# Geometric Approach to the Weil-Petersson Symplectic Form

Reynir Axelsson  Háskóli Íslands, HjarÐarhaga 6, IS-107 Reykjavík, Ísland
<EMAIL_ADDRESS>and  Georg Schumacher  Fachbereich Mathematik und
Informatik, Philipps-Universität Marburg, Lahnberge, Hans-Meerwein-Strasse,
D-35032 Marburg,Germany<EMAIL_ADDRESS>
###### Abstract.

In a family of compact, canonically polarized, complex manifolds the first
variation of the lengths of closed geodesics is computed. As an application,
we show the coincidence of the Fenchel-Nielsen and Weil-Petersson symplectic
forms on the Teichmüller spaces of compact Riemann surfaces in a purely
geometric way. The method can also be applied to situations like moduli spaces
of weighted punctured Riemann surfaces, where the methods of Kleinian groups
are not available.

##  1\. Introduction

In 1958 André Weil introduced the methods of deformation theory of compact
complex manifolds to the study of Teichmüller spaces [WE]. In particular, he
used the $L^{2}$-inner product, induced by a metric of constant negative
curvature, to the harmonic representatives of Kodaira-Spencer classes. He
conjectured that the resulting Hermitian inner product on the Teichmüller
space would be of Kähler type and of negative curvature. The resulting Kähler
form is since known as the Weil-Petersson form.

Beginning in 1982, Wolpert analyzed the Fenchel-Nielsen structure of the
Teichmüller space and showed that the standard symplectic form
$\omega^{FN}=-\sum d\tau^{i}\wedge d\ell^{i}$, defined in terms of Fenchel-
Nielsen coordinates, actually coincides with the Weil-Petersson form
$\omega^{WP}$. This was achieved in a sequence of papers [WO2, WO3, WO4],
which also contain important arithmetic results. Most of the arguments use in
an essential way the theory of Kleinian groups. For example, one key idea in
Wolpert’s approach is to solve Beltrami equations on the upper half plane, to
compute derivatives of cross-ratios of the solutions, and then use Poincaré
series, in order to descend to the Riemann surfaces.

Our aim is to extend these results to cases where the original methods do not
apply, i.e. to cases where the hyperbolic structure is not necessarily induced
by the universal covering, such as weighted punctured Riemann surfaces. Our
methods also provide a purely differential geometric approach to a key result
in Wolpert’s approach, namely the Weil-Petersson duality of the twist vector
fields $\partial/\partial\tau^{j}$ and the differentials $d\ell^{j}$ of the
corresponding length functions. We use path integrals along closed geodesics
of harmonic Beltrami differentials and holomorphic quadratic differentials.
More generally, the first variation of the length of a closed geodesic in a
holomorphic family of Kähler-Einstein manifolds of negative curvature can be
expressed as the integral of the harmonic Kodaira-Spencer tensor along a
geodesic; this is our Theorem 3.2.

We consider the Teichmüller space of (possibly weighted) punctured Riemann
surfaces of genus $p$ with $n$ punctures. The existence of pants
decompositions and Fenchel-Nielsen coordinates was shown by Zhang in [ZH]
under the natural assumption that the cone angles do not exceed the value of
$\pi$, i.e. the weights are in the interval $[1/2,1]$ (c.f. Section 8). Let
$s^{\kappa};\kappa=1,\ldots,N$ be local holomorphic coordinates on the
Teichmüller space near a fixed point $s_{0}$ corresponding to a Riemann
surface $X$; then the tangent vectors $\partial/\partial s^{\kappa}|_{s_{0}}$
correspond to harmonic Beltrami differentials $\mu_{\kappa}$ on $X$. An
immediate consequence of Theorem 3.2 is the following result:

######  Theorem 1.1.

The complex derivative of a length coordinate with respect to a holomorphic
coordinate equals half the integral of the corresponding harmonic Beltrami
differential along the closed geodesic:

(1) $\left.\frac{\partial\ell^{i}(s)}{\partial
s^{\kappa}}\right|_{s_{0}}=\frac{1}{2}\int_{\gamma_{i}}\mu_{\kappa}.$

The above formula is independent of the choice of a set of Fenchel-Nielsen
coordinates and holds for the length function $\ell^{\gamma}(s)$ of a
differentiable family $\gamma(s)$ of closed geodesics.

In the classical case, closely related results, where the first variation of
the length of a closed geodesic is expressed in terms of integrals, can be
found in [WO1, WO2].

The shearing technique for twists along closed geodesics plays an important
role in Teichmüller theory. We handle the twist parameters $\tau^{i}$ in the
context of deformation theory and cocycles of vertical automorphisms. This
reflects only one aspect of the complex earthquake map associated to a given
closed geodesic in the sense of McMullen [Mc] (cf. also [E-M]). The tangent
vector $\partial/\partial\tau^{i}|_{s_{0}}$ of the Teichmüller space is
uniquely determined by its action on the cotangent space, i.e. the space of
holomorphic quadratic differentials. (Here we use the type decomposition of
the real tangent space of the Teichmüller space as embedded into its
complexification.)

######  Theorem 1.2.

The tangent vector $\partial/\partial\tau^{i}|_{s_{0}}$ applied to a quadratic
holomorphic differential $\varphi=\varphi(z)dz^{2}$ is given by the
integration of the quadratic differential along the corresponding closed
geodesic.

(2)
$\left(\left.\frac{\partial}{\partial\tau^{i}}\right|_{s_{0}}\\!\\!,\varphi(z)dz^{2}\right)=\frac{\sqrt{-1}}{2}\int_{\gamma_{i}}\varphi.$

Note that also the above formula is independent of the choice of Fenchel-
Nielsen coordinates. It holds for the twist vector
$\partial/\partial\tau^{\gamma}$ for any given closed geodesic $\gamma$.

Observe that the above inner product is actually a product of the $(1,0)$-part
of $(\partial/\partial\tau^{i})$ and the quadratic holomorphic differential
$\varphi$. As a corollary we obtain a key result proved by Wolpert for the
classical case in [WO2].

######  Theorem 1.3.

The Fenchel-Nielsen symplectic form on the Teichmüller space is equal to the
Weil-Petersson form:

(3) $\omega^{FN}=\omega^{WP}\text{, i.e. }-\sum_{i=1}^{N}d\tau^{i}\wedge
d\ell^{i}=\sqrt{-1}\;G_{\kappa{\overline{\lambda}}}^{WP}\;ds^{\kappa}\wedge
ds^{\overline{\lambda}}.$

In fact the formulas (1), (2) and (3) are intimately related in the sense that
any two of them imply the third one.

In the situation of Fuchsian groups, the above geodesic integral (2) seems to
be related to the monodromy integral of the quotient of the given quadratic
differential by a certain Abelian differential in [WO2] (cf. Section 6).

Our approach can also be used to compute second variations – our computations
and applications will appear elsewhere.

Acknowledgements. The second named author would like to thank Inkang Kim for
important discussions. The authors would like to thank the referee for
pointing out the reference [E-M].

This paper was written at the University of Iceland. The second named author
would like to express his thanks for the kind hospitality and support.

##  2\. Families of Kähler-Einstein Manifolds

Let $\\{{\mathcal{X}}_{s}\\}_{s\in S}$ be a holomorphic family of canonically
polarized compact complex manifolds parameterized by a (connected) complex
space $S$. It is given by a proper, smooth, holomorphic mapping
$f:{\mathcal{X}}\to S$ such that ${\mathcal{X}}_{s}=f^{-1}(s)$ for all $s\in
S$. For simplicity we will assume that the base $S$ is smooth, although our
results can also be given a meaning for possibly non-reduced singular base
spaces.

We denote by $X={\mathcal{X}}_{s_{0}}$; $s_{0}\in S$ a distinguished fiber.
Let $n=\dim_{\mathbb{C}}X$, and denote by $z^{\alpha}$; $\alpha=1,\ldots,n$
local holomorphic coordinates on $X$.

A Kähler form on $X$ will be denoted by

$\omega_{X}=\sqrt{-1}{g_{\alpha{\overline{\beta}}}}dz^{\alpha}\wedge
dz^{\overline{\beta}}$

On the fiber $X$ we are using the summation convention together with the
$\nabla$-notation for covariant derivatives. A $|$–symbol will denote an
ordinary derivative. Also, $\partial_{\alpha}$ and
$\partial_{\overline{\beta}}$ will stand for $\partial/\partial z^{\alpha}$
and $\partial/\partial z^{\overline{\beta}}$ respectively. The raising and
lowering of indices is defined as usual in terms of covariant derivatives.

For the Ricci tensor $R_{\alpha{\overline{\beta}}}$ of $X$ we will use the
sign convention

$R_{\alpha{\overline{\beta}}}=-\log(g(z))_{|{\alpha{\overline{\beta}}}},$

where $g(z)=\det({g_{\alpha{\overline{\beta}}}}(z))$.

Accordingly we set $g(z,s)=\det(g_{\alpha{\overline{\beta}}}(z,s))$ for the
family $f:{\mathcal{X}}\to S$, where we equip the fibers ${\mathcal{X}}_{s}$
with Kähler-Einstein forms

$\omega_{{\mathcal{X}}_{s}}=\sqrt{-1}g_{\alpha{\overline{\beta}}}(z,s)dz^{\alpha}\wedge
dz^{\overline{\beta}}$

of constant negative curvature $-1$, i.e.
$R_{\alpha{\overline{\beta}}}(z,s)=-g_{\alpha{\overline{\beta}}}(z,s)$.

Let

$\rho:T_{s_{0}}S\to H^{1}(X,{\mathcal{T}}_{X})$

be the Kodaira-Spencer map for the corresponding deformation of $X$ over $S$
at the point $s_{0}\in S$.

A natural inner product on the space $H^{1}(X,{\mathcal{T}}_{X})$ of
infinitesimal deformations of $X$, the Weil-Petersson Hermitian inner product
on $T_{s_{0}}S$, is induced by the Kähler-Einstein metric $\omega_{X}$ on $X$.
Namely, given tangent vectors $u,v\in T_{s_{0}}S$, we denote by
$A_{u}=A_{u{\overline{\beta}}}^{\alpha}\partial_{\alpha}dz^{\overline{\beta}}$
and $A_{v}$ the harmonic representatives of $\rho(u)$ and $\rho(v)$
respectively. Then the inner product of $u$ and $v$ equals

$\langle
u,v\rangle_{WP}=\int_{X}A_{u{\overline{\beta}}}^{\alpha}A_{{\overline{v}}\gamma}^{\overline{\delta}}g_{\alpha{\overline{\delta}}}g^{{\overline{\beta}}\gamma}g\/dV,$

where $A_{\overline{v}}$ denotes the adjoint (conjugate) tensor of $A_{v}$,
and $g\/dV$ the volume element.

We note that the Weil-Petersson inner product is positive definite at a given
point of the base, if the induced deformation is effective.

Let $s_{j}$; $j=1,\ldots,N$ be local coordinates on $S$, and set
$A_{j}=A_{\partial/\partial s_{j}}$. Then the Weil-Petersson form on $S$
equals

$\omega^{WP}=\sqrt{-1}G_{i{\overline{\jmath}}}^{WP}(s)ds^{i}\wedge
ds^{\overline{\jmath}},$

where we use the notation

$G_{i{\overline{\jmath}}}^{WP}(s)=\langle\partial/\partial
s^{i},\partial/\partial
s^{j}\rangle_{WP}=\int_{X}A_{i{\overline{\beta}}}^{\alpha}A_{{\overline{\jmath}}\gamma}^{\overline{\delta}}g_{\alpha{\overline{\delta}}}g^{{\overline{\beta}}\gamma}g\/dV.$

A closed real $(1,1)$-form on the total space ${\mathcal{X}}$ of the given
family is defined by

(4) $\omega_{\mathcal{X}}=-\sqrt{-1}\partial{\overline{\partial}}\log g(z,s).$

The Kähler-Einstein condition for the fibers implies that the restrictions of
$\omega_{\mathcal{X}}$ to the fibers are the given Kähler-Einstein metrics:

$\omega_{\mathcal{X}}|_{{\mathcal{X}}_{s}}=\omega_{{\mathcal{X}}_{s}}$

for all $s\in S$.

An important fact is that the family of Kähler-Einstein metric tensors
contains the harmonic representatives of Kodaira-Spencer classes. The short
exact sequence

$0\to{\mathcal{T}}_{{\mathcal{X}}/S}\to{\mathcal{T}}_{{\mathcal{X}}}\to
f^{*}{\mathcal{T}}_{S}\to 0$

induces the Kodaira-Spencer map via the edge homomorphism for direct images.
Again, we use $\nabla$-notation for covariant derivatives as well as raising
and lowering of indices in fiber direction for families. A lift of a tangent
vector $\partial/\partial s^{i}$ of $S$ at $s_{0}$ as a differentiable vector
field of ${\mathcal{X}}$ on $X$ is of the form

$\partial/\partial s^{i}+b_{i}^{\alpha}\partial_{\alpha}.$

Its exterior ${\overline{\partial}}$-derivative
$B_{i{\overline{\beta}}}^{\alpha}\partial_{\alpha}dz^{\overline{\beta}}$,
where
$B_{i{\overline{\beta}}}^{\alpha}=\nabla_{\overline{\beta}}b_{i}^{\alpha}$, is
interpreted as a ${\overline{\partial}}$-closed $(0,1)$-form on $X$ with
values in the tangent bundle of $X$. It represents the obstruction against the
existence of a holomorphic lift of the given tangent vector, i.e. the
infinitesimal triviality of the deformation. Moreover,

(5) $\rho(\partial/\partial
s^{i})=[B_{i{\overline{\beta}}}^{\alpha}\partial_{\alpha}dz^{\overline{\beta}}]\in
H^{1}(X,{\mathcal{T}}_{X}).$

Our argument is based upon the notion of canonical lifts in the sense of Siu
(cf. [SIU]), which turned out to be $\omega_{\mathcal{X}}$-horizontal: The
form $\omega_{\mathcal{X}}$ is positive definite in fiber directions but need
not be positive on ${\mathcal{X}}$. Its components in the directions of
$z^{\alpha}$ and $s^{j}$ are denoted by $g_{\alpha{\overline{\jmath}}}$ etc.
However, horizontal lifts of tangent vectors are well-defined and of the form

(6) $\partial/\partial s^{i}+a_{i}^{\alpha}\partial_{\alpha},$

where

(7) $a_{i}^{\alpha}=-{g^{{\overline{\beta}}\alpha}}g_{i{\overline{\beta}}}.$

We set
$A_{i{\overline{\beta}}}^{\alpha}=\nabla_{\overline{\beta}}a_{i}^{\alpha}$ and
obtain (cf. [SCH2, SCH3]) the following fact:

######  Proposition 2.1.

The horizontal lifts with respect to $\omega_{\mathcal{X}}$ of tangent vectors
induce harmonic representatives of Kodaira-Spencer classes: The harmonic
representative of $\rho(\partial/\partial s^{i})$ equals
$A_{i{\overline{\beta}}}^{\alpha}\partial_{\alpha}dz^{\overline{\beta}}$.
These satisfy the following properties

  * (i)

$\nabla_{\overline{\delta}}A^{\alpha}_{i{\overline{\beta}}}=\nabla_{\overline{\beta}}A^{\alpha}_{i{\overline{\delta}}},$

  * (ii)

$\nabla_{\gamma}A_{i{\overline{\beta}}}^{\alpha}g^{{\overline{\beta}}\gamma}=0,$

  * (iii)

$A_{i{\overline{\beta}}{\overline{\delta}}}=A_{i{\overline{\delta}}{\overline{\beta}}}.$

The conditions (i) and (ii) above correspond to harmonicity, whereas condition
(iii) reflects the relationship with the metric tensor.

##  3\. Families of closed geodesics

Let $(f:{\mathcal{X}}\to S,\omega_{\mathcal{X}})$ be a family of Kähler-
Einstein manifolds with constant negative Ricci curvature equal to $-1$, where
$\omega_{\mathcal{X}}$ is given by (4).

We denote by $\gamma_{s}$ a differentiable family of closed geodesics
contained in the fibers ${\mathcal{X}}_{s}$. In order to describe the
variation of the length of closed geodesics in a family, we use the notion of
integrating a tensor along a geodesic. Exemplarily we define:

######  Definition 3.1.

Let $C=C_{{\overline{\beta}}{\overline{\delta}}}$ be a tensor on the Kähler
manifold $X$, and $\gamma$ be a geodesic of length $\ell$, parameterized by
$u(t)=(u^{1}(t),\ldots,u^{n}(t))$, such that $\|\dot{u}(t)\|=1$. Then

$\int_{\gamma}C=\int_{\gamma}C_{{\overline{\beta}}{\overline{\delta}}}dz^{\overline{\beta}}dz^{\overline{\delta}}:=\int_{0}^{\ell}C_{{\overline{\beta}}{\overline{\delta}}}(u(t))\dot{u}^{\overline{\beta}}\dot{u}^{\overline{\delta}}dt.$

For contravariant tensors of order one this notation coincides with the
integration of a differential form along the curve $\gamma$.

The main result of this section is the following theorem:

######  Theorem 3.2.

Let ${\mathcal{X}}\to S$ be a family of canonically polarized, compact,
complex manifolds equipped with Kähler-Einstein metrics. Let $\gamma_{s}$ be a
differentiable family of closed geodesics on the fibers ${\mathcal{X}}_{s}$,
having the lengths $\ell(s)$. Let $\partial/\partial s^{i}|_{s=s_{0}}\in
T_{s_{0}}S$ be a tangent vector. Then the first variation of the length of the
closed geodesic with respect to the holomorphic parameter of the family is
half the integral of the symmetric, harmonic Kodaira-Spencer tensor along the
geodesic:

$\left.\frac{\partial}{\partial
s^{i}}\right|_{s=s_{0}}\\!\\!\\!\ell(s)=\frac{1}{2}\int_{\gamma(s_{0})}A_{i{\overline{\beta}}{\overline{\delta}}}dz^{\overline{\beta}}dz^{\overline{\delta}}.$

In order to prove the theorem, we may assume that $S=\\{s\\}$ is a disk in
$\mathbb{C}$, and $s_{0}=0$. The closed geodesic curves $\gamma_{s}$ are
solutions of the equation for geodesics

(8)
$\ddot{u}^{\alpha}(t,s)+\Gamma^{\alpha}_{\gamma\sigma}(u(t,s))\dot{u}^{\gamma}(t,s)\dot{u}^{\sigma}(t,s)=0.$

The solution is unique up to the constant value of the speed

$\|\dot{u}(t,s)\|^{2}={{g_{\alpha{\overline{\beta}}}}(u(t,s),s)\dot{u}^{\alpha}(t,s)\dot{u}^{\overline{\beta}}(t,s)}.$

For $s=0$ we chose $\|\dot{u}\|=1$, for the remaining values of $s$ the value
of $\|\dot{u}\|$ will be determined by the fact that the parameter $t$ assumes
values in the interval $[0,\ell_{0}]$, where $\ell_{0}$ is the length of
$\gamma_{0}$. Now

$\ell(s)=\int_{0}^{\ell_{0}}\|\dot{u}(t,s)\|dt$

so that

(9)
$\left.\frac{d\ell(s)}{ds}\right|_{s=0}=\frac{1}{2}\int_{0}^{\ell_{0}}\frac{d}{ds}\|\dot{u}(t,s)\|^{2}dt.$

We need to study various tensors on ${\mathcal{X}}$ along the geodesics. In
particular, we consider the vector field

$U_{s}(t,s)=\left.\left(\frac{\partial}{\partial
s}+u_{s}^{\alpha}(t,s)\frac{\partial}{\partial
z^{\alpha}}+u_{s}^{\overline{\beta}}(t,s)\frac{\partial}{\partial
z^{\overline{\beta}}}\right)\right|_{u(t,s)},$

where

$u_{s}^{\alpha}(t,s)=\frac{\partial u^{\alpha}(t,s)}{\partial s}\text{ and
}u_{s}^{\overline{\beta}}(t,s)=\frac{\partial
u^{\overline{\beta}}(t,s)}{\partial s}\,.$

The function

$\chi(t,s)=\langle U_{s},\dot{u}\rangle$

is defined as the inner product of vector fields along the geodesics with
values on the total space ${\mathcal{X}}$. Concerning $\omega_{\mathcal{X}}$
we indicate the direction of the one-dimensional space $S$ by $s$. We have

$\chi(t,s)=g_{s{\overline{\beta}}}\dot{u}^{\overline{\beta}}+{g_{\alpha{\overline{\beta}}}}u_{s}^{\alpha}\dot{u}^{\overline{\beta}}+{g_{\alpha{\overline{\beta}}}}\dot{u}^{\alpha}u_{s}^{\overline{\beta}}.$

Using (8) we simplify the expression for the derivative $\dot{\chi}$ and get:

(10)
$\begin{split}\dot{\chi}(t,s)=g_{s{\overline{\beta}}|\alpha}\dot{u}^{\alpha}\dot{u}^{\overline{\beta}}+g_{s{\overline{\beta}}|{\overline{\delta}}}\dot{u}^{\overline{\beta}}\dot{u}^{\overline{\delta}}+g_{s{\overline{\beta}}}\ddot{u}^{\overline{\beta}}+g_{\alpha{\overline{\beta}}|\gamma}\dot{u}^{\gamma}u_{s}^{\alpha}\dot{u}^{\overline{\beta}}+\\\
{g_{\alpha{\overline{\beta}}}}\dot{u}_{s}^{\alpha}\dot{u}^{\overline{\beta}}+g_{\alpha{\overline{\beta}}|{\overline{\delta}}}\dot{u}^{\alpha}u_{s}^{\overline{\beta}}\dot{u}^{\overline{\delta}}+{g_{\alpha{\overline{\beta}}}}\dot{u}^{\alpha}\dot{u}_{s}^{\overline{\beta}}.\end{split}$

######  Lemma 3.3.

$\frac{\partial}{\partial
s}\|\dot{u}\|^{2}-\dot{\chi}=A_{s{\overline{\beta}}{\overline{\delta}}}\dot{u}^{\overline{\beta}}\dot{u}^{\overline{\delta}}.$

###### Proof.

We use (8) and (10) together with (6) and (7) and apply Proposition 2.1. The
statement now follows from a calculation. ∎

The proof of Theorem 3.2 concludes with (9) and Lemma 3.3.

##  4\. Cocycles of vertical automorphisms

Let $X$ be a compact complex manifold, and $(S,s_{0})$ a germ of a complex
space. We follow the approach of Forster and Knorr [F-K] and assign to any
deformation of $X$ over a $(S,s_{0})$ an element of the first cohomology
$H^{1}(X,\mathfrak{G}_{S})$ of $X$ with values in the sheaf $\mathfrak{G}_{S}$
of vertical automorphisms. Its derivative with respect to the base space
yields the Kodaira-Spencer map. For any deformation of $X$ given by a
cartesian diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(S,s_{0})}$

there exist open coverings $\\{U_{i}\\}$ of $X$ and $\\{Z_{i}\\}$ of
${\mathcal{X}}$ together with isomorphisms $\psi_{i}:U_{i}\times S\to Z_{i}$,
compatible with the cartesian diagram, and equal to the identity for $s=0$.
The actual cocycles $\gamma_{ij}$ are defined on $(U_{i}\cap U_{j})\times S$
by $\psi_{j}^{-1}\circ\psi_{i}$. We look at the derivatives of the
$\gamma_{ij}$ with respect to tangent vectors of the base and values in the
sheaf of holomorphic vector fields. These define the Kodaira-Spencer classes
and will be compared with suitable $(0,1)$-forms representing the same
classes.

We take a differentiable trivialization $\zeta:{\mathcal{X}}\to X\times S$ of
the family, which equals the identity for $s=s_{0}$. Then, according to (5),
the class $\rho(\partial/\partial s^{\kappa})|_{s=s_{0}}$ is represented by

(11) $\left.\frac{\partial^{2}\zeta^{\alpha}(z,s)}{\partial s^{\kappa}\partial
z^{\overline{\beta}}}\right|_{s=s_{0}}\\!\frac{\partial}{\partial
z^{\alpha}}dz^{\overline{\beta}}.$

We now consider the situation where the covering that is involved in the
deformation only consists of only two elements, $U_{1}$ and $U_{2}$, say. A
differentiable trivialization of the holomorphic family can be chosen to be
equal to the identity on the complement of any given compact neighborhood of
$Z_{1}\cap Z_{2}\subset{\mathcal{X}}$.

##  5\. Application to Riemann surfaces

For families of Riemann surfaces (with notation from Section 1) the above
Proposition 2.1 states that harmonic Beltrami differentials are of the form

(12) $\mu_{\kappa}=\mu_{\kappa}(z)\frac{\partial}{\partial
z}{\overline{dz}}=-\frac{\partial}{{\overline{\partial
z}}}\left(\frac{1}{g}\frac{\partial^{2}\log g}{\partial
s^{\kappa}{\overline{\partial z}}}\right)\frac{\partial}{\partial
z}{\overline{dz}}.$

We write these also in the form

(13) $\mu_{\kappa}(z)\frac{\partial}{\partial
z}{\overline{dz}}=\frac{\varphi_{\overline{\kappa}}(z)}{g(z)}\frac{\partial}{\partial
z}{\overline{dz}}$

for certain quadratic holomorphic differentials
$\varphi_{\kappa}=\varphi_{\kappa}(z)dz^{2}$, with
$\varphi_{\overline{\kappa}}:={\overline{\varphi_{\kappa}}}$.

Now Theorem 1.1 is an immediate consequence of Theorem 3.2.

##  6\. Twist coordinates vs. holomorphic coordinates

In this section we will prove Theorem 1.2. We will first treat the case of
Riemann surfaces $X$ equipped with the hyperbolic metrics $g=g(z)|dz|^{2}$.
Given the context, we normalize a metric of constant curvature $-1$ so that
the Ricci curvature equals $-1$, i.e. $\partial^{2}\log g(z)/\partial
z{\overline{\partial z}}=g(z)$.

Our point is to give a direct proof, which only uses the hyperbolic geometry
of Riemann surfaces.

We consider a closed geodesic $\gamma$ on $X$ parameterized by $u(t)$ with
$\|\dot{u}(t)\|=1$. Let the length of $\gamma$ be $\ell_{0}=2\pi r_{0}$. The
twist along $\gamma$ gives rise to a real, one parameter family of Riemann
surfaces, in particular it gives rise to a real (analytic) curve in the
appropriate Teichmüller space.

As local coordinate set we use an annulus
$W=\\{z\in\mathbb{C};r_{1}<|z|<r_{2}\\}$ embedded into $X$, such that $\gamma$
corresponds to a circle $\\{|z|=r_{0}\\}$ for $r_{1}<r_{0}<r_{2}$, with
parameterization $u(t)=r_{0}e^{\sqrt{-1}t/r_{0}}$ for $0\leq t\leq\ell_{0}$.

For $\tau\in{\mathbb{R}}$ a family of holomorphic automorphisms of $W$ is
given by the twist $z\mapsto e^{\sqrt{-1}\tau/r_{0}}z$. In the sense of
Section 4 we interpret these as a family of vertical automorphisms with
respect to an open covering of $X$ consisting of two elements. Now a
differentiable trivialization (which is equal to the identity on the
complement of $W$) is given by

(14) $\zeta(z,\tau)=e^{\sqrt{-1}\frac{\tau}{r_{0}}h(r)}z,$

where $r=|z|$ and $h\in C^{\infty}((r_{1},r_{2}),{\mathbb{R}})$ denotes a
differentiable function which is identically zero near $r_{1}$, and
identically equal to one near $r_{2}$. The value $\tau=\ell_{0}$ represents a
Dehn twist.

We take a deformation theoretic standpoint, and we use in this context the
shearing technique [WO2, Section 1]: A Beltrami differential will again stand
for an infinitesimal deformation. If we replace in (11) a tangent vector
$\partial/\partial s^{\kappa}$ by a real tangent vector
$\partial/\partial\tau$, we obtain the image of its $(1,0)$-component under
the Kodaira-Spencer map. The resulting Beltrami differential
$\mu=\mu(z)(\partial/\partial z){\overline{dz}}$ is identically zero on the
complement of $W$. For $z\in W$ it follows immediately that

(15) $\mu(z)=\frac{\sqrt{-1}}{2}\frac{z^{2}}{rr_{0}}h^{\prime}(r).$

Now we can prove Theorem 1.2. The idea is to determine a Beltrami differential
as a linear form on the cotangent space of the Teichmüller space.

The duality $H^{1}(X,{\mathcal{T}}_{X})\times
H^{0}(X,{\mathcal{O}}_{X}(2K_{X}))\to\mathbb{C}$ of tangent and cotangent
spaces of ${\mathcal{T}}$ is given by

$\left(\mu(z)\frac{\partial}{\partial
z}{\overline{dz}},\varphi(z)dz^{2}\right)\mapsto\int_{X}\mu(z)\varphi(z)dA,$

where $dA=\sqrt{-1}dz\wedge{\overline{dz}}$ is the Euclidean area element.
Observe that $\mu$ need not be harmonic in this expression, in fact we get the
same value for any function $h(r)$ with the above properties. The claim of the
theorem follows from (15) in principle by Stokes’ theorem – we can also let
$h$ tend to the function $h_{0}$, which is defined to be equal to zero for
$r<r_{0}$ and equal to one for $r\geq r_{0}$. Then $h^{\prime}(r)$ converges
to a delta function, and integration over $r$ implies that that for any
quadratic holomorphic differential $\varphi$

$(\mu,\varphi)=\frac{\sqrt{-1}}{2}\;r_{0}\int_{0}^{2\pi}e^{2\sqrt{-1}\vartheta}\varphi(r_{0}e^{\sqrt{-1}\vartheta})\/\/d\vartheta.$

Now the necessary reparameterization $\vartheta=t/r_{0}$ yields the claim of
the theorem. ∎

We mention the monodromy integrals from [WO2]. If $A$ is a hyperbolic element
in $PSL(2,{\mathbb{R}})$, lifted to $\widehat{A}\in SL(2,{\mathbb{R}})$, the
monodromy integral is

${\mathcal{P}}(\psi,\widehat{A})=\frac{1}{2}\int_{s}^{As}\psi\Omega_{\widehat{A}}$

where $\psi$ is a quadratic holomorphic differential, and
$\Omega_{\widehat{A}}^{-1}$ a suitable Abelian differential. Then by [WO2,
Lemma 4.2] the following formula holds for the twist tangent vector $\mu_{0}$
applied to a holomorphic quadratic differential

$\int_{\Delta}\mu_{0}\psi=-\sqrt{-1}\sigma_{\widehat{A}}{\eta(\widehat{A})}^{-1}{\mathcal{P}}(\psi,\widehat{A}),$

where $\Delta$ is a fundamental domain, $\sigma_{\widehat{A}}=\pm 1$, and
$\eta(\widehat{A})=(({\mathrm{\,tr\,}}\widehat{A})^{2}-4)^{1/2}$.

##  7\. Computation of the Fenchel-Nielsen symplectic form

The results of this section are valid for the Teichmüller space of (possibly
weighted) punctured Riemann surfaces.

Combining our geodesic integral formulas (1) and (2) we immediately obtain a
result, which was proved in the classical case by Wolpert ([WO2, Theorem
2.10]).

######  Corollary 7.1.

The vector fields $\partial/\partial\tau^{j}$ are Hamiltonian with respect to
the Weil-Petersson form:

(16) $\displaystyle\frac{\partial\ell^{j}}{\partial s^{\overline{\kappa}}}$
$\displaystyle=$
$\displaystyle-\sqrt{-1}\left\langle\frac{\partial}{\partial\tau^{j}},\mu_{\kappa}\right\rangle,$
(17) $\displaystyle\frac{\partial\ell^{j}}{\partial s^{\kappa}}$
$\displaystyle=$
$\displaystyle\sqrt{-1}\left\langle\mu_{\kappa},\frac{\partial}{\partial\tau^{j}}\right\rangle.$

Here, the Weil-Petersson inner product, which is defined initially on the
complex tangent space, is extended in a natural way to the complexified real
tangent space (which also contains the complex tangent vectors of type
$(0,1)$).

Next, we prove the coincidence (3) of the Fenchel-Nielsen and Weil-Petersson
symplectic forms using (1) and (2). We will give a short computational
argument:

Proof of Theorem 1.3. We write $\omega^{WP}$ in terms of Fenchel-Nielsen
coordinates and begin with the contribution of the twist parameter involving
$d\tau^{j}\wedge d\tau^{k}$. Since we are dealing with real forms, we need to
show:

Claim. The real part

$\text{\rm Re}\left(\sqrt{-1}\;\left\langle\frac{\partial}{\partial
s^{\alpha}},\frac{\partial}{\partial s^{\beta}}\right\rangle\frac{\partial
s^{\alpha}}{\partial\tau^{j}}\frac{\partial
s^{\overline{\beta}}}{\partial\tau^{k}}\right)$

vanishes, in particular, it is symmetric in $j$ and $k$.

Proof of the claim:

$\displaystyle\sqrt{-1}\;\left\langle\frac{\partial}{\partial
s^{\alpha}},\frac{\partial}{\partial s^{\beta}}\right\rangle\frac{\partial
s^{\alpha}}{\partial\tau^{j}}\frac{\partial
s^{\overline{\beta}}}{\partial\tau^{k}}$ $\displaystyle=$
$\displaystyle\sqrt{-1}\;\left\langle\frac{\partial}{\partial
s^{\alpha}},\frac{\partial
s^{\beta}}{\partial\tau^{k}}\frac{\partial}{\partial
s^{\beta}}\right\rangle\frac{\partial s^{\alpha}}{\partial\tau^{j}}$
$\displaystyle=$ $\displaystyle\sqrt{-1}\;\left\langle\frac{\partial}{\partial
s^{\alpha}},\frac{\partial}{\partial\tau^{k}}\right\rangle\frac{\partial
s^{\alpha}}{\partial\tau^{j}}\text{\footnotesize\quad(type consideration) }$
$\displaystyle=\atop\eqref{eq:ltaudual2}$
$\displaystyle\frac{\partial\ell^{k}}{\partial s^{\alpha}}\cdot\frac{\partial
s^{\alpha}}{\partial\tau^{j}}.$

The real part of this quantity equals

$None$ $\frac{1}{2}\frac{\partial\ell^{k}}{\partial\tau^{j}}=0.$

We conclude the proof of Theorem 1.3. The vanishing of the coefficients of
$d\ell^{k}\wedge d\ell^{m}$ is an immediate consequence of the “dual” equation
(18), which may not be available at this point. However, we can revert here to
Wolpert’s purely (differential) geometric argument [WO4, Corollary 1.2], which
involves orientation reversing (anti-holomorphic) symmetries.

Finally, the coefficient of $d\tau^{j}\wedge d\ell^{m}$ in $\omega^{WP}$
equals

$\displaystyle\sqrt{-1}\;\left(\left\langle\frac{\partial
s^{\alpha}}{\partial\tau^{j}}\frac{\partial}{\partial
s^{\alpha}},\frac{\partial
s^{\beta}}{\partial\ell^{m}}\frac{\partial}{\partial
s^{\beta}}\right\rangle-\left\langle\frac{\partial
s^{\alpha}}{\partial\ell^{m}}\frac{\partial}{\partial
s^{\alpha}},\frac{\partial
s^{\beta}}{\partial\tau^{j}}\frac{\partial}{\partial
s^{\beta}}\right\rangle\right)$ $\displaystyle=$
$\displaystyle\sqrt{-1}\;\left(\left\langle\frac{\partial}{\partial\tau^{j}},\frac{\partial
s^{\beta}}{\partial\ell^{m}}\frac{\partial}{\partial
s^{\beta}}\right\rangle-\left\langle\frac{\partial
s^{\alpha}}{\partial\ell^{m}}\frac{\partial}{\partial
s^{\alpha}},\frac{\partial}{\partial\tau^{j}}\right\rangle\right)$
$\displaystyle=\atop{\eqref{eq:ltaudual1}\eqref{eq:ltaudual2}}$
$\displaystyle-\left(\frac{\partial\ell^{j}}{\partial
s^{\overline{\beta}}}\frac{\partial
s^{\overline{\beta}}}{\partial\ell^{m}}+\frac{\partial\ell^{j}}{\partial
s^{\alpha}}\frac{\partial s^{\alpha}}{\partial\ell^{m}}\right)$
$\displaystyle=$ $\displaystyle-\delta_{\ell m}.\hskip 170.71652pt\qed$

Now we obtain the following formula of Wolpert from the equality
$\omega^{FN}=\omega^{WP}$:

######  Corollary 7.2.

The length coordinates $\ell^{j}$ give rise to Hamiltonian vector fields
$\partial/\partial\ell^{j}$:

(18) $\frac{\partial\tau^{j}}{\partial
s^{\alpha}}=-\sqrt{-1}\left\langle\frac{\partial}{\partial
s^{\alpha}},\frac{\partial}{\partial\ell^{j}}\right\rangle.$

The above equation (17) also implies the following corollary, whose classical
version is Theorem 2.11 from [WO2]: We first observe that (17) holds in a
slightly more general situation. Namely, as usual, we pick local holomorphic
coordinates $(s_{1},\ldots,s_{N})$ on the Teichmüller space near a point
$s_{0}$ and denote by $\mu_{\alpha}$ the harmonic Beltrami differentials
corresponding to the $\partial/\partial s^{\alpha}|_{s_{0}}$. Then for any
closed geodesic $\gamma$ on $X$ we have the length function $\ell(\gamma)$,
and also $\partial/\partial\tau(\gamma)$ is defined at the given point $s_{0}$
without a choice of a system of Fenchel-Nielsen coordinates. Now (17) reads

(19) $\frac{\partial\ell(\gamma)}{\partial
s^{\alpha}}=\sqrt{-1}\;\left\langle\frac{\partial}{\partial
s^{\alpha}},\frac{\partial}{\partial\tau(\gamma)}\right\rangle.$

for all $\alpha=1,\ldots,N$.

######  Corollary 7.3.

Let $\gamma$ and $\delta$ be closed geodesics on $X$. Then

(20)
$\frac{\partial\ell(\gamma)}{\partial\tau(\delta)}=-\frac{\partial\ell(\delta)}{\partial\tau(\gamma)}.$

###### Proof.

We show that the left-hand side is skew symmetric in $\gamma$ and $\delta$.

$\displaystyle\frac{\partial\ell(\gamma)}{\partial\tau(\delta)}$
$\displaystyle=$ $\displaystyle 2\text{\rm
Re}\left(\frac{\partial\ell(\gamma)}{\partial s^{\alpha}}\cdot\frac{\partial
s^{\alpha}}{\partial\tau(\delta)}\right)$
$\displaystyle=\atop\eqref{eq:ltaudualvar}$ $\displaystyle 2\text{\rm
Re}\left(\sqrt{-1}\;\frac{\partial
s^{\alpha}}{\partial\tau(\delta)}\cdot\left\langle\frac{\partial}{\partial
s^{\alpha}},\frac{\partial}{\partial\tau(\gamma)}\right\rangle\right)$
$\displaystyle=$ $\displaystyle 2\text{\rm Re}\left(\sqrt{-1}\;\frac{\partial
s^{\alpha}}{\partial\tau(\delta)}\cdot\left\langle\frac{\partial}{\partial
s^{\alpha}},\frac{\partial
s^{\beta}}{\partial\tau(\gamma)}\frac{\partial}{\partial
s^{\beta}}\right\rangle\right)\text{\footnotesize\quad(type consideration) }$
$\displaystyle=$ $\displaystyle 2\text{\rm
Re}\left(\sqrt{-1}\;G^{WP}_{\alpha{\overline{\beta}}}\frac{\partial
s^{\alpha}}{\partial\tau(\delta)}\frac{\partial
s^{\overline{\beta}}}{\partial\tau(\gamma)}\right)$ $\displaystyle=$
$\displaystyle\sqrt{-1}\;G^{WP}_{\alpha{\overline{\beta}}}\left(\frac{\partial
s^{\alpha}}{\partial\tau(\delta)}\frac{\partial
s^{\overline{\beta}}}{\partial\tau(\gamma)}-\frac{\partial
s^{\alpha}}{\partial\tau(\gamma)}\frac{\partial
s^{\overline{\beta}}}{\partial\tau(\delta)}\right).$

∎

##  8\. Weighted punctured Riemann surfaces and conical metrics

Let $X$ be a compact Riemann surface of genus $p$ and ${\bf
a}=\sum_{i=1}^{n}a_{i}p_{i}$ be an ${\mathbb{R}}$-divisor, with $0<a_{i}\leq
1$ for all $i$. Then $(X,\bf a)$ is called a weighted punctured Riemann
surface. We always assume that the degree of the ${\mathbb{R}}$-divisor
$K_{(X,{\bf a})}:=K_{X}+{\bf a}$, is positive. It was shown by Hulin, McOwen
and Troyanov [H-T, McO, T] that such surfaces possess unique hyperbolic
conical metrics. These are characterized as follows:

  * (i)

For $a_{j}=1$ the metric $g$ satisfies a Poincaré growth condition at $p_{j}$,
i.e. $g=\frac{\rho(z)}{|z|^{2}\log^{2}(|z|^{2})}|dz|^{2}$ in some local
holomorphic coordinate system, where $p_{j}$ corresponds to $z=0$, and $\rho$
is a continuous positive function.

  * (ii)

For $0<a_{j}<1$ the metric is of the form
$g=\frac{\rho(z)}{|z|^{2a_{j}}}|dz|^{2}$, where $\rho$ is continuous.

In a holomorphic family, in particular regarding Teichmüller and moduli
spaces, punctures are labeled. The complex structure of the resulting
Teichmüller space ${\mathcal{T}}_{\bf a}$ of weighted punctured surfaces (with
weights being fixed) can be identified with ${\mathcal{T}}_{p,n}$. However,
weights give rise to a hierarchy of compactifications of the corresponding
moduli spaces, which were extensively investigated by Hassett [HA].

On the other hand, hyperbolic conical metrics induce different Kähler
structures on Teichmüller and moduli spaces. Generalized Weil-Petersson
metrics were introduced and studied in [S-T1, S-T2].

From the viewpoint of hyperbolic geometry, the Teichmüller space of weighted
punctured Riemann surfaces were studied by Zhang in [ZH]. Under the seemingly
necessary assumption that the weights are between $1/2$ and $1$, Zhang showed
the existence of Fenchel-Nielsen coordinates so that the Fenchel-Nielsen
symplectic form $\omega^{FN}$ becomes meaningful. The above range of weights
contains the interesting weights of the form $1-1/k$ that arise from finite
quotients. Any pants decomposition of a Riemann surface with conical
singularities with the above restriction avoids the punctures. So Theorems 1.1
and 1.2 are still valid.

Concerning the notion of harmonic Beltrami differentials, Hilbert space theory
is not available in general. Instead we use equation (13) as a definition for
the space $H^{1}(X,\bf a)$ of such differentials on a weighted punctured
Riemann surface $(X,\bf a)$. In [S-T1] the existence of a duality

(21) $H^{1}(X,{\bf a})\times
H^{0}(X,{\mathcal{O}}_{X}(2K_{X}+D))\to\mathbb{C},$

was shown, where $D=\sum_{i}p_{i}$, is the underlying divisor to $\bf a$ with
weights one. Also (12) holds. The $L^{2}$ inner product on $H^{1}(X,{\bf a})$
induced by the conical hyperbolic metric gives rise to a generalized Weil-
Petersson metric, which is of Kähler type [S-T1, S-T2]. The statement and
proof of the Theorem 1.3 can be carried over litterally to this case. So all
necessary conditions for the coincidence of $\omega^{FN}$ and $\omega^{WP}$
are satisfied for weighted punctured Riemann surfaces.

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