal{N}}}\simeq
H^{1}(L)\otimes{\mathcal{O}}_{S_{0}}$

induced by $f^{+}|_{S_{0}}$ has entries in the maximal ideal, so its image is
not everything.

Case $h^{0}(L)$ is odd. Since the locus of local freeness is open, it is
enough to consider the case $h^{0}(L)=1$ (since the set of such points is
dense in the odd component of the moduli space). In this case
$f^{+}|_{S_{0}}=0$, so $f^{+}$ is contained in ${\mathcal{N}}^{2}$. Hence, the
sheaf

${\mathcal{F}}_{\mathcal{N}}=H^{0}(\omega_{C})\otimes{\mathcal{O}}_{S_{0}}\oplus
H^{0}(L)\otimes{\mathcal{O}}_{S_{0}}$

is locally free of rank $g+1$ (we count both odd and even generators). Let us
also consider ${\mathcal{F}}_{{\mathcal{N}}^{2}}$, the annihilator of
${\mathcal{N}}^{2}$ in ${\mathcal{F}}$. We have

${\mathcal{F}}_{{\mathcal{N}}^{2}}=\ker((H^{0}(\omega_{C})\otimes{\mathcal{O}}_{S})_{{\mathcal{N}}^{2}}\rTo{f^{-}}(H^{1}(L)\otimes{\mathcal{O}}_{S})_{\mathcal{N}})\oplus(H^{0}(L)\otimes{\mathcal{O}}_{S})_{{\mathcal{N}}^{2}},$

so that

${\mathcal{F}}_{{\mathcal{N}}^{2}}/{\mathcal{F}}_{\mathcal{N}}\simeq\ker(W^{\vee}\otimes
H^{0}(\omega_{C})\otimes{\mathcal{O}}_{S_{0}}\rTo{f^{-}}H^{1}(L)\otimes{\mathcal{O}}_{S_{0}})\oplus
W^{\vee}\otimes{\mathcal{O}}_{S_{0}}.$

It follows that
${\mathcal{F}}_{{\mathcal{N}}^{2}}/{\mathcal{F}}_{\mathcal{N}}$ is locally
free over ${\mathcal{O}}_{S_{0}}$ of rank $(g+1)\dim(W)-1$. But for a free
${\mathcal{O}}_{S}$-module ${\mathcal{G}}$, we would have

$\operatorname{rk}({\mathcal{G}}_{{\mathcal{N}}^{2}}/{\mathcal{G}}_{\mathcal{N}})=\dim(W)\cdot\operatorname{rk}({\mathcal{G}}_{\mathcal{N}}).$

Hence, ${\mathcal{F}}$ is not locally free over ${\mathcal{O}}_{S}$ near $s$.
∎

##  4\. Symplectic picture

###  4.1. Support of a complex of bundles

Let $Q^{\bullet}$ be a bounded complex of vector bundles over a superscheme
$S$.

######  Definition 4.1.

Let us define the quasicoherent sheaf of ideals
${\mathcal{I}}(Q^{\bullet})\subset{\mathcal{O}}_{S}$ as the kernel of the
canonical morphism

${\mathcal{O}}_{S}\to\underline{H}^{0}((Q^{\bullet})^{\vee}\otimes
Q^{\bullet}).$

Equivalently, this ideal consists of $f\in{\mathcal{O}}$ such that the
morphism $f\cdot\operatorname{id}$ of $Q^{\bullet}$ is locally homotopic to
zero.

We need some simple properties of this definition, which in particular imply
that ${\mathcal{I}}(Q^{\bullet})$ defines a subscheme structure on the support
of $Q^{\bullet}$.

######  Lemma 4.2.

(i) If complexes $Q^{\bullet}$ and $R^{\bullet}$ are quasi-isomorphic then
${\mathcal{I}}(Q^{\bullet})={\mathcal{I}}(R^{\bullet})$. In particular, one
has ${\mathcal{I}}(Q^{\bullet})={\mathcal{O}}_{S}$ if and only if
$Q^{\bullet}$ is acyclic.

(ii) One has
${\mathcal{I}}(Q^{\bullet})\subset\cap_{i}\operatorname{Ann}(\underline{H}^{i}Q^{\bullet})$.

(iii) For an open subset $U\subset S$ one has
${\mathcal{I}}(Q^{\bullet})|_{U}={\mathcal{I}}(Q^{\bullet}|_{U})$.

(iv) Consider a $2$-term complex $A\rTo{\delta}B$, such that $\delta$ is an
isomorphism on a Zariski open subset $U\subset S$. Assume that $S$ is affine,
and the homomorphism ${\mathcal{O}}(S)\to{\mathcal{O}}(U)$ is injective. Then
$f\in H^{0}(S,{\mathcal{I}}(A\to B))$ if and only if the morphism
$f|_{U}\delta^{-1}:B|_{U}\to A|_{U}$ has a regular extension to $S$.

###### Proof.

Part (i) follows from the fact that any quasi-isomorphism between bounded
complexes of vector bundles locally becomes a homotopy equivalence. Parts (ii)
and (iii) are straightforward. For part (iv), we have to observe that a
homotopy $h:B\to A$ between $f\cdot\operatorname{id}$ and $0$ satisfies
$h\delta=f\operatorname{id}_{A}$, so $h$ is an extension of $f\delta^{-1}$. ∎

###  4.2. Isotropic intersections/co-intersections

Let ${\mathcal{V}},(\cdot,\cdot)$ be a symplectic vector bundle of rank
$2m|2n$ over an irreducible superscheme $S$, and let
$L_{1},L_{2}\subset{\mathcal{V}}$ be a pair of maximal isotropic subbundles of
rank $m|n$ (this means that $L_{i}$ is isotropic and the morphism
${\mathcal{V}}/L_{i}\to L_{i}^{\vee}$ induced by the pairing is an
isomorphism). Then we have a complex of bundles over $S$, concentrated in
degrees $0$ and $1$,

$C^{\bullet}(L_{1},L_{2})=C^{\bullet}({\mathcal{V}};L_{1},L_{2}):[L_{1}\to{\mathcal{V}}/L_{2}]$

that controls the behavior of the intersections $f^{*}L_{1}\cap
f^{*}L_{2}=\underline{H}^{0}f^{*}C^{\bullet}$ for arbitrary morphisms $f:T\to
S$, and of the co-intersection
${\mathcal{V}}/(L_{1}+L_{2})\simeq\underline{H}^{1}C^{\bullet}$. Note that the
symplectic form induces an isomorphism ${\mathcal{V}}/L_{2}\simeq
L_{2}^{\vee}$, so we can also write the above complex as

$[L_{1}\to L_{2}^{\vee}],$

which in particular shows that $C^{\bullet}(L_{2},L_{1})$ is dual to
$C^{\bullet}(L_{1},L_{2})[-1]$. On the other hand, we have a natural quasi-
isomorphism to $C^{\bullet}(L_{1},L_{2})$ from the complex

$[L_{1}\oplus L_{2}\to{\mathcal{V}}].$

Thus, as an object in the derived category, it is self-dual up to a shift.

We say that $L_{1}$ and $L_{2}$ are transversal if the natural map
$L_{1}\oplus L_{2}\to{\mathcal{V}}$ is an isomorphism. This is equivalent to
exactness of the complex $C^{\bullet}(L_{1},L_{2})$.

######  Proposition 4.3.

(i) Assume that $L^{\prime}_{2}\subset{\mathcal{V}}$ is a maximal isotropic
subbundle transversal to both $L_{1}$ and $L_{2}$. Let us identify
$L^{\prime}_{2}$ with $L_{2}^{\vee}$ using the pairing between $L_{2}$ and
$L^{\prime}_{2}$ given by the symplectic form. Then $L_{1}$ is the graph of a
symmetric morphism $\phi:L_{2}\to L^{\prime}_{2}\simeq L_{2}^{\vee}$ (i.e.,
$\phi^{*}=\phi$), and the complex $C^{\bullet}(L_{1},L_{2})$ is isomorphic to

$[L_{2}\rTo{\phi}L^{\prime}_{2}].$

(ii) Assume that the complex $C^{\bullet}(L_{1},L_{2})$ is generically exact.
If $n$ (the dimension of $L_{i}^{-}$) is even then locally there exists a
maximal isotropic subbundle transversal to both $L_{1}$ and $L_{2}$. For any
$n$ there exists a symmetric representative of $C^{\bullet}(L_{1},L_{2})$,
i.e., there exists an quasi-isomorphic complex of the form

$M\rTo{\phi}M^{\vee}$

with $\phi^{*}=\phi$. Furthermore, if $n$ is even then we can find such $M$ of
rank $m|n$ (same as the rank of $L_{i}$), while for odd $n$ we can find $M$ of
rank $m|(n+1)$.

###### Proof.

(i) The first assertion is standard. Using the identification of
$L_{2}^{\vee}$ with $L^{\prime}_{2}$, we can identify $C^{\bullet}$ with
$[L_{1}\to L^{\prime}_{2}]$, where the map is given by the projection along
$L_{2}$. Since the projection $L_{1}\to L_{2}$ along $L^{\prime}_{2}$ is an
isomorphism, we get an isomorphism of complexes

$[L_{1}\to L^{\prime}_{2}]\simeq[L_{2}\rTo{\phi}L^{\prime}_{2}]$

as claimed.

(ii) Assume first that $n$ is even. By part (i), in this case it is enough to
check that locally we can choose a maximal isotropic subbundle
$L^{\prime}_{2}\subset{\mathcal{V}}$, transversal to both $L_{1}$ and $L_{2}$.
Indeed, it is easy to see that one can choose a maximal isotropic subbundle
$\widetilde{L}_{2}\subset{\mathcal{V}}$, complementary to $L_{2}$. Next, for
every point $x\in S$, we have

${\mathcal{V}}|_{x}={\mathcal{V}}^{+}_{x}\oplus{\mathcal{V}}^{-}_{x},\ \
L_{i}|_{x}=L_{ix}^{+}\oplus L_{ix}^{-},$

where $L_{ix}^{+}$ are Lagrangian subspaces of the symplectic vector space
${\mathcal{V}}^{+}|_{x}$ and $L_{ix}^{-}$ are maximal isotropic subspaces of
the orthogonal vector space ${\mathcal{V}}^{-}|_{x}$. Since generically
$L_{1}|_{x}$ and $L_{2}|_{x}$ have trivial intersection in
${\mathcal{V}}|_{x}$, it follows that at any point the intersection
$L^{-}_{1}|_{x}\cap L^{-}_{2}|_{x}$ has even dimension.

Then an easy linear algebra argument (using the fact that $n-\dim
L_{1x}^{-}\cap L_{2x}^{-}$ is even) shows that there exists a skew-symmetric
map $\psi_{x}^{-}:\widetilde{L}^{-}_{2x}\to L_{2x}^{-}$ such that the graph of
$\psi_{x}^{-}$ has trivial intersection with $L_{1x}^{-}$. Similarly, there
exists a symmetric map $\psi_{x}^{+}:\widetilde{L}^{+}_{2x}\to L_{2x}^{+}$
such that the graph of $\psi_{x}^{+}$ has trivial intersection with
$L_{1x}^{+}$. Now we view $\psi_{x}=(\psi_{x}^{+},\psi_{x}^{-})$ as a
symmetric map of vector spaces, extend it to a symmetric morphism of bundles
$\psi:\widetilde{L}_{2}\to L_{2}$ in a neighborhood of $x$ and define
$L^{\prime}_{2}$ to be the graph of $\psi$.

Now let us consider the case when $n$ is odd. Then we can set
${\mathcal{V}}^{\prime}={\mathcal{V}}\oplus({\mathcal{O}}e_{1}\oplus{\mathcal{O}}e_{2})$,
where $e_{1}$ and $e_{2}$ are odd, and equip ${\mathcal{V}}^{\prime}$ with a
symplectic pairing using the pairing on ${\mathcal{V}}$ and the standard
symmetric pairing on ${\mathcal{O}}e_{1}\oplus{\mathcal{O}}e_{2}$, so that
$e_{1}$ and $e_{2}$ are isotropic and $(e_{1},e_{2})=1$. Set us also set

$L^{\prime}_{1}=L_{1}\oplus{\mathcal{O}}e_{1},\ \
L^{\prime}_{2}=L_{2}\oplus{\mathcal{O}}e_{2}.$

Then $L^{\prime}_{1}$ and $L^{\prime}_{2}$ are maximal isotropic in
${\mathcal{V}}^{\prime}$, and the complex $L_{1}\to{\mathcal{V}}/L_{2}$ is
quasi-isomorphic to

$[L_{1}\oplus{\mathcal{O}}e_{1}\to{\mathcal{V}}/L_{2}\oplus{\mathcal{O}}e_{1}]\simeq[L^{\prime}_{1}\to{\mathcal{V}}^{\prime}/L^{\prime}_{2}].$

It remains to apply the case of even $n$. ∎

We have the following nice reduction property of the complexes
$C^{\bullet}({\mathcal{V}};L_{1},L_{2})$.

######  Lemma 4.4.

Let $L_{1},L_{2}\subset{\mathcal{V}}$ be a pair of maximal isotropic
subbundles in a symplectic bundle of rank $2m|2n$. Let $M\subset L_{1}$ be a
subbundle such that the map $L_{2}\to M^{\vee}$, induced by the pairing, is
surjective (equivalently, ${\mathcal{V}}=L_{2}+M^{\perp}$). Then the bundle
$\overline{{\mathcal{V}}}:=M^{\perp}/M$ has an induced symplectic structure
and induced maximal isotropic subbundles $\overline{L}_{1}=L_{1}/M$,
$\overline{L}_{2}=L_{2}\cap M^{\perp}$ in $\overline{{\mathcal{V}}}$.
Furthermore, there is a natural quasi-isomorphism of complexes

$C^{\bullet}({\mathcal{V}};L_{1},L_{2})\to
C^{\bullet}(\overline{{\mathcal{V}}};\overline{L}_{1},\overline{L}_{2}).$

###### Proof.

Note that $\overline{L}_{2}$ embeds into $\overline{{\mathcal{V}}}$ since
$L_{2}\cap M\subset(L_{2}+M^{\perp})^{\perp}=0$. One can easily check that
$\overline{L}_{1}$ and $\overline{L}_{2}$ are maximal isotropic in
$\overline{{\mathcal{V}}}$. Then we use a quasi-isomorphism of complexes

$[L_{1}\to{\mathcal{V}}/L_{2}]\to[L_{1}/M\to{\mathcal{V}}/(M+L_{2})],$

and observe that the natural map

$\overline{{\mathcal{V}}}/\overline{L}_{2}\simeq M^{\perp}/(M+L_{2}\cap
M^{\perp})\to{\mathcal{V}}/(M+L_{2})$

is an isomorphism. ∎

###  4.3. Determinants

Recall that when a complex of vector bundles of even rank,
$C^{\bullet}=[C^{0}\rTo{d}C^{1}]$, is generically exact then we have a natural
section

$\theta_{C^{\bullet}}=\det(d)\in\operatorname{Det}(C^{\bullet})^{-1}=\operatorname{Det}(C_{0})^{-1}\otimes\operatorname{Det}(C_{1}),$

which vanishes precisely on the locus where $C^{\bullet}$ fails to be exact.
Similarly, for an arbitrary complex of (super)vector bundles
$C^{\bullet}=[C^{0}\rTo{d}C^{1}]$, exact over an open subset $U$, there is a
natural invertible section

$\theta=\theta_{C^{\bullet}}=\operatorname{ber}(d)\in
H^{0}(U,\operatorname{Ber}(C^{\bullet})^{-1}).$

Now let $L_{1},L_{2}\subset{\mathcal{V}}$ be a pair of maximal isotropic
subbundles in a symplectic bundle. Assume that $L_{1}$ and $L_{2}$ are
transversal over a nonempty open $U\subset S$. Then we define

$\theta(L_{1},L_{2})\in\operatorname{Ber}(C^{\bullet}({\mathcal{V}};L_{1},L_{2}))^{-1}(U)\simeq\operatorname{Ber}(L_{1})^{-1}\otimes\operatorname{Ber}(L_{2})^{-1}|_{U}$

to be the canonical (even) nonvanishing section corresponding to the exact
complex $C^{\bullet}({\mathcal{V}};L_{1},L_{2})|_{U}$. Equivalently,
$\theta(L_{1},L_{2})$ is the Berezinian of the isomorphism of bundles

$L_{1}|_{U}\to L_{2}^{\vee}|_{U}$

induced by the symplectic form. Note that the line bundle
$\operatorname{Ber}(C^{\bullet}({\mathcal{V}};L_{1},L_{2}))$ has rank $1|0$,
so its even sections are given locally by even functions.

Similarly to the even case, a quasi-isomorphism of exact complexes leads to an
isomorphism of their Berezinians compatible with their canonical sections
coming from the Berezinian of the differential. Thus, in the context of Lemma
4.4, the sections $\theta(L_{1},L_{2})$ and
$\theta(\overline{L}_{1},\overline{L}_{2})$ get identified under the
isomorphism between the corresponding line bundles.

######  Theorem 4.5.

Let $L_{1},L_{2}\subset{\mathcal{V}}$ be a pair of maximal isotropic
subbundles in a symplectic bundle over $S$, transversal over some open
$j:U\hookrightarrow S$ such that the natural map ${\mathcal{O}}_{S}\to
j_{*}{\mathcal{O}}_{U}$ is injective. Assume that for some point $s\in S$, one
has

$L_{1s}^{+}\cap L_{2s}^{+}=0.$

Then locally near $s$ there exists a trivialization of $\operatorname{Ber}(C)$
such that $\theta^{-1}(L_{1},L_{2})$ corresponds to $f^{2}$, where $f$ is a
regular function. Furthermore, $f$ belongs to the ideal
${\mathcal{I}}(C({\mathcal{V}},L_{1},L_{2}))$. In other words, $f\delta^{-1}$
is regular near $s$, where $\delta:L_{1}\to{\mathcal{V}}/L_{2}$ is the natural
map.

###### Proof.

Locally near $s$ we can define a decomposition $L_{1}=L_{1}^{+}\oplus
L_{1}^{-}$ where $L_{1}^{+}$ has rank $m|0$ and $L_{2}^{-}$ has rank $0|n$.
Let us consider the isotropic subbundle $L_{1}^{+}\subset{\mathcal{V}}$. Then
our assumption implies that in a neighborhood of $s$ the morphism
$L_{2}\to(L_{1}^{+})^{\vee}$ is surjective, so by Lemma 4.4, we have a quasi-
isomorphism of $C^{\bullet}({\mathcal{V}};L_{1},L_{2})$ with
$C^{\bullet}(\overline{{\mathcal{V}}},\overline{L}_{1},\overline{L}_{2})$,
where $\overline{{\mathcal{V}}}=(L_{1}^{+})^{\perp}/L_{1}^{+}$ and
$\overline{L}_{1}=L_{1}/L_{1}^{+}$ has rank $0|n$. Now by Proposition 4.3,
there exists a quasi-isomorphism

$C^{\bullet}(\overline{{\mathcal{V}}},\overline{L}_{1},\overline{L}_{2})\simeq[M\rTo{\phi}M^{\vee}],$

with $M$ of rank $0|n^{\prime}$ and $\phi$ selfdual. But this means that
actually $\theta^{-1}(L_{1},L_{2})=\operatorname{ber}(\phi)^{-1}$ is given by
the determinant of a skew-symmetric matrix with entries in
${\mathcal{O}}_{S}^{+}$. It remains to set

$f=\operatorname{Pf}(\phi),$

where $\operatorname{Pf}(?)$ denotes the Pfaffian.

For the last assertion, we recall that by Lemma 4.2(i), one has

${\mathcal{I}}(C({\mathcal{V}},L_{1},L_{2}))\simeq{\mathcal{I}}(M\rTo{\phi}M^{\vee}).$

Now the assertion follows from Lemma 4.2(iv) together with the well known fact
that $\operatorname{Pf}(\phi)\phi^{-1}$ is regular (see e.g., [10, Eq. (10)]).
∎

We need a simple observation that allows to study $\theta(L_{1},L_{2})$ in the
presence of a Lagrangian splitting $V=\Lambda\oplus\Lambda^{\prime}$.

######  Lemma 4.6.

Let $V=\Lambda\oplus\Lambda^{\prime}$ be a splitting of a symplectic bundle
into Lagrangian subbundles, and let $L_{1},L_{2}\subset V$ be Lagrangian
subbundles. Assume that the pairs $(L_{1},L_{2})$, $(L_{1},\Lambda)$ and
$(L_{2},\Lambda)$ are transversal, and let

$\tau_{i}:\Lambda^{\prime}\to\Lambda,\ i=1,2,$

be symmetric (with respect to the duality
$\Lambda^{\prime}\simeq\Lambda^{\vee}$) morphisms such that $L_{i}$ is the
graph of $\tau_{i}$. Then under the identification
$\operatorname{Ber}(\Lambda^{\prime})\simeq\operatorname{Ber}(\Lambda)^{-1}$,
we have equality of sections of
$\operatorname{Ber}(L_{1})^{-1}\otimes\operatorname{Ber}(L_{2})^{-1}$,

$\theta(L_{1},L_{2})=\pm\theta(L_{1},\Lambda)\theta(L_{2},\Lambda)\operatorname{ber}(\tau_{1}-\tau_{2}).$

###### Proof.

By the definition,
$\pm\theta(L_{1},L_{2})\theta(L_{1},\Lambda)^{-1}\theta(L_{2},\Lambda)^{-1}$
is the Berezinian of the pairing on $\Lambda^{\prime}$ given as the
composition

$\Lambda^{\prime}\otimes\Lambda^{\prime}\rTo{\alpha_{1}\otimes\alpha_{2}}L_{1}\otimes
L_{2}\rTo{(\cdot,\cdot)}{\mathcal{O}},$

where $(\cdot,\cdot)$ is the symplectic form on $V$, and
$\alpha_{i}:\Lambda^{\prime}\to L_{i}$ are isomorphisms given by

$\alpha_{i}:\Lambda^{\prime}\to
L_{i}:\lambda^{\prime}\mapsto\tau_{i}(\lambda^{\prime})+\lambda^{\prime}.$

Using the fact that $\tau_{2}$ is symmetric, we compute

$\displaystyle(\alpha_{1}(\lambda^{\prime}_{1}),\alpha_{2}(\lambda^{\prime}_{2}))=(\tau_{1}(\lambda^{\prime}_{1})+\lambda^{\prime}_{1},\tau_{2}(\lambda^{\prime}_{2})+\lambda^{\prime}_{2})=(\tau_{1}(\lambda^{\prime}_{1}),\lambda^{\prime}_{2})-(\tau_{2}(\lambda^{\prime}_{2}),\lambda^{\prime}_{1})$
$\displaystyle=((\tau_{1}-\tau_{2})(\lambda^{\prime}_{1}),\lambda^{\prime}_{2}).$

This implies our formula. ∎

###  4.4. Classical picture for theta-characteristic

We are inspired by the following classical isotropic intersection picture for
theta-characteristics (see e.g., [12], [8]).

Let $C$ be a curve with a theta-characteristic ${\mathcal{L}}$, and let $p\in
C$ be a point, $D$ a formal punctured disk around $p$. Then we have a natural
symmetric pairing on $H^{0}(D,{\mathcal{L}})$ given by

$b(s,t)=\operatorname{Res}(st),$

where we use the isomorphism ${\mathcal{L}}^{\otimes 2}\simeq\omega_{C}$ to
view $st$ is a $1$-form on $D$. This induces a nondegenerate pairing on

$V=H^{0}(C,{\mathcal{L}}(np)/{\mathcal{L}}(-np))$

for any $n>0$, such that the subspace

$L_{2}:=H^{0}(C,{\mathcal{L}}/{\mathcal{L}}(-np))\subset V$

is maximal isotropic. Furthermore, for $n$ large enough, the map
$H^{0}(C,{\mathcal{L}}(np))\to V$ is injective, and one can check that

$L_{1}=H^{0}(C,{\mathcal{L}}(np))\subset V$

is also maximal isotropic. Now the complex
$C^{\bullet}(V;L_{1},L_{2})=[L_{1}\to V/L_{2}]$ can be identified with the
complex

$H^{0}(C,{\mathcal{L}}(np))\to H^{0}(C,{\mathcal{L}}(np)/{\mathcal{L}})$

computing the cohomology $H^{*}(C,{\mathcal{L}})$.

This also makes sense for families $(C,{\mathcal{L}})/S$. If we further assume
that generically ${\mathcal{L}}$ has no nonzero sections then Proposition 4.3
implies that locally we can find a quasi-isomorphism of
$R\pi_{*}{\mathcal{L}}$ with a complex of the form

$M\rTo{\phi}M^{\vee}$

on $S$, with $\phi^{*}=-\phi$. It follows that we can use the Pfaffian of
$\phi$ as a local equation of the locus where ${\mathcal{L}}$ acquires nonzero
sections.

Applying this to a universal family over the moduli space ${\mathcal{M}}^{+}$
of curves with even theta-characteristics, we get a canonical structure of an
effective Cartier divisor on the locus of $(C,{\mathcal{L}})$ with
$h^{0}({\mathcal{L}})>0$. Furthermore, the first order computation in [13,
Sec. 1] implies that this divisor is smooth at all points $(C,{\mathcal{L}})$
where $h^{0}({\mathcal{L}})=2$.

###  4.5. Symplectic structure for the usual family of curves

For clarity let us consider the even picture first.

Note that for a family of curves $\pi:C\to S$ the Hodge bundle
$\pi_{*}\omega_{C}$ embeds as a Lagrangian subbundle into the symplectic
bundle $R^{1}\pi_{*}\operatorname{\mathbb{C}}_{C/S}$, where
$\operatorname{\mathbb{C}}_{C/S}=\pi^{-1}{\mathcal{O}}_{S}$. Assuming that we
have a marked point $p:S\to C$, we will construct for every $n\gg 0$, a bigger
symplectic bundle ${\mathcal{V}}$, equipped with an isotropic subbundle
$M\subset{\mathcal{V}}$ and a Lagrangian subbundle $L\subset{\mathcal{V}}$
such that the reduced symplectic bundle $M^{\perp}/M$ is identified with
$R^{1}\pi_{*}\operatorname{\mathbb{C}}_{C/S}$ and $L\cap M^{\perp}$ gets
identified with $\pi_{*}\omega_{C}$.

We have a natural skew-symmetric form $B_{0}$ on $H^{0}(D,{\mathcal{O}})$,
where $D$ is a formal punctured disk around $p$, given by

$B_{0}(f,g)=\operatorname{Res}_{p}(fdg).$

######  Lemma 4.7.

For every $n>0$, such that $R^{1}\pi_{*}({\mathcal{O}}(np))=0$, the form
$B_{0}$ induces a symplectic form on the bundle

${\mathcal{V}}={\mathcal{V}}_{n}:=\pi_{*}({\mathcal{O}}_{C}(np)/(\operatorname{\mathbb{C}}_{C/S}+{\mathcal{O}}_{C}(-(n+1)p)))\simeq\operatorname{coker}({\mathcal{O}}_{S}\to\pi_{*}({\mathcal{O}}_{C}(np)/{\mathcal{O}}_{C}(-(n+1)p)))$

of rank $2n$. Furthermore,
$L_{\operatorname{can}}:=\pi_{*}({\mathcal{O}}_{C}/{\mathcal{O}}(-(n+1)p))/{\mathcal{O}}_{S}$
is a Lagrangian subbundle in ${\mathcal{V}}$.

###### Proof.

Let us denote by $\omega_{C/S}(np)^{ex}\subset\omega_{C/S}(np)$ the subsheaf
of locally exact forms. Then we have an exact sequence

$0\to\operatorname{\mathbb{C}}_{C/S}\to{\mathcal{O}}_{C}(np)\rTo{d}\omega_{C/S}((n+1)p)^{ex}\to
0$

which induces an isomorphism of sheaves

${\mathcal{O}}_{C}(np)/(\operatorname{\mathbb{C}}_{C/S}+{\mathcal{O}}_{C}(-(n+1)p))\rTo{\sim}\omega_{C/S}((n+1)p)^{ex}/\omega_{C/S}(-np).$

On the other hand, arguing locally near $p$ we see that the differential $d$
induces a surjective map

$\pi_{*}({\mathcal{O}}_{C}(np)/{\mathcal{O}}_{C}(-(n+1)p)))\to\pi_{*}(\omega_{C/S}((n+1)p)^{ex}/\omega_{C/S}(-np))$

with the kernel ${\mathcal{O}}_{S}$. This shows that the two definitions of
${\mathcal{V}}$ agree. The fact that $B_{0}(f,g)$ is symplectic and
$L_{\operatorname{can}}$ is Lagrangian is clear from the second definition of
${\mathcal{V}}$. ∎

######  Lemma 4.8.

Assume that $n>0$ is such that $R^{1}\pi_{*}({\mathcal{O}}(np))=0$. Then the
natural map

$\widetilde{L}_{\operatorname{can}}:=\pi_{*}({\mathcal{O}}_{C}(np)/\operatorname{\mathbb{C}}_{C/S})\to{\mathcal{V}}$

is an embedding of a subbundle of rank $n+g$.

###### Proof.

We have an exact sequence

$0\to{\mathcal{O}}_{S}\to\pi_{*}({\mathcal{O}}_{C}(np))\to\widetilde{L}_{\operatorname{can}}\rTo{\gamma}R^{1}\pi_{*}\operatorname{\mathbb{C}}_{C/S}\to
0$ (4.1)

which shows that $\widetilde{L}_{\operatorname{can}}$ is a bundle of rank
$n+g$. Now the map $\widetilde{L}_{\operatorname{can}}\to{\mathcal{V}}$ can be
identified with

$\pi_{*}(\omega_{C/S}((n+1)p)^{ex})\to\pi_{*}(\omega_{C/S}((n+1)p)^{ex}/\omega_{C/S}(-np)).$

The fact that it is injective with locally free quotient can be deduced from
the long exact sequence associated with the exact sequence

$0\to\omega_{C/S}(-np)\to\omega_{C/S}((n+1)p)^{ex}\to\omega_{C/S}((n+1)p)^{ex}/\omega_{C/S}(-np)\to
0$

Indeed, it is enough to see that $\pi_{*}(\omega_{C/S}(-np))=0$ and the
sheaves $R^{1}\pi_{*}(\omega_{C/S}(-np))$ and
$R^{1}\pi_{*}(\omega_{C/S}((n+1)p)^{ex})$ are locally free. Since
$R^{1}\pi_{*}({\mathcal{O}}_{C}(np))=0$, by Serre duality, we obtain that
$\pi_{*}(\omega_{C/S}(-np))=0$ and $R^{1}\pi_{*}(\omega_{C/S}(-np))$ is
locally free. On the other hand, the exact sequence

$0\to\omega_{C/S}((n+1)p)^{ex}\to\omega_{C/S}((n+1)p)\to{\mathcal{O}}_{p}\to
0$

gives a long exact sequence

$\pi_{*}(\omega_{C/S}((n+1)p))\to{\mathcal{O}}_{S}\to
R^{1}\pi_{*}(\omega_{C/S}((n+1)p)^{ex})\to
R^{1}\pi_{*}(\omega_{C/S}((n+1)p))=0$

where the first map is zero by the residue theorem, so
$R^{1}\pi_{*}(\omega_{C/S}((n+1)p)^{ex})\simeq{\mathcal{O}}_{S}$. ∎

Note that the bundle $R^{1}\pi_{*}\operatorname{\mathbb{C}}_{C/S}$ has a
symplectic structure induced by the multiplication

$R^{1}\pi_{*}\operatorname{\mathbb{C}}_{C/S}\otimes
R^{1}\pi_{*}\operatorname{\mathbb{C}}_{C/S}\to
R^{2}\pi_{*}\operatorname{\mathbb{C}}_{C/S}\simeq{\mathcal{O}}_{S}.$

We are going to show that the symplectic bundle
$R^{1}\pi_{*}\operatorname{\mathbb{C}}_{C/S}$ is obtained from $V$ by a
reduction with respect to an isotropic subbundle. We will deduce this from a
more general result concerning the skew-symmetric form on

${\mathcal{W}}:=\pi_{*}({\mathcal{O}}_{C}(\infty
p)/\operatorname{\mathbb{C}}_{C/S})$

induced by $B_{0}(f,g)=\operatorname{Res}_{p}(fdg)$.

######  Proposition 4.9.

(i) The map

$\gamma:{\mathcal{W}}\to R^{1}\pi_{*}\operatorname{\mathbb{C}}_{C/S}$

is compatible with the skew-symmetric forms.

(ii) Assume that $n>0$ is such that $R^{1}\pi_{*}({\mathcal{O}}(np))=0$. Then
the subbundle $M:=\pi_{*}({\mathcal{O}}_{C}(np))/{\mathcal{O}}_{S}$ in
${\mathcal{V}}={\mathcal{V}}_{n}$ is isotropic and one has
$\widetilde{L}_{\operatorname{can}}=M^{\perp}$, so that the restriction of
$\gamma$ to $\widetilde{L}_{\operatorname{can}}\subset{\mathcal{W}}$ induces
an isomorphism

$M^{\perp}/M\rTo{\sim}R^{1}\pi_{*}\operatorname{\mathbb{C}}_{C/S}.$

The main idea is to use the following resolution of the relative constant
sheaf, $\operatorname{\mathbb{C}}_{C/S}$,

$\operatorname{\mathbb{C}}_{C/S}\to[{\mathcal{O}}_{C}(\infty
p)\rTo{d}\omega_{C/S}(\infty p)^{ex}].$

We will use also the following second version of the above resolution, due to
the quasi-isomorphism where we identify $p$ with the subscheme $p(S)\subset
C$.

The map $\gamma$ is induced by the morphism in the derived category

$\widetilde{\gamma}:{\mathcal{O}}_{C}(\infty
p)/\operatorname{\mathbb{C}}_{C/S}\to\operatorname{\mathbb{C}}_{C/S}[1],$

represented by the map of complexes Without loss of generality we can assume
$S$ to be affine. Given a section $\overline{f}\in H^{0}(S,{\mathcal{W}})$
$=H^{0}(C,{\mathcal{O}}_{C}(\infty p)/\operatorname{\mathbb{C}}_{C/S})$, we
want to describe the corresponding morphism in the derived category

$\operatorname{\mathbb{C}}_{C/S}\rTo{\cdot\gamma(\overline{f})}\operatorname{\mathbb{C}}_{C/S}[1]$
(4.2)

given by the multiplication with $\gamma(\overline{f})$.

We need the following routine technical result.

######  Lemma 4.10.

Let ${\mathcal{A}}=[{\mathcal{A}}^{0}\rTo{d}{\mathcal{A}}^{1}]$ be a sheaf of
dg-algebras on a topological space $Y$, and let $c$ be a Cech $1$-cocycle with
coefficients in ${\mathcal{A}}$ (viewed as complex of sheaves), with respect
to some open covering $(U_{i})$ over $Y$:

$c=((a^{0}_{ij}),(a^{1}_{i})),$

with $a^{0}_{ij}\in{\mathcal{A}}^{0}(U_{i}\cap U_{j})$,
$a^{1}_{i}\in{\mathcal{A}}^{1}(U_{i})$, so that
$d(a^{0}_{ij})=a^{1}_{j}-a^{1}_{i}$, $a^{0}_{ij}+a^{0}_{jk}=a^{0}_{ik}$. We
can think of the cohomology class $[c]$ as a morphism
$\operatorname{\mathbb{Z}}_{Y}\to{\mathcal{A}}[1]$ in derived category of
sheaves, so we have the induced map

$r_{[c]}:{\mathcal{A}}[-1]\to{\mathcal{A}}$

given by the right multiplication with $c$. Let $C({\mathcal{A}})$ be the
sheaf Cech complex associated with ${\mathcal{A}}$, equipped with the standard
quasi-isomorphism

$\iota:{\mathcal{A}}\to C({\mathcal{A}}).$

Then $r_{[c]}$ is represented by the chain map

$r_{c}:{\mathcal{A}}[-1]\to C({\mathcal{A}})$

with the components

${\mathcal{A}}^{0}\rTo{\cdot c}C^{1}({\mathcal{A}}^{0})\oplus
C^{0}({\mathcal{A}}^{1})$
${\mathcal{A}}^{1}\rTo{(0,\cdot(-a^{0}_{ij}))}C^{2}({\mathcal{A}}^{0})\oplus
C^{1}({\mathcal{A}}^{1}).$

###### Proof.

The map $r_{[c]}$ is the composition

${\mathcal{A}}[-1]\simeq{\mathcal{A}}\otimes\operatorname{\mathbb{Z}}_{Y}[-1]\rTo{\operatorname{id}\otimes[c]}{\mathcal{A}}\otimes{\mathcal{A}}\rTo{\mu}{\mathcal{A}},$

where $\mu$ is the multiplication map. Since $[c]$ is represented by the chain
map $c:\operatorname{\mathbb{Z}}_{Y}[-1]\to C({\mathcal{A}})$, it is enough to
construct a chain map $\mu^{\prime}:{\mathcal{A}}\otimes C({\mathcal{A}})\to
C({\mathcal{A}})$, such that we have a commutative diagram Then as $r_{c}$ we
would take the composition of chain maps

${\mathcal{A}}[-1]\simeq{\mathcal{A}}\otimes\operatorname{\mathbb{Z}}_{Y}[-1]\rTo{\operatorname{id}\otimes
c}{\mathcal{A}}\otimes C({\mathcal{A}})\rTo{\mu^{\prime}}C({\mathcal{A}}).$

The map $\mu^{\prime}$ is constructed in a straightforward way:

$\mu^{\prime}(a\otimes(a_{i_{1}\ldots i_{k}}))=(a|_{U_{i_{1}\ldots
i_{k}}}\cdot a_{i_{1}\ldots i_{k}}).$

This gives the claimed map for $r_{c}$. ∎

######  Lemma 4.11.

The morphism (4.2) is represented by the chain map of resolutions

###### Proof.

Note that since the natural map of complexes

$[{\mathcal{O}}_{C}\to\omega_{C/S}]\to[{\mathcal{O}}_{C}(\infty
p)\to\omega_{C/S}(\infty p)^{ex}]$

is a quasi-isomorphism, it is enough to prove a similar result using the
resolution $[{\mathcal{O}}_{C}\to\omega_{C/S}]$ in the source of the map.

Now let us consider the resolution for $\operatorname{\mathbb{C}}_{C/S}$
obtained by first replacing it with $[{\mathcal{O}}\to\omega_{C/S}]$ and then
taking the sheaf Cech resolution corresponding to the covering $C=U_{0}\cup
U_{1}$, where $U_{0}$ is a neighborthood of $p$ and $U_{1}=C\setminus p$. This
resolution looks like

$C^{0}({\mathcal{O}})\to C^{1}({\mathcal{O}})\oplus C^{0}(\omega_{C/S})\to
C^{1}(\omega_{C/S}).$

Given a closed cycle $c=(f_{01};\alpha_{0},\alpha_{1})\in
C^{1}({\mathcal{O}})\oplus C^{0}(\omega_{C/S})$, so that
$\alpha_{1}-\alpha_{0}=df_{01}$ on $U_{0}\cap U_{1}$, the multiplication by
$[c]$ map
$\operatorname{\mathbb{C}}_{C/S}\to\operatorname{\mathbb{C}}_{C/S}[1]$ is
represented by the chain map

$\begin{diagram}$ (4.3)

(see Lemma 4.10). Now, we observe that there is a natural quasi-isomorphism

$\begin{diagram}$ (4.4)

induced by the natural projections
$C^{0}({\mathcal{O}})\to{\mathcal{O}}_{C}(\infty p)$,
$C^{0}(\omega_{C/S})\to\omega_{C/S}(\infty p)$ and by the residue map
$C^{1}(\omega_{C/S})\to{\mathcal{O}}_{p}$. Composing it with the map (4.3) we
get that the multiplication is represented by the map

Next, we claim that the class $\gamma(\overline{f})$ is represented by the
closed cycle

$c_{\overline{f}}=(f_{01};0,d\overline{f})\in C^{1}({\mathcal{O}})\oplus
C^{0}(\omega_{C/S}),$

where $f_{01}\in{\mathcal{O}}(U_{0}\cap U_{1})$ is such that
$f_{01}\equiv\overline{f}\mod\operatorname{\mathbb{C}}_{C/S}$. Indeed, this
immediately follows from the fact that the image of $c_{\overline{f}}$ under
the quasi-isomorphism (4.4) is the cycle $d\overline{f}$ in the middle term.
Applying the above calculation to $c_{\overline{f}}$ we get the result. ∎

###### Proof of Proposition 4.9.

(i) Given $\overline{f}\in H^{0}(C,{\mathcal{O}}_{C}(\infty
p)/\operatorname{\mathbb{C}}_{C/S})$, the pairing with $\overline{f}$ on
$H^{0}(S,{\mathcal{W}})$ is induced by the morphism of sheaves

$R(\overline{f}):{\mathcal{O}}_{C}(\infty
p)/\operatorname{\mathbb{C}}_{C/S}\rTo{\operatorname{Res}_{p}(?\cdot
d\overline{f})}{\mathcal{O}}_{p}.$

The resolution

$\operatorname{\mathbb{C}}_{C/S}\to[{\mathcal{O}}_{C}(\infty
p)\rTo{d}\omega_{C/S}(\infty p)\rTo{\operatorname{Res}_{p}}{\mathcal{O}}_{p}]$

gives a morphism
$\delta:{\mathcal{O}}_{p}\to\operatorname{\mathbb{C}}_{C/S}[2]$ which induces
an isomorphism

${\mathcal{O}}_{S}=\pi_{*}{\mathcal{O}}_{p}\rTo{\sim}R^{2}\pi_{*}(\operatorname{\mathbb{C}}_{C/S}).$

Thus, it is enough to check the commutativity of the following square in the
derived category: By Lemma 4.11, the composition of
$\cdot\gamma(\overline{f})$ with $\widetilde{\gamma}$ corresponds to the
following composition of maps of complexes Hence, this composition is equal to
the composition of $\delta$ with the map

${\mathcal{O}}_{C}(\infty
p)/\operatorname{\mathbb{C}}_{C/S}\to{\mathcal{O}}_{p}:\overline{g}\mapsto-\operatorname{Res}_{p}(\overline{f}d\overline{g}).$

But this is equal to
$\overline{g}\mapsto\operatorname{Res}_{p}(\overline{g}d\overline{f})$ with is
exactly $R(\overline{f})$.

(ii) The exact sequence (4.1) shows that
$M\subset\widetilde{L}_{\operatorname{can}}$ is the kernel of the surjective
map

$\gamma:\widetilde{L}_{\operatorname{can}}\to
R^{1}\pi_{*}\operatorname{\mathbb{C}}_{C/S}.$

Hence, by (i), we have an inclusion $\widetilde{L}_{\operatorname{can}}\subset
M^{\perp}$. Since these are subbundles of the same rank in ${\mathcal{V}}$, it
follows that $\widetilde{L}_{\operatorname{can}}=M^{\perp}$. ∎

For a Lagrangian subbundle $\Lambda\subset
R^{1}\pi_{*}\operatorname{\mathbb{C}}_{C/S}$, let us set

$L_{\Lambda}:=\gamma^{-1}(\Lambda)\subset\widetilde{L}_{\operatorname{can}}\subset{\mathcal{V}}.$
(4.5)

######  Proposition 4.12.

Assume that $n>0$ is such that $R^{1}\pi_{*}({\mathcal{O}}(np))=0$. Given a
Lagrangian subbundle $\Lambda\subset
R^{1}\pi_{*}\operatorname{\mathbb{C}}_{C/S}$, let us define $L_{\Lambda}$ by
(4.5) and $L_{\operatorname{can}}$ as in Lemma 4.7. Then $L_{\Lambda}$ and
$L_{\operatorname{can}}$ are Lagrangian subbundles in ${\mathcal{V}}$, and the
complex $C^{\bullet}({\mathcal{V}};L_{\Lambda},L_{\operatorname{can}})$ is
quasi-isomorphic to

$[\Lambda\to R^{1}\pi_{*}{\mathcal{O}}_{C}],$

where the morphism is the composition $\Lambda\hookrightarrow
R^{1}\pi_{*}\operatorname{\mathbb{C}}_{C/S}\to R^{1}\pi_{*}{\mathcal{O}}_{C}$.

###### Proof.

The first assertion follows from Proposition 4.9(ii) and Lemma 4.7. To deduce
the second, we use Lemma 4.4. We need to check that the natural map

$\alpha:M^{\perp}=\pi_{*}({\mathcal{O}}_{C}(np)/\operatorname{\mathbb{C}}_{C/S})\to{\mathcal{V}}/L_{\operatorname{can}}\simeq\pi_{*}({\mathcal{O}}_{C}(np)/{\mathcal{O}}_{C})$

is surjective. This map fits into the following commutative diagram with exact
rows: Now we observe that the map $\gamma$ fits into an exact sequence

$R^{1}\pi_{*}(\operatorname{\mathbb{C}}_{C/S})\to
R^{1}\pi_{*}({\mathcal{O}}_{C})\rTo{\gamma}R^{1}\pi_{*}({\mathcal{O}}_{C}/\operatorname{\mathbb{C}}_{C/S})$

which implies that $\gamma=0$ since the previous map in the exact sequence is
surjective. Hence, in the above commutative diagram we should have $\beta=0$,
and so $\alpha$ is surjective, as claimed.

Furthermore, we have

$L_{\operatorname{can}}\cap
M^{\perp}=L_{\operatorname{can}}\cap\widetilde{L}_{\operatorname{can}}=\pi_{*}({\mathcal{O}}_{C}/\operatorname{\mathbb{C}}_{C/S})\simeq\pi_{*}(\omega_{C/S}).$

Thus, from Lemma 4.4 we get a quasi-isomorphism

$C^{\bullet}({\mathcal{V}};L_{\Lambda},L_{\operatorname{can}})\simeq
C^{\bullet}(R^{1}\pi_{*}\operatorname{\mathbb{C}}_{C/S};\Lambda,\pi_{*}(\omega_{C/S})).$

Now the result follows from the exact sequence

$0\to\pi_{*}(\omega_{C/S})\to R^{1}\pi_{*}\operatorname{\mathbb{C}}_{C/S}\to
R^{1}\pi_{*}{\mathcal{O}}_{C}\to 0.$ (4.6)

∎

Note that the bundle $R^{1}\pi_{*}\operatorname{\mathbb{C}}_{C/S}$ comes from
a local system over $S$ equipped with a real structure. If we choose $\Lambda$
coming from a real Lagrangian local subsystem then the map $\Lambda\to
R^{1}\pi_{*}{\mathcal{O}}_{C}$ will be an isomorphism, hence in this case the
complex $C^{\bullet}({\mathcal{V}};L_{\Lambda},L_{\operatorname{can}})$ is
acyclic.

###  4.6. Symplectic structures in supercase

Assume now that $\pi:X\to S$ is a family of supercurves, and we have an
effective Cartier divisor $p\subset X$ of codimension $1|0$ with the support
$\overline{p}$ (as observed in [6], the choice of $p$ is equivalent to a
choice of section $S\to X$ supported on $\overline{p}$, see also [20, Sec.
9.3]). Then we can define sheaves ${\mathcal{O}}_{X}(mp)$ for
$m\in\operatorname{\mathbb{Z}}$. Now we would like to mimic the definition of
a symplectic bundle ${\mathcal{V}}={\mathcal{V}}_{n}$ from Sec. 4.5, where
$n>0$ is such that

$R^{1}\pi_{*}({\mathcal{O}}_{X}(np))=\pi_{*}({\mathcal{O}}_{X}(-np))=0.$

We can start by taking the vector bundle

$\widetilde{{\mathcal{V}}}=\pi_{*}({\mathcal{O}}_{X}(np)/{\mathcal{O}}_{X}(-Np))$

for some $N\gg n$, and observe that the form

$B(f,g)=\operatorname{Res}_{p}(f\delta g)$

descends to a skew-symmetric form on $\widetilde{{\mathcal{V}}}$ (recall that
$\delta:{\mathcal{O}}_{X}\to\omega_{X/S}$ is the canonical derivation). Now we
denote by ${\mathcal{V}}$ the quotient of $\widetilde{{\mathcal{V}}}$ by the
kernel of this form. Locally over $S$ we can choose an isomorphism of a formal
neighborhood of $p$ in $X$ with the split formal superdisk $D$ over $S$ with
the coordinate $(z,\theta)$ (see Remark 2.8). Then

$\widetilde{{\mathcal{V}}}|_{D}=z^{-n}{\mathcal{O}}_{S}[\\![z]\\!]/z^{N}{\mathcal{O}}_{S}[\\![z]\\!]\oplus
z^{-n}{\mathcal{O}}_{S}[\\![z]\\!]\theta/z^{N}{\mathcal{O}}_{S}[\\![z]\\!]\theta,$

and the kernel of the form gets identified with

$z^{n+1}{\mathcal{O}}_{S}[\\![z]\\!]/z^{N}{\mathcal{O}}_{S}[\\![z]\\!]\oplus
z^{n}{\mathcal{O}}_{S}[\\![z]\\!]\theta/z^{N}{\mathcal{O}}_{S}[\\![z]\\!]\theta,$

so that

${\mathcal{V}}|_{D}=\operatorname{coker}({\mathcal{O}}_{S}\to
z^{-n}{\mathcal{O}}_{S}[\\![z]\\!]/z^{n+1}{\mathcal{O}}_{S}[\\![z]\\!]\oplus
z^{-n}{\mathcal{O}}_{S}[\\![z]\\!]\theta/z^{n}{\mathcal{O}}_{S}[\\![z]\\!]\theta).$

This shows that ${\mathcal{V}}$ is a symplectic bundle of rank $2n|2n$.

Furthermore, we can still define the Lagrangian subbundle
$L_{\operatorname{can}}\subset{\mathcal{V}}$ by

$L_{\operatorname{can}}:=\pi_{*}({\mathcal{O}}_{X}/{\mathcal{O}}_{X}(-(n+1)p))/{\mathcal{O}}_{S},$

and the Lagrangian subbundle $L_{\Lambda}\subset{\mathcal{V}}$ as the preimage
of a Lagrangian subbundle $\Lambda\subset
R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/S}$ under the map

$\widetilde{L}_{\operatorname{can}}:=\pi_{*}({\mathcal{O}}_{X}(np)/\operatorname{\mathbb{C}}_{X/S})\to
R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/S}.$

We also have an isotropic subbundle

$M:=\pi_{*}({\mathcal{O}}_{X}(np))/{\mathcal{O}}_{S}\subset\widetilde{L}_{\operatorname{can}}.$

Most of the arguments of Section 4.5 go through in the supercase. However, the
assertion of Proposition 4.12 only holds over an open subset $U\subset S$
where ${\mathcal{O}}_{S}\to\pi_{*}{\mathcal{O}}_{X}$ is an isomorphism since
we want to use the exact sequence (4.6).

To describe the situation globally, let us consider the complex (concentrated
in degrees $0$ and $1$),

$\widetilde{\Lambda}=[\pi_{*}{\mathcal{O}}_{X}(np)\to L_{\Lambda}].$

It has locally free cohomology, so locally

$\widetilde{\Lambda}\simeq{\mathcal{O}}_{S}\oplus\Lambda[-1].$

Then we have a natural morphism in derived category

$\widetilde{\Lambda}\to R\pi_{*}{\mathcal{O}}_{X}$

represented by the natural map of complexes

$[\pi_{*}{\mathcal{O}}_{X}(np)\to
L_{\Lambda}]\to[\pi_{*}{\mathcal{O}}_{X}(np)\to\pi_{*}({\mathcal{O}}_{X}(np)/{\mathcal{O}}_{X})].$

Now we have the following replacement of Proposition 4.12.

######  Proposition 4.13.

For large enough $n$, and a Lagrangian subbundle $\Lambda\subset
R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/S}$, the subbundles $L_{\Lambda}$ and
$L_{\operatorname{can}}$ in ${\mathcal{V}}$, defined above are Lagrangian.
Furthermore, we have $M^{\perp}=\widetilde{L}_{\operatorname{can}}$ and the
natural map

$M^{\perp}/M\to R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/S}$

is an isomorphism and identifies $L_{\Lambda}/M$ with $\Lambda\subset
R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/S}$. Over an open subset $U\subset S$
where the theta-characteristics associated with $X$ has vanishing push-
forward, we have $L_{\operatorname{can}}+M^{\perp}={\mathcal{V}}$ and the
reduced Lagrangian $L_{\operatorname{can}}\cap M^{\perp}$ gets identified with
$\pi_{*}\omega_{X/S}\subset R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/S}$.
There is an isomorphism in the derived category of $S$,

$\operatorname{Cone}(\widetilde{\Lambda}\to
R\pi_{*}{\mathcal{O}}_{X})[-1]\simeq
C^{\bullet}({\mathcal{V}};L_{\Lambda},L_{\operatorname{can}}).$

###### Proof.

Most of the assertions are checked as in the even case. To prove the last
assertion, we note that the complex
$C^{\bullet}({\mathcal{V}};L_{\Lambda},L_{\operatorname{can}})$ has form

$[L_{\Lambda}\to{\mathcal{V}}/L_{\operatorname{can}}]\simeq[L_{\Lambda}\to\pi_{*}({\mathcal{O}}_{X}(np)/{\mathcal{O}}_{X})].$

Now we have a natural surjective morphism of complexes

$\operatorname{Cone}(\widetilde{\Lambda}\to[\pi_{*}{\mathcal{O}}_{X}(np)\to\pi_{*}{\mathcal{O}}_{X}(np)])[-1]\to[L_{\Lambda}\to\pi_{*}({\mathcal{O}}_{X}(np)/{\mathcal{O}}_{X})]$

with the kernel
$[\pi_{*}{\mathcal{O}}_{X}(np)\rTo{\operatorname{id}}\pi_{*}{\mathcal{O}}_{X}(np)][1]$.
This immediately implies our claim. ∎

###  4.7. Application to the moduli space of supercurves and poles of the
period matrix

Let ${\mathcal{M}}$ denote the moduli space of supercurves of fixed genus
$g\geq 2$, and let ${\mathcal{M}}_{0}$ denote the corresponding reduced space.
Let ${\mathcal{M}}^{+}$ denote the connected component over the component
${\mathcal{M}}_{0}^{+}$ corresponding to even spin structures. There is a
natural reduced divisor ${\mathcal{D}}_{0}\subset{\mathcal{M}}_{0}^{+}$,
corresponding to spin structures with a nonzero section.

Most of the time we work over some affine $S$ equipped with an étale morphism
$S\to{\mathcal{M}}^{+}$. Abusing the notation we will still denote by
${\mathcal{D}}_{0}$ the preimage of
${\mathcal{D}}_{0}\subset{\mathcal{M}}_{0}^{+}$ in $S$. We have a well defined
line bundle $\operatorname{Ber}(R\pi_{*}{\mathcal{O}}_{X})$ of rank $1|0$,
where $\pi:X\to{\mathcal{M}}^{+}$ is the universal supercurve. Let $U\subset
S$ be the open complement to ${\mathcal{D}}_{0}$. Then over $U$,
$R^{1}\pi_{*}{\mathcal{O}}_{X}$ is a vector bundle of rank $g|0$ and
$\pi_{*}{\mathcal{O}}_{X}={\mathcal{O}}_{S}$ (see Proposition 3.2). Thus, we
have a canonical isomorphism

$\operatorname{Ber}(R\pi_{*}{\mathcal{O}}_{X})|_{U}\simeq\operatorname{Det}(R^{1}\pi_{*}{\mathcal{O}}_{X})^{-1}|_{U}.$
(4.7)

Now assume that over $S$ we have a Lagrangian subbundle $\Lambda\subset
R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/S}$ such that the induced map
$\Lambda|_{U}\to R^{1}\pi_{*}{\mathcal{O}}_{X}|_{U}$ is an isomorphism. Then
the pair of Lagrangian subbundles $\pi_{*}\omega_{X/S}|_{U}$ and
$\Lambda|_{U}$ is transversal, so using the construction of Section 4.3, we
have the corresponding nonvanishing section

$\theta_{\Lambda}:=\theta(\pi_{*}\omega_{X/S},\Lambda)\in\Gamma(U,\operatorname{Det}(\Lambda)^{-1}\otimes\operatorname{Ber}(R\pi_{*}{\mathcal{O}}_{X})^{-1}|_{U}),$

where we used the isomorphism (4.7). Note that the line bundle on the right is
defined not just over $U$, so it makes sense to ask about the behavior of
$\theta_{\Lambda}$ near the divisor ${\mathcal{D}}_{0}$.

Assume that $\Lambda^{\prime}\subset
R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/S}$ is a Lagrangian complement to
$\Lambda$ defined over $S$, so that we have a decomposition

$R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/S}=\Lambda\oplus\Lambda^{\prime}.$

Then over $U$ the subbundle $\pi_{*}(\omega_{X/S})\subset
R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/S}$ is the graph of a morphism

$\Omega:\Lambda^{\prime}\to\Lambda.$

######  Theorem 4.14.

Assume that $\Lambda$ is transversal to

$\pi_{*}\omega_{C/S_{0}}\subset
R^{1}\pi_{*}\operatorname{\mathbb{C}}_{C/S_{0}}=R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/S}.$

Then locally near any point of the divisor ${\mathcal{D}}_{0}\subset S$ there
exists a trivialization of
$\operatorname{Det}(\Lambda)^{-1}\otimes\operatorname{Ber}(R\pi_{*}{\mathcal{O}}_{X})^{-1}$,
such that $\theta_{\Lambda}^{-1}=f^{2}$ for some regular function $f$
vanishing on ${\mathcal{D}}_{0}$. Furthermore, $f\Omega$ is regular on $S$. At
a generic point of ${\mathcal{D}}_{0}$, $f$ projects to a local equation of
${\mathcal{D}}_{0}$.

###### Proof.

We will deduce this from Theorem 4.5. Over $U$ the complex
$\operatorname{Cone}(\widetilde{\Lambda}\to R\pi_{*}{\mathcal{O}}_{X})$ is
exact, and $\theta_{\Lambda}$ can be identified with the canonical
nonvanishing section

$\widetilde{\theta}\in\operatorname{Ber}\operatorname{Cone}(\widetilde{\Lambda}\to
R\pi_{*}{\mathcal{O}}_{X})^{-1}|_{U}\simeq\operatorname{Det}(\Lambda)^{-1}\otimes\operatorname{Ber}(R\pi_{*}{\mathcal{O}}_{X})^{-1}|_{U}.$

Now by Proposition 4.13, we have a quasi-isomorphism of
$\operatorname{Cone}(\widetilde{\Lambda}\to R\pi_{*}{\mathcal{O}}_{X})[-1]$
with the complex
$C^{\bullet}({\mathcal{V}};L_{\Lambda},L_{\operatorname{can}})$, so that
$\widetilde{\theta}$ gets identified with $\theta(L_{\Lambda},L_{can})$. We
claim that the assumptions of Theorem 4.5 are satisfied for the pair of
Lagrangians $(L_{\Lambda},L_{\operatorname{can}})$. This will immediately
imply the first assertion. It is easy to see that the fibers of our
Lagrangians in ${\mathcal{V}}$ can be identified with the similarly defined
subspaces of the symplectic vector space

$V:=H^{0}({\mathcal{O}}_{C}(np)/{\mathcal{O}}_{C}(-(n+1)p))/\operatorname{\mathbb{C}}\oplus
H^{0}(L(np)/L(-np)),$

where $(C,{\mathcal{O}}_{C}\oplus L)$ is the split supercurve corresponding to
$s$. Namely, the fiber of $L_{\Lambda}$ is the preimage of $\Lambda_{s}\subset
H^{1}(C,\operatorname{\mathbb{C}})$ under the map

$H^{0}({\mathcal{O}}_{C}(np)/\operatorname{\mathbb{C}}_{C})\oplus
H^{0}(L(np))\to H^{1}(C,\operatorname{\mathbb{C}}),$

so its even part is the preimage of $\Lambda_{s}$ in
$H^{0}({\mathcal{O}}_{C}(np)/\operatorname{\mathbb{C}}_{C})\subset V^{+}$. On
the other hand, the fiber of $L_{\operatorname{can}}$ is the subspace

$H^{0}({\mathcal{O}}_{C}/{\mathcal{O}}_{C(-(n+1)p)}/\operatorname{\mathbb{C}})\oplus
H^{0}(L/L(-np))\subset V,$

so its even part is
$H^{0}({\mathcal{O}}_{C}/{\mathcal{O}}_{C(-(n+1)p)}/\operatorname{\mathbb{C}})\subset
V^{+}$. Thus, the intersection of even parts of the Lagrangians is the
intersection of the preimage of $\Lambda_{s}$ with
$H^{0}({\mathcal{O}}_{C}/\operatorname{\mathbb{C}}_{S})$. But the latter space
projects to $H^{0}(\omega_{C})\subset H^{1}(C,\operatorname{\mathbb{C}})$
which intersects trivially with $\Lambda_{s}$.

Let us choose a splitting $\Lambda\to L_{\Lambda}$ of the projection
$L_{\Lambda}\to\Lambda$. Then we get a composed morphism

$\Lambda\to L_{\Lambda}\to\pi_{*}({\mathcal{O}}_{X}(np)/{\mathcal{O}}_{X})$

such that there is a quasi-isomorphism of complexes

$[\Lambda\oplus\pi_{*}({\mathcal{O}}_{X}(np))/{\mathcal{O}}_{S}\rTo{\delta}\pi_{*}({\mathcal{O}}_{X}(np)/{\mathcal{O}}_{X})]\simeq\operatorname{Cone}(\widetilde{\Lambda}\to
R\pi_{*}{\mathcal{O}}_{X})[-1].$

Hence, combining Proposition 4.13 with Theorem 4.14 we obtain that
$f\delta^{-1}$ is regular on ${\mathcal{M}}^{+}$.

Now let $L^{\prime}_{1}\subset\widetilde{L}_{\operatorname{can}}$ be the
preimage of $\Lambda^{\prime}$, so that we have a resolution

$[\pi_{*}({\mathcal{O}}_{X}(np))/{\mathcal{O}}_{S}\to
L^{\prime}_{1}]\rTo{\sim}\Lambda^{\prime}.$

Now to show that $f\Omega$ is regular, it remains to check that $\Omega$ is
induced by the composed map

$L^{\prime}_{1}\to\pi_{*}({\mathcal{O}}_{X}(np)/\operatorname{\mathbb{C}}_{X/S})\to\pi_{*}({\mathcal{O}}_{X}(np)/{\mathcal{O}}_{X})\rTo{\delta^{-1}}\Lambda\oplus\pi_{*}({\mathcal{O}}_{X}(np))/{\mathcal{O}}_{S}\rTo{p_{\Lambda}}\Lambda$

which vanishes on $\pi_{*}({\mathcal{O}}_{X}(np))/{\mathcal{O}}_{S}\subset
L^{\prime}_{1}$ (where $p_{\Lambda}$ is the projection onto $\Lambda$).

To this end, we note that $-\Omega$ is the composition of the maps of bundles
over $U$,

$\Lambda^{\prime}\to R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/S}\to
R^{1}\pi_{*}{\mathcal{O}}\to\Lambda,$

where the last arrow is the inverse of the isomorphism $\Lambda\to
R^{1}\pi_{*}{\mathcal{O}}$ over $U$. Now we observe that the maps

$\Lambda^{\prime}\to R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/S}\to
R^{1}\pi_{*}{\mathcal{O}}$

are represented by the maps of resolutions (given by columns) in the following
diagram Furthermore, over $U$, the inverse to the isomorphism $\Lambda\to
R^{1}\pi_{*}{\mathcal{O}}$ is precisely the composition of $\delta^{-1}$ with
the projection to $\Lambda$. Together with the previous observation this
proves our claim.

The fact that $f$ projects to a local equation of ${\mathcal{D}}_{0}$ at a
generic point follows from the comparison with the construction of Sec. 4.4. ∎

##  5\. Supermeasure on the moduli space

###  5.1. Complex conjugation and the pairing

For a complex manifold ${\mathcal{X}}$ we denote by $\overline{{\mathcal{X}}}$
conjugate complex manifold. In other words,
$\overline{{\mathcal{X}}}={\mathcal{X}}$ as a $C^{\infty}$-manifold, but the
holomorphic functions on $\overline{{\mathcal{X}}}$ are antiholomorphic
functions on ${\mathcal{X}}$. For every local holomorphic function $f$ on
${\mathcal{X}}$ we can view $\overline{f}$ as a holomorphic function on
$\overline{{\mathcal{X}}}$. It follows that for every holomorphic vector
bundle ${\mathcal{V}}$ over ${\mathcal{X}}$ we have a holomorphic vector
bundle $\overline{{\mathcal{V}}}$ over $\overline{{\mathcal{X}}}$ (defined by
the complex conjugate transition matrices), so that we have a
$\operatorname{\mathbb{C}}$-antilinear isomorphism

$H^{0}({\mathcal{X}},{\mathcal{V}})\rTo{\sim}H^{0}(\overline{{\mathcal{X}}},\overline{{\mathcal{V}}}):s\mapsto\overline{s}.$

More generally, the correspondence
${\mathcal{X}}\mapsto\overline{{\mathcal{X}}}$ extends to a functor on the
category of complex manifolds, and for a morphism
$f:{\mathcal{X}}\to{\mathcal{Y}}$, we have

$\overline{f_{*}{\mathcal{V}}}\simeq\overline{f}_{*}(\overline{{\mathcal{V}}}).$

Given a holomorphic $n$-form $\eta$ on ${\mathcal{X}}$, we can view
$\overline{\eta}$ as a holomorphic $n$-form on $\overline{{\mathcal{X}}}$.
Hence, $p_{1}^{*}\eta\wedge p_{2}^{*}\overline{\eta}$ is a holomorphic
$2n$-form on ${\mathcal{X}}\times\overline{{\mathcal{X}}}$, where
$p_{1}:{\mathcal{X}}\times\overline{{\mathcal{X}}}\to{\mathcal{X}}$ and
$p_{2}:{\mathcal{X}}\times\overline{{\mathcal{X}}}\to\overline{{\mathcal{X}}}$
are the projections. Now let us consider the diagonal map
$\Delta:{\mathcal{X}}\to{\mathcal{X}}\times\overline{{\mathcal{X}}}$ (which is
neither holomorphic nor antiholomorphic). Then we have

$\eta\wedge\overline{\eta}=\Delta^{*}(p_{1}^{*}\eta\wedge
p_{2}^{*}\overline{\eta}).$

All of the above is also valid for complex supermanifolds, with the convention
that odd coordinates are self-conjugate. In local coordinates
$(x_{1},\ldots,x_{n},$ $\theta_{1},\ldots,\theta_{m})$ holomorphic functions
on $\overline{{\mathcal{X}}}$ form an exterior algebra in $\theta_{i}$ over
the ring of antiholomorphic functions in $x_{1},\ldots,x_{n}$.

If $\eta$ is a holomorphic section of the holomorphic Berezinian on
${\mathcal{X}}$ then $p_{1}^{*}\eta\wedge p_{2}^{*}\overline{\eta}$ will be a
holomorphic section of the holomorphic Berezinian on
${\mathcal{X}}\times\overline{{\mathcal{X}}}$. However, if we view
${\mathcal{X}}\times\overline{{\mathcal{X}}}$ as a real analytic supermanifold
then $p_{1}^{*}\eta\wedge p_{2}^{*}\overline{\eta}$ will be an integral form
of codimension $n|0$, where the complex dimension of ${\mathcal{X}}$ is $n|m$.

Now let $V$ be a local system of $\operatorname{\mathbb{R}}$-vector spaces of
finite rank over ${\mathcal{X}}$ with a bilinear pairing $b:V\otimes
V\to\operatorname{\mathbb{R}}$ (where we denote by $\operatorname{\mathbb{R}}$
the corresponding constant sheaf). Assume that

$\iota:{\mathcal{V}}\to
V\otimes_{\operatorname{\mathbb{R}}}{\mathcal{O}}_{{\mathcal{X}}}$

is a morphism of holomorphic vector bundles on ${\mathcal{X}}$. Then it
induces a morphism $\overline{\iota}:\overline{{\mathcal{V}}}\hookrightarrow
V\otimes_{\operatorname{\mathbb{R}}}{\mathcal{O}}_{\overline{{\mathcal{X}}}}$
of holomorphic vector bundles on $\overline{{\mathcal{X}}}$. Hence, on
${\mathcal{X}}\times\overline{{\mathcal{X}}}$ we get morphisms

$p_{1}^{*}\iota:p_{1}^{*}{\mathcal{V}}\hookrightarrow
p_{1}^{-1}V\otimes_{\operatorname{\mathbb{R}}}{\mathcal{O}}_{{\mathcal{X}}\times\overline{{\mathcal{X}}}},\
\ p_{2}^{*}\overline{\iota}:p_{2}^{*}\overline{{\mathcal{V}}}\hookrightarrow
p_{2}^{-1}V\otimes_{\operatorname{\mathbb{R}}}{\mathcal{O}}_{{\mathcal{X}}\times\overline{{\mathcal{X}}}}.$

Now we observe that the restrictions of the local systems $p_{1}^{-1}V$ and
$p_{2}^{-1}V$ to the diagonal in ${\mathcal{X}}\times\overline{{\mathcal{X}}}$
are naturally isomorphic: both restrictions are identified with $V$.
Therefore, there exists a unique isomorphism

$p_{1}^{-1}V\simeq p_{2}^{-1}V$

defined in a neighborhood of the diagonal in
${\mathcal{X}}\times\overline{{\mathcal{X}}}$. Thus, on this neighborhood we
get an ${\mathcal{O}}$-bilinear pairing

$p_{1}^{*}{\mathcal{V}}\otimes p_{2}^{*}\overline{{\mathcal{V}}}\to
p_{1}^{-1}V\otimes_{\operatorname{\mathbb{R}}}p_{1}^{-1}V\otimes_{\operatorname{\mathbb{R}}}{\mathcal{O}}_{{\mathcal{X}}\times\overline{{\mathcal{X}}}}\rTo{b\otimes\operatorname{id}}{\mathcal{O}}_{{\mathcal{X}}\times\overline{{\mathcal{X}}}}.$

###  5.2. Definition of the supermeasure and regularity theorem

We can apply the above construction to the embedding
$\pi_{*}\omega_{X/{\mathcal{U}}}\to
R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/{\mathcal{U}}}$ (recall that
${\mathcal{U}}\subset{\mathcal{M}}^{+}$ is the locus corresponding to theta-
characteristics with trivial $H^{0}$) and the identification

$R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/{\mathcal{M}}^{+}}=R^{1}\pi_{*}(\operatorname{\mathbb{R}}_{X})\otimes_{\operatorname{\mathbb{R}}}{\mathcal{O}}_{{\mathcal{M}}^{+}}.$
(5.1)

A slight modification we can make in this case is that instead of the diagonal
in ${\mathcal{M}}^{+}\times\overline{{\mathcal{M}}^{+}}$ it is more natural to
consider the locus in the reduced space
${\mathcal{M}}^{+}_{0}\times\overline{{\mathcal{M}}^{+}}_{0}$ corresponding to
the pairs $((C,L),(C^{\prime},L^{\prime}))$ with $C=C^{\prime}$ (but possibly
$L\neq L^{\prime}$). Let us call this locus the quasi-diagonal in
${\mathcal{M}}_{0}^{+}\times\overline{{\mathcal{M}}^{+}}_{0}$ and denote it by

$\Delta^{qu}({\mathcal{M}}^{+}_{0})\subset{\mathcal{M}}^{+}_{0}\times\overline{{\mathcal{M}}^{+}}_{0}.$

Since the local sistem $R^{1}\pi_{*}\operatorname{\mathbb{R}}_{X}$ comes from
the usual moduli space of curves, we can apply the construction of Section 5.1
in our case, working on a neighborhood of the quasi-diagonal (instead of the
diagonal). Thus, we get a pairing between vector bundles,

$p_{1}^{*}\pi_{*}\omega_{X/{\mathcal{U}}}\otimes
p_{2}^{*}\overline{\pi_{*}\omega_{X/{\mathcal{U}}}}\to{\mathcal{O}}_{{\mathcal{U}}\times\overline{{\mathcal{U}}}}$
(5.2)

defined near the quasi-diagonal $\Delta^{qu}({\mathcal{U}}_{0})$ in
${\mathcal{U}}_{0}\times\overline{{\mathcal{U}}}_{0}$.

We claim that this pairing is nondegenerate in a neighborhood of the quasi-
diagonal. Indeed, it is enough to check this on points of the quasi-diagonal,
where it follows from the nonvanishing of $\int_{C}\eta\wedge\overline{\eta}$
for a nonzero holomorphic $1$-form on a usual complex curve.

From the pairing (5.2) we get a morphism of holomorphic bundles

$p_{1}^{*}\pi_{*}\omega_{X/{\mathcal{U}}}\to
p_{2}^{*}\overline{\pi_{*}\omega_{X/{\mathcal{U}}}}^{\vee},$

or passing to determinants,

$p_{1}^{*}\operatorname{Ber}_{1}|_{{\mathcal{U}}}\to
p_{2}^{*}\overline{\operatorname{Ber}_{1}}^{-1}|_{{\mathcal{U}}},$

where
$\operatorname{Ber}_{j}:=\operatorname{Ber}(R\pi_{*}\omega_{X/{\mathcal{M}}}^{\otimes
j})$ (recall that
$R^{1}\pi_{*}\omega_{X/{\mathcal{U}}}\simeq{\mathcal{O}}_{\mathcal{U}}$, so
$\operatorname{Ber}_{1}|_{{\mathcal{U}}}\simeq\operatorname{Det}(\pi_{*}\omega_{X/{\mathcal{U}}})$).
Furthermore, this is an isomorphism in a neighborhood of
$\Delta^{qu}({\mathcal{U}}_{0})$.

In other words, considering the inverse of the above isomorphism, we get a
canonical holomorphic nonvanishing section

$s\in p_{1}^{*}\operatorname{Ber}_{1}\otimes
p_{2}^{*}\overline{\operatorname{Ber}_{1}}$

in a neighborhood of $\Delta^{qu}({\mathcal{U}}_{0})$ in
${\mathcal{U}}\times\overline{{\mathcal{U}}}$. Note that by definition, $s$ is
obtained by applying the construction of Section 4.3 to the pair of
Lagrangians
$(p_{1}^{*}\pi_{*}\omega_{X/{\mathcal{U}}},p_{2}^{*}\overline{\pi_{*}\omega_{X/{\mathcal{U}}}})$:

$s=\theta(p_{1}^{*}\pi_{*}\omega_{X/{\mathcal{U}}},p_{2}^{*}\overline{\pi_{*}\omega_{X/{\mathcal{U}}}})^{-1}.$

The Mumford isomorphism in the supercase (see [18]) is an isomorphism from
$\operatorname{Ber}_{1}^{5}$ to $\operatorname{Ber}_{3}$, which is the
holomorphic Berezinian on ${\mathcal{M}}$. Using it, from $s^{5}$ we get a
canonical holomorphic nonvanishing section of the holomorphic Berezinian on a
neighborhood of $\Delta^{qu}({\mathcal{U}}_{0})$ in
${\mathcal{U}}\times\overline{{\mathcal{U}}}$. Below we will show that it
extends to a meromorphic section on a neighborhood of
$\Delta^{qu}({\mathcal{M}}^{+}_{0})$ in
${\mathcal{M}}^{+}\times\overline{{\mathcal{M}}^{+}}$. Moreover, we have the
following regularity statement.

######  Theorem 5.1.

The image of the section $s^{5}$ under the Mumford isomorphism extends to a
meromorphic section of the holomorphic Berezinian on a neighborhood of the
quasi-diagonal in ${\mathcal{M}}^{+}\times\overline{{\mathcal{M}}^{+}}$. If
$g\leq 11$ then the extended section is regular on a neighborhood of the
quasi-diagonal.

###  5.3. Setup for studying the poles

To study poles of $s$ near the point corresponding to a supercurve with the
underlying curve $C$, we choose a Lagrangian splitting of
$H^{1}(C,\operatorname{\mathbb{R}})$. This defines a local Lagrangian
splitting of the local system $R^{1}\pi_{*}(\operatorname{\mathbb{R}}_{X})$,
and hence a Lagrangian splitting

$R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/{\mathcal{M}}^{+}}=W\oplus
W^{\prime}.$ (5.3)

In this case we have conjugate subbundles $\overline{W}$,
$\overline{W^{\prime}}$ in
$R^{1}\pi_{*}\operatorname{\mathbb{C}}_{\overline{X}/\overline{{\mathcal{M}}^{+}}}\simeq
H^{1}(C,\operatorname{\mathbb{R}})\otimes_{\operatorname{\mathbb{R}}}{\mathcal{O}}_{\overline{{\mathcal{M}}^{+}}}$.
As before, we work in a neighborhood of the quasi-diagonal in
${\mathcal{M}}^{+}\times\overline{{\mathcal{M}}^{+}}$, where we have the
identification

$p_{1}^{-1}R^{1}\pi_{*}(\operatorname{\mathbb{R}})\simeq
p_{2}^{-1}R^{1}\pi_{*}(\operatorname{\mathbb{R}}).$

With respect to this identification, we have

$p_{1}^{*}W=p_{2}^{*}\overline{W},\ \
p_{1}^{*}W^{\prime}=p_{2}^{*}\overline{W^{\prime}}$

as subbundles in
$H^{1}(C,\operatorname{\mathbb{R}})\otimes_{\operatorname{\mathbb{R}}}{\mathcal{O}}_{{\mathcal{M}}^{+}\times\overline{{\mathcal{M}}^{+}}}$.

Recall that the choice of $W$ leads to a nonvanishing section over the
complement to the divisor ${\mathcal{D}}_{0}\subset{\mathcal{M}}^{+}_{0}$,

$\theta^{-1}_{W}=\theta(\pi_{*}\omega_{X/{\mathcal{U}}},W)^{-1}\in\operatorname{Det}(W)\otimes\operatorname{Det}(R^{1}\pi_{*}{\mathcal{O}})^{-1}\simeq\operatorname{Det}(W)\otimes\operatorname{Ber}_{1}.$

Applying Lemma 4.6 to the triple of Lagrangians

$(p_{1}^{*}\pi_{*}\omega_{X/{\mathcal{U}}},p_{2}^{*}\overline{\pi_{*}\omega_{X/{\mathcal{U}}}},p_{1}^{*}W=p_{2}^{*}\overline{W})$

we get

$\begin{array}[]{l}\pm
s=\theta(p_{1}^{*}\pi_{*}\omega_{X/{\mathcal{U}}},p_{1}^{*}W)^{-1}\theta(p_{2}^{*}\overline{\pi_{*}\omega_{X/{\mathcal{U}}}},p_{2}^{*}\overline{W})^{-1}\det(p_{1}^{*}\tau-
p_{2}^{*}\overline{\tau})^{-1}\\\ =p_{1}^{*}\theta^{-1}_{W}\cdot
p_{2}^{*}\overline{\theta}^{-1}_{W}\cdot\det(p_{1}^{*}\tau-
p_{2}^{*}\overline{\tau})^{-1},\end{array}$ (5.4)

where

$\tau:W^{\prime}\to W\simeq(W^{\prime})^{\vee}$

is a symmetric morphism such that $\pi_{*}\omega_{X/{\mathcal{U}}}$ is the
graph of $\tau$ (so $\tau$ is the period matrix with respect to the above
splitting). Formula (5.4) will allow us to use information on the behavior of
$\theta_{W}$ near the divisor ${\mathcal{D}}_{0}$, to deduce the regularity of
$s$ for small genus.

###  5.4. Proof of Theorem 5.1

It is enough to study the situation in a neighborhood of the supercurve
corresponding to a curve with an even theta-characteristic $(C_{0},L_{0})$
such that $H^{0}(C_{0},L_{0})\neq 0$. In addition to the splitting (5.3) into
real Lagrangians, we can also define locally a different splitting. Namely,
let ${\mathcal{M}}^{+}_{0}$ be the usual moduli stack underlying
${\mathcal{M}}^{+}$ (classifying curves with even theta-characteristics).
Since $R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/{\mathcal{M}}^{+}}$ comes from
a local system, it is identified with the pull-back of a similar bundle
$R^{1}\pi_{0*}\operatorname{\mathbb{C}}_{C/{\mathcal{M}}^{+}_{0}}$ on
${\mathcal{M}}^{+}_{0}$, where $\pi_{0}:C\to{\mathcal{M}}^{+}_{0}$ is the
universal curve over ${\mathcal{M}}^{+}_{0}$, with respect to some local
splitting of the supermanifold ${\mathcal{M}}^{+}$.

Now, locally over ${\mathcal{M}}^{+}_{0}$ we can choose a Lagrangian splitting
of the form

$R^{1}\pi_{0*}\operatorname{\mathbb{C}}_{C/{\mathcal{M}}^{+}_{0}}=\Lambda^{\prime}\oplus\Lambda,$

with $\Lambda^{\prime}=\pi_{0*}\omega_{C/{\mathcal{M}}^{+}_{0}}$. Let us
denote still by $\Lambda$ and $\Lambda^{\prime}$ the induced Lagrangian
subbundles of rank $g|0$ in
$R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/{\mathcal{M}}^{+}}$. Furthermore, a
convenient choice of the complement $\Lambda$ is

$\Lambda=W.$

Over the open subset ${\mathcal{U}}\subset{\mathcal{M}}^{+}$,
$\pi_{*}\omega_{X/{\mathcal{U}}}$ is a Lagrangian subbundle of
$R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/{\mathcal{U}}}$, whose restriction
to ${\mathcal{U}}_{0}={\mathcal{U}}\cap{\mathcal{M}}^{+}_{0}$ is
$\Lambda^{\prime}$. Hence, there is a unique symmetric even morphism

$\Omega:\Lambda^{\prime}\to\Lambda$

over ${\mathcal{U}}$, whose graph is $\pi_{*}\omega_{X/{\mathcal{U}}}$. Note
that $\Omega|_{{\mathcal{U}}_{0}}=0$, so the entries of $\Omega$ belong to
${\mathcal{N}}^{2}$ (where ${\mathcal{N}}$ is the nilradical) on
${\mathcal{U}}$.

By Theorem 4.14, locally there exists a function $f$ on ${\mathcal{M}}^{+}$
such that $f\Omega$ extends regularly to ${\mathcal{M}}^{+}$ and $f^{2}$
differs from $\theta_{\Lambda}^{-1}=\theta_{W}^{-1}$ by a unit.

Now let us compare our two splittings of
$R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/{\mathcal{M}}^{+}}$. There exist a
regular even morphism $A:W^{\prime}\to W$ over ${\mathcal{M}}^{+}$ (locally)
such that $\Lambda^{\prime}$ is the graph of $A$. Let us consider the
composition

$\widetilde{\Omega}:W^{\prime}\rTo{x\mapsto
Ax+x}\Lambda^{\prime}\rTo{\Omega}\Lambda=W$

Note that the entries of $\widetilde{\Omega}$ still belong to
${\mathcal{N}}^{2}$ and $f\widetilde{\Omega}$ is regular on
${\mathcal{M}}^{+}$. Now any element of $\pi_{*}\omega_{X/{\mathcal{U}}}$ has
form $y+\Omega y$ for some $y\in\Lambda^{\prime}$, so using the decomposition
$y=Ax+x$, where $x\in W^{\prime}$, we get

$y+\Omega y=x+Ax+\widetilde{\Omega}(x)=x+(A+\widetilde{\Omega})x.$

This implies that $\pi_{*}\omega_{X/{\mathcal{U}}}$ is the graph of
$\tau:W^{\prime}\to W$, with

$\tau=A+\widetilde{\Omega}.$

Now let us denote by

${\mathcal{A}}(f)\subset
j_{*}{\mathcal{O}}_{{\mathcal{U}}\times\overline{{\mathcal{U}}}}$

the
${\mathcal{O}}_{{\mathcal{M}}^{+}\times\overline{{\mathcal{M}}^{+}}}$-subalgebra
generated by elements $p_{1}^{*}(x/f)$,
$p_{2}^{*}(\overline{x}/\overline{f})$, with $x\in{\mathcal{N}}^{2}$. Note
that the entries of $p_{1}^{*}\tau$ and $p_{2}^{*}\overline{\tau}$ are in
${\mathcal{A}}(f)$. Furthermore, since on ${\mathcal{M}}^{+}_{0}$ the
Lagrangian subbundles $\pi_{0*}\omega_{C/{\mathcal{M}}_{0}^{+}}$ and
$\overline{\pi_{0*}\omega_{CC/{\mathcal{M}}^{+}_{0}}}$ in
$R^{1}\pi_{*}(\operatorname{\mathbb{C}}_{C/{\mathcal{M}}^{+}_{0}})$ are
transversal and come from the usual moduli space of curves, we get that
$p_{1}^{*}A-p_{2}^{*}\overline{A}$ is invertible near the quasi-diagonal. This
implies that

${\det}(p_{1}^{*}\tau-p_{2}^{*}\overline{\tau})^{-1}\in{\mathcal{A}}(f).$

Finally, since $\theta_{W}^{-1}$ differs from $f^{2}$ by an invertible
function, using (5.4) we obtain

$s=p_{1}^{*}f^{2}\cdot p_{2}^{*}\overline{f}^{2}\cdot a,$

with $a\in{\mathcal{A}}(f)$. This immediately shows that $s$ is meromorphic.
Furthermore, since

$s^{5}=p_{1}^{*}f^{10}\cdot p_{2}^{*}\overline{f}^{10}\cdot a^{5},$

using the fact that $a^{5}\in{\mathcal{A}}(F)$, we see that $s^{5}$ is regular
near the quasi-diagonal in
${\mathcal{M}}^{+}\times\overline{{\mathcal{M}}^{+}}$ provided
${\mathcal{N}}^{22}=0$. Recall that the number of odd variables on
${\mathcal{M}}^{+}$ is $2g-2$. Thus, ${\mathcal{N}}^{22}=0$ as long as
$2g-2\leq 20$, i.e., $g\leq 11$. This finishes the proof. ∎

###  5.5. The pole of the superperiod map along the divisor

Let $S$ be a superscheme, $j:U\hookrightarrow S$ a complement to a Cartier
divisor $D^{{\operatorname{red}}}\subset S^{{\operatorname{red}}}$, so
$D^{{\operatorname{red}}}$ is locally given by an equation $f=0$, where $f$ is
a non-zero-divisor. Let us assume in addition that each local equation of
$D^{{\operatorname{red}}}$ is a non-zero-divisor in $\bigoplus_{n\geq
0}{\mathcal{N}}^{n}/{\mathcal{N}}^{n+1}$, where
${\mathcal{N}}\subset{\mathcal{O}}_{S}$ is the nilradical (this condition is
automatic in the case of a smooth superscheme). This implies that the natural
homomorphism ${\mathcal{O}}_{S}\to j_{*}{\mathcal{O}}_{U}$ is injective.

Now assume that $D^{{\operatorname{red}}}$ is extended to a Cartier divisor
$D\subset S$ (locally given by an even equation, reducing to an equation of
$D^{{\operatorname{red}}}$). Let us define the sheaf of rings on $S$,
${\mathcal{A}}_{D}\subset j_{*}{\mathcal{O}}_{U}$ by

${\mathcal{A}}_{D}:={\mathcal{O}}_{S}+{\mathcal{N}}^{2}(D)+{\mathcal{N}}^{4}(2D)+\ldots.$

Note that locally any extension $\widetilde{f}\in{\mathcal{O}}_{S}$ of an
equation $f$ of $D^{{\operatorname{red}}}$ is a non-zero-divisor in
${\mathcal{O}}_{S}$, due to our assumption on $f$, so it gives a Cartier
divisor on $S$ extending $D^{{\operatorname{red}}}$. We will show below that
in fact ${\mathcal{A}}_{D}$ depends only on $D^{{\operatorname{red}}}$.

Note that ${\mathcal{A}}_{D}$ is a coherent sheaf of
${\mathcal{O}}_{S}$-algebras, so it defines a superscheme $S_{D}$ with a
finite morphism to $S$, so that the embedding of $U$ into $S$ factors as the
composition

$U\hookrightarrow S_{D}\to S.$

Furthermore, we have $S_{D}^{{\operatorname{red}}}=S^{{\operatorname{red}}}$.

Let us also set

${\mathcal{I}}_{D}:={\mathcal{N}}{\mathcal{A}}_{D}={\mathcal{N}}+{\mathcal{N}}^{3}(D)+{\mathcal{N}}^{5}(2D)+\ldots\subset{\mathcal{A}}_{D}.$

It is clear that ${\mathcal{I}}_{D}$ is an ideal in ${\mathcal{A}}_{D}$.

######  Lemma 5.2.

(i) The sheaves ${\mathcal{A}}_{D}$ and ${\mathcal{I}}_{D}$ depend only on the
reduced divisor $D^{red}\subset S^{red}$.

(ii) There is a natural isomorphism of sheaves of rings

$\overline{{\mathcal{A}}}_{D}:={\mathcal{A}}_{D}/{\mathcal{I}}_{D}\simeq{\mathcal{O}}_{S}/{\mathcal{N}}\oplus\bigoplus_{n\geq
1}{\mathcal{N}}^{2n}/{\mathcal{N}}^{2n+1}(nD^{{\operatorname{red}}})|_{D^{{\operatorname{red}}}}.$

Thus, the closed sub-superscheme $\overline{S}_{D}\subset S_{D}$, associated
with ${\mathcal{I}}_{D}$, is equipped with a natural ${\mathbb{G}}_{m}$-action
such that the invariant locus is precisely $S^{{\operatorname{red}}}$.

###### Proof.

(i) We need to check that the sheaf
${\mathcal{O}}_{S}+\frac{{\mathcal{N}}^{2}}{f}+\frac{{\mathcal{N}}^{4}}{f^{2}}+\ldots$
does not change if we replace the local equation $f$ of $D$ by $f+n_{2}$ where
$n_{2}\in{\mathcal{N}}^{2}$. But this follows easily from the formula

$\frac{1}{f+n_{2}}=\frac{1}{f}(1-\frac{n_{2}}{f}+\frac{n_{2}^{2}}{f^{2}}-\ldots).$

(ii) First, let us check that the map

$\begin{array}[]{l}{\mathcal{A}}_{D}/{\mathcal{I}}_{D}\to{\mathcal{O}}_{S}/{\mathcal{N}}\oplus\bigoplus_{n\geq
1}{\mathcal{N}}^{2n}/({\mathcal{N}}^{2n+1}+{\mathcal{N}}^{2n}(-D)):\\\
a_{0}+\frac{a_{2}}{f}+\frac{a_{4}}{f^{2}}+\ldots\mapsto(a_{0}\mod{\mathcal{N}},(a_{2n}\mod{\mathcal{N}}^{2n+1}+{\mathcal{N}}^{2n}(-D))_{n\geq
1})\end{array}$ (5.5)

is well defined (here $a_{2i}\in{\mathcal{N}}^{2i}$ and $f$ is a local
equation of $D$). Indeed, suppose

$a_{0}+\frac{a_{2}}{f}+\frac{a_{4}}{f^{2}}+\ldots+\frac{a_{2n}}{f^{n}}=a_{1}+\frac{a_{3}}{f}+\ldots+\frac{a_{2n+1}}{f^{n}}\in{\mathcal{I}}_{D},$

where $a_{i}\in{\mathcal{N}}^{i}$. We need to check that
$a_{0}\in{\mathcal{N}}$ and
$a_{2i}\in{\mathcal{N}}^{2i+1}+f{\mathcal{N}}^{2i}$ for $i\geq 1$.

Note that since the multiplication by $f$ is injective on all the quotients
${\mathcal{N}}^{i}/{\mathcal{N}}^{i+1}$, for $i\geq 0$, it is injective on
${\mathcal{O}}_{S}/{\mathcal{N}}^{i}$. Hence, for $a\in{\mathcal{O}}$, if
$f^{m}a\in{\mathcal{N}}^{i}$ then $a\in{\mathcal{N}}^{i}$ (for $m\geq 0$). We
have $f^{n}a_{0}\in{\mathcal{N}}$, hence $a_{0}\in{\mathcal{N}}$. Next, for
every $i\geq 1$, we have

$f^{2n-i}[f^{i}(a_{0}-a_{1})+f^{i-1}(a_{2}-a_{3})+\ldots+f(a_{2i-2}-a_{2i-1})+a_{2i}]\in{\mathcal{N}}^{2i+1},$

hence, we get that $fa+a_{2i}\in{\mathcal{N}}^{2i+1}$ for some
$a\in{\mathcal{O}}$. But this implies that $fa\in{\mathcal{N}}^{2i}$, so
$a\in{\mathcal{N}}^{2i}$, therefore,
$a_{2i}\in{\mathcal{N}}^{2i+1}+f{\mathcal{N}}^{2i}$, as claimed.

Thus, the map (5.5) is well defined. It is clearly surjective. Now suppose
$a\in{\mathcal{A}}_{D}$ gives an element in its kernel. Then we have

$a=a_{1}+\frac{a_{3}+fb_{2}}{f}+\frac{a_{5}+fb_{4}}{f^{2}}+\ldots=(a_{1}+b_{2})+\frac{a_{3}+b_{4}}{f}+\ldots$

which is in ${\mathcal{I}}_{D}$, so $a\equiv 0\mod{\mathcal{I}}_{D}$. ∎

######  Definition 5.3.

For a Cartier divisor $D^{{\operatorname{red}}}\subset
S^{{\operatorname{red}}}$, whose local equations are non-zero-divisor in
$\bigoplus_{i\geq 0}{\mathcal{N}}^{i}/{\mathcal{N}}^{i+1}$, we define the
sheaf of rings on $S$, ${\mathcal{A}}_{D^{{\operatorname{red}}}}\subset
j_{*}{\mathcal{O}}_{U}$, by choosing locally an extension of
$D^{{\operatorname{red}}}$ to a Cartier divisor $D\subset S$ and setting

${\mathcal{A}}_{D^{{\operatorname{red}}}}:={\mathcal{A}}_{D}.$

We also have a well defined ideal
${\mathcal{I}}_{D^{{\operatorname{red}}}}\subset{\mathcal{A}}_{D^{{\operatorname{red}}}}$,
given locally by ${\mathcal{I}}_{D}\subset{\mathcal{A}}_{D}$. We denote by
$S_{D^{{\operatorname{red}}}}$ the superscheme over $S$ with the structure
sheaf ${\mathcal{A}}_{D^{{\operatorname{red}}}}$, and by
$\overline{S}_{D^{{\operatorname{red}}}}\subset S_{D^{{\operatorname{red}}}}$
the sub-superscheme associated with
${\mathcal{I}}_{D^{{\operatorname{red}}}}$.

Now assume that $f:U\to X$ is a morphism of superschemes, such that
corresponding morphism $f_{0}:U^{red}\to X$ extends to a regular morphism
$f^{{\operatorname{red}}}:S^{red}\to X$.

######  Definition 5.4.

We say that $f$ has regular pole along $D^{{\operatorname{red}}}$ if locally
the pull-backs of regular functions from $X$ belong to
${\mathcal{A}}_{D^{{\operatorname{red}}}}$. Equivalently, we require that $f$
extends to a regular morphism $\widetilde{f}:S_{D^{{\operatorname{red}}}}\to
X$.

Note that if $f$ has a regular pole along $D^{{\operatorname{red}}}$ then we
have the residual map

$\operatorname{Res}_{D^{{\operatorname{red}}}}(f):\overline{S}_{D^{{\operatorname{red}}}}\to
X$

obtained as the composition

$\overline{S}_{D^{{\operatorname{red}}}}\to
S_{D^{{\operatorname{red}}}}\rTo{\widetilde{f}}X.$

Furthermore, the restriction of
$\operatorname{Res}_{D^{{\operatorname{red}}}}(f)$ to
$S^{{\operatorname{red}}}$ is $f^{{\operatorname{red}}}$.

Now let ${\mathcal{M}}^{+}$ denote the even component of the moduli of
supercurves, and let ${\mathcal{U}}\subset{\mathcal{M}}^{+}$ be the open
substack corresponding to theta-characteristics with vanishing cohomology.
Then we have a well defined superperiod map

$\operatorname{per}:\widetilde{{\mathcal{U}}}\to{\mathcal{L}}{\mathcal{G}},$
(5.6)

where ${\mathcal{L}}{\mathcal{G}}$ is the Lagrangian Grassmannian of a
$2g$-dimensional symplectic space over $\operatorname{\mathbb{C}}$,
$\widetilde{{\mathcal{U}}}\to{\mathcal{U}}$ is the covering corresponding to a
choice of a symplectic basis in cohomology.

Our results show that the superperiod map has a regular pole along the divisor
${\mathcal{D}}_{0}$ where the theta-characteristic has nontrivial global
sections.

######  Proposition 5.5.

The superperiod map (5.6) has a regular pole along the preimage of the divisor
${\mathcal{D}}_{0}\subset{\mathcal{M}}^{+}_{0}$.

###### Proof.

This follows from the proof of Theorem 5.1. We have seen in that proof that
locally near a point on ${\mathcal{D}}_{0}$, the superperiod map is given by a
symmetric matrix $\Omega$ with entries in ${\mathcal{N}}^{2}$ such that
$f\Omega$ is regular, where $f$ reduces to an equation of ${\mathcal{D}}_{0}$.
∎

######  Remark 5.6.

In the case when $S$ is smooth and the divisor
$D^{{\operatorname{red}}}\subset S^{{\operatorname{red}}}$ is smooth, we can
define a superorbifold $\widetilde{S}_{D^{{\operatorname{red}}}}$ such that
$S_{D^{{\operatorname{red}}}}$ is the coarse moduli space of
$\widetilde{S}_{D}$. Namely, let $f$ be a local function on $S$ such that
$f\mod{\mathcal{N}}$ defines $D^{{\operatorname{red}}}$. Let us consider the
double covering $Z_{D}\to S$ given by $t^{2}=f$, and let $\widetilde{Z}_{D}$
be the blow-up of $Z_{D}$ at the ideal $(t)+{\mathcal{N}}$. If
$(x_{1},x_{2},\ldots,x_{n},(\theta_{j}))$ are local (even and odd) coordinates
on $S$, such that $x_{1}=f$ then
$(t,x_{2},\ldots,x_{n},(\frac{\theta_{j}}{t}))$ are local coordinates on
$\widetilde{Z}_{D}$, so $\widetilde{Z}_{D}$ is a supermanifold. We have a
natural action of $\operatorname{\mathbb{Z}}/2$ on $\widetilde{Z}_{D}$,
sending $t$ to $-t$, and we consider the quotient-stack
$\widetilde{Z}_{D}/(\operatorname{\mathbb{Z}}/2)$. One can check that the
resulting stacks glue into a global stack
$\widetilde{S}_{D^{{\operatorname{red}}}}$ depending only on
$D^{{\operatorname{red}}}$. Thus, away from the singular locus of
${\mathcal{D}}$, the superperiod map extends to a regular map from a
superorbifold over ${\mathcal{M}}^{+}$.

##  6\. Second variation of the period map as a Massey product

###  6.1. Second variation map

Let $X$ be a smooth supervariety. Let us denote by

$\Omega^{\prime}_{X}\subset\Omega_{X}$

the ${\mathcal{O}}_{X}$-submodule generated by all $df$, where $f$ is an even
function. Locally, $\Omega^{\prime}_{X}$ is generated by $dx_{i}$ and
$d(\theta_{i}\theta_{j})$ for $i<j$. The restriction
$\Omega^{\prime}_{X}|_{X^{{\operatorname{red}}}}$ is locally free and fits
into an exact sequence

$0\to{\mathcal{N}}^{2}/{\mathcal{N}}^{3}\to\Omega^{\prime}_{X}|_{X^{{\operatorname{red}}}}\to\Omega_{X^{{\operatorname{red}}}}\to
0.$

Note that if $x\in X^{{\operatorname{red}}}$ is a point,
${\mathfrak{m}}\subset{\mathcal{O}}_{X,x}$ the corresponding maximal ideal,
then the fiber of $\Omega^{\prime}_{X}|_{X^{{\operatorname{red}}}}$ at $x$ can
be identified with ${\mathfrak{m}}^{ev}/({\mathfrak{m}}^{ev})^{2}$, where
${\mathfrak{m}}^{ev}$ is the even part of ${\mathfrak{m}}$.

Now assume that we have a morphism $f:X\to Y$, where $Y$ is reduced (hence,
purely even). Let $f^{{\operatorname{red}}}=f|_{X^{{\operatorname{red}}}}$ be
the corresponding map $X^{{\operatorname{red}}}\to Y$. Then we have the
induced map $f^{*}\Omega_{Y}\to\Omega^{\prime}_{X}$, and hence,

$(f^{{\operatorname{red}}})^{*}\Omega_{Y}\to\Omega^{\prime}_{X}|_{X^{{\operatorname{red}}}}.$
(6.1)

Furthermore, its composition with the projection to
$\Omega_{X^{{\operatorname{red}}}}$ is exactly the map induced by
$df^{{\operatorname{red}}}$. Dually, we get a diagram It follows that there is
a unique map of ${\mathcal{O}}_{X^{\operatorname{red}}}$-modules,

$d^{(2)}f:{\bigwedge}^{2}({\mathcal{N}}/{\mathcal{N}}^{2})^{\vee}\simeq({\mathcal{N}}^{2}/{\mathcal{N}}^{3})^{\vee}\to\operatorname{coker}(df^{{\operatorname{red}}})$

induced by the vertical arrow in the above diagram. We call $d^{(2)}f$ the
second variation map.

Given a point $x\in X^{{\operatorname{red}}}$, to compute $d^{(2)}f$ at $x$,
we observe that the map (6.1) at $x$ is given by the natural map

${\mathfrak{m}}_{y}/{\mathfrak{m}}_{y}^{2}\to{\mathfrak{m}}_{x}^{ev}/({\mathfrak{m}}_{x}^{ev})^{2}.$
(6.2)

Note that the latter map is well defined also when $X$ is replaced by its sub-
superscheme. Thus, we can choose coordinates
$(x_{1},\ldots,x_{n};\theta_{1},\ldots,\theta_{m})$ near $x$ and consider the
closed subscheme

$X^{odd,\leq 2}(x)\subset X$

given by the ideal generated $(x_{1},\ldots,x_{n})+{\mathfrak{m}}_{x}^{3}$.
Note that the algebra of functions on $X^{odd,\leq 2}(x)$ is the truncated
exterior algebra $\bigwedge(\theta_{1},\ldots,\theta_{m})/\bigwedge^{\geq 3}$.
Let $\overline{{\mathfrak{m}}}_{x}$ denote the maximal ideal in this algebra.
Then it is easy to see that the composition

$({\mathcal{N}}^{2}/{\mathcal{N}}^{3})|_{x}\simeq{\mathcal{N}}^{2}/({\mathcal{N}}^{3}+{\mathfrak{m}}{\mathcal{N}}^{2})\to{\mathfrak{m}}_{x}^{ev}/({\mathfrak{m}}_{x}^{ev})^{2}\to\overline{{\mathfrak{m}}}_{x}^{ev}$

is an isomorphism. Thus, the dual of the map $d^{(2)}f$ at $x$ can be
identified with the restriction to $\ker((df^{\operatorname{red}})^{*})$ of
the map

${\mathfrak{m}}_{y}/{\mathfrak{m}}_{y}^{2}\to\overline{{\mathfrak{m}}}_{x}^{ev},$

induced by the map $X^{odd,\leq 2}(x)\to X\to Y$.

######  Remark 6.1.

It is clear that the map (6.2) can also be interpreted as the differential of
the induced map of usual schemes

$X/\Gamma\to Y,$

where $X/\Gamma$ is the scheme $(X,{\mathcal{O}}_{X}^{ev})$ called bosonic
quotient of $X$ in [2].

###  6.2. Second variation of a map to the Grassmannian

Assume that we have an exact sequence of vector bundles of even rank over a
supervariety $X$,

$0\to\SS\to V\otimes{\mathcal{O}}_{X}\to{\mathcal{Q}}\to 0,$

so that we have an associated morphism $f:X\to\operatorname{Gr}(V)$ to the
Grassmannian of $V$.

Given a point $x\in X$, we have the induced embedding of the fibers at $x$,
$\SS|_{x}\hookrightarrow V$. Now the composition

$\SS|_{x}\hookrightarrow V\hookrightarrow
V\otimes{\mathcal{O}}_{X}\to{\mathcal{Q}}$

factors through a map

$\SS|_{x}\to{\mathfrak{m}}_{x}{\mathcal{Q}}.$

The induced map

$\SS|_{x}\to{\mathfrak{m}}_{x}{\mathcal{Q}}/{\mathfrak{m}}_{x}^{2}{\mathcal{Q}}\simeq({\mathfrak{m}}_{x}/{\mathfrak{m}}_{x}^{2})\otimes{\mathcal{Q}}|_{x}$

corresponds to a map

$T_{x}X\to\operatorname{Hom}(\SS|_{x},{\mathcal{Q}}|_{x})$

which is precisely the tangent map to $f$ at $x$.

To calculate $d^{(2)}f_{x}$, we apply a similar to procedure to the
restriction

$0\to\overline{\SS}\to V\otimes{\mathcal{O}}_{X^{odd,\leq
2}(x)}\to\overline{{\mathcal{Q}}}\to 0$

of our sequence of vector bundles to a sub-superscheme $X^{odd,\leq 2}(x)$
(which depends on a choice of coordinates near $x$). Namely, as above, from
the sequence of bundles on $X^{odd,\leq 2}(x)$ we get a map

$\SS|_{x}\hookrightarrow
V\to(\overline{{\mathfrak{m}}}_{x}\overline{{\mathcal{Q}}})^{ev}\simeq\overline{{\mathfrak{m}}}^{ev}_{x}\otimes{\mathcal{Q}}|_{x}.$

Now $d^{(2)}f_{x}$ is the composition of the corresponding map

$({\mathcal{N}}^{2}/{\mathcal{N}}^{3})|_{x}^{\vee}\simeq(\overline{{\mathfrak{m}}}^{ev}_{x})^{\vee}\to\operatorname{Hom}(\SS|_{x},{\mathcal{Q}}|_{x})$

with the projection to $\operatorname{coker}(df^{{\operatorname{red}}}_{x})$.

Note that the analog of the map $d^{(2)}f_{x}$ makes sense if we start with an
exact sequence

$0\to\SS\to{\mathcal{V}}\to{\mathcal{Q}}\to 0$

of bundles on a supermanifold, where ${\mathcal{V}}$ is equipped with an
integrable connection. Namely, the connection gives a trivialization of
${\mathcal{V}}/{\mathfrak{m}}_{x}^{3}{\mathcal{V}}$, and then we can apply the
same construction.

###  6.3. Second variation of the period map

Let $\pi:X\to S={\mathcal{M}}^{+}$ be the universal supercurve over the even
component of the moduli space of supercurves. Then over the open subset
corresponding to theta-characteristics with no global sections, the exact
sequence (2.3) induces an exact sequence of bundles over $S$,

$0\to\pi_{*}\omega_{X/S}\to R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/S}\to
R^{1}\pi_{*}{\mathcal{O}}_{X}\to 0$ (6.3)

Furthermore, the bundle in the middle carries a Gauss-Manin connection, so we
have the corresponding period map,
$\operatorname{per}:\widetilde{S}\to\operatorname{Gr}$, from a covering of $S$
to the Grassmannian. We would like to calculate the corresponding second
variation map. As we observed above, the second variation map depends only on
the flat connection on $R^{1}\pi_{*}\operatorname{\mathbb{C}}_{X/S}$, so it
can be calculated over $S$.

######  Theorem 6.2.

Let $C$ be a smooth projective curve of genus $g$, $L$ a theta-characteristic
on $C$ with $H^{0}(C,L)=0$. Let $[C,L]$ be the corresponding point of $S$.

(i) The tangent map

$\kappa=d\operatorname{per}^{{\operatorname{red}}}_{[C,L]}:H^{1}(C,\omega_{C}^{-1})\to\operatorname{Hom}(H^{0}(C,\omega_{C}),H^{1}(C,{\mathcal{O}}_{C}))$
(6.4)

is given by the cup-product $H^{1}(C,\omega_{C}^{-1})\otimes
H^{0}(C,\omega_{C})\to H^{1}(C,{\mathcal{O}}_{C})$.

(ii) The negative of the second variation map,

$-d^{(2)}\operatorname{per}_{[C,L]}:{\bigwedge}^{2}H^{1}(C,L^{-1})\to\operatorname{Hom}(H^{0}(C,\omega_{C}),H^{1}(C,{\mathcal{O}}_{C}))/\operatorname{im}(\kappa),$

is given by the skew-symmetrization of the Massey product map

$H^{1}(C,L^{-1})\otimes
H^{1}(C,L^{-1})\to\operatorname{Hom}(H^{0}(\omega_{C}),H^{1}({\mathcal{O}}_{C}))/\operatorname{im}(\kappa)$
(6.5)

sending $x_{1}\otimes x_{2}$ to the map

$\alpha\mapsto d^{-1}(x_{1}\cdot x_{2})\cdot\alpha+x_{1}\cdot
d^{-1}(x_{2}\cdot\alpha)$

viewed modulo the image of $\kappa$ (where $d$ is the differential in an
appropriate dg-model).

###### Proof.

The main idea is to realize explicitly the exact sequence (6.3) using Cech
realization of cohomology coming from a covering $C=U_{0}\cup U_{1}$, where
$U_{0}$ is the formal neighborhood of a point $p\in C$ and $U_{1}=C\setminus
p$. In addition, we use an explicit realization of the universal curve over
the subscheme ${\mathcal{M}}^{odd,\leq 2}([C,L])$.

For any family of supercurves $X/S$, where $S$ is affine, with the underlying
usual curve ${\mathcal{C}}/S^{{\operatorname{red}}}$, and a marked point
$p:S^{{\operatorname{red}}}\to{\mathcal{C}}$, we can consider the covering of
$X$ by ${\mathcal{U}}_{1}=X\setminus p(S^{\operatorname{red}})$ and
${\mathcal{U}}_{0}$, the formal neighborhood of $p(S^{\operatorname{red}})$.
For brevity, we will denote $p(S^{\operatorname{red}})$ simply as $p$. We can
represent $R\pi_{*}{\mathcal{O}}_{X}$ by the complex

${\mathcal{P}}^{0}\rTo{d_{{\mathcal{O}}_{X}}}{\mathcal{P}}^{1}$

where ${\mathcal{P}}^{0}={\mathcal{O}}_{X}({\mathcal{U}}_{1})$,
${\mathcal{P}}^{1}={\mathcal{O}}_{X}({\mathcal{U}}_{0}\setminus
p)/{\mathcal{O}}({\mathcal{U}}_{0})$, and $d_{{\mathcal{O}}_{X}}$ is induced
by the restriction from ${\mathcal{U}}_{1}$ to ${\mathcal{U}}_{0}\setminus
p={\mathcal{U}}_{0}\cap{\mathcal{U}}_{1}$.

On the other hand, using the embedding
$j:{\mathcal{U}}_{1}\hookrightarrow{\mathcal{C}}$, we obtain quasi-
isomorphisms

$\operatorname{\mathbb{C}}_{X/S}\to[{\mathcal{O}}_{X}\to\omega_{X/S}]\to[j_{*}{\mathcal{O}}_{{\mathcal{U}}_{1}}\to(j_{*}\omega_{{\mathcal{U}}_{1}/S})^{ex}],$

where $(j_{*}\omega_{{\mathcal{U}}_{1}/S})^{ex}\subset
j_{*}\omega_{{\mathcal{U}}_{1}/S}$ denotes the subsheaf of forms that are
exact near $p$ (in other words, this is the sheaf image of $\delta$). Note
that the latter subsheaf is not acyclic for $\pi$. However, since
$j_{*}{\mathcal{O}}_{{\mathcal{U}}_{1}}$ is $\pi$-acyclic, we can still
represent $\tau_{\leq 1}R\pi_{*}\operatorname{\mathbb{C}}_{X/S}$ by the two-
term complex

${\mathcal{P}}^{0}\rTo{\delta}{\mathcal{Q}}^{1}$

where ${\mathcal{Q}}^{1}=\omega_{X/S}({\mathcal{U}}_{1})^{ex}$ is the space of
forms on ${\mathcal{U}}_{1}$, exact near $p$.

Now the exact sequence (6.3) can be represented explicitly as

$0\to\omega_{X/S}(X)\to\operatorname{coker}({\mathcal{P}}^{0}\rTo{\delta}{\mathcal{Q}}^{1})\to\operatorname{coker}({\mathcal{P}}^{0}\rTo{d_{{\mathcal{O}}_{X}}}{\mathcal{P}}^{1})\to
0$

where the first map sends $\alpha\in\omega_{X/S}(X)$ to