# Iterated residue, toric forms and Witten genus

Fei Han, Hao Li and Zhi Lü

###### Abstract.

We introduce the notion of iterated residue to study generalized Bott
manifolds. When applying the iterated residues to compute the Borisov-Gunnells
toric form and the Witten genus of certain toric varieties as well as complete
intersections, we obtain interesting vanishing results and some theta function
identities, one of which is a twisted version of a classical Rogers–Ramanujan
type formula.

##  1\. Introduction

The residue of a meromorphic function $f$ at an isolated singularity $a$,
often denoted by $\mathrm{Res}_{a}\ f$, is the coefficient $a_{-1}$ of its
Laurent series expansion at $a$. According to the Cauchy integral theorem, we
have

$\mathrm{Res}_{a}\ f=\frac{1}{2\pi i}\oint_{\gamma}f(z)dz,$

where $\gamma$ is a circle small enough around $a$ in a counterclockwise
manner. This residue has an interesting simple topological application on
complex projective spaces. Actually, as
$H^{*}(\mathbb{C}P^{n},\mathbb{C})\cong\mathbb{C}[z]/\langle z^{n+1}\rangle$,
for any cohomology class $g$ in $H^{*}(\mathbb{C}P^{n},{\mathbb{C}})$, we have

$\langle g,\,[\mathbb{C}P^{n}]\rangle=\mathrm{Res}_{0}\
\frac{g}{z^{n+1}}=\frac{1}{2\pi i}\oint_{\gamma}\frac{g}{z^{n+1}}dz.$

For the case of multiple complex variables, there exists similar story. Let
$U$ be the ball $\\{\textbf{z}\in\mathbb{C}^{n}:|\textbf{z}|<\epsilon\\}$ and
$f_{1},\cdots,f_{n}\in\mathcal{O}(\overline{U})$ be holomorphic functions in a
neighborhood of the closure $\overline{U}$. Assume $f_{i}$’s have the origin
as the isolated common zero and their Jacobian determinant is nonzero at the
origin. The residue of meromorphic $n$-form

$\omega=\frac{g(\textbf{z})dz_{1}\wedge\cdots\wedge
dz_{n}}{f_{1}(\textbf{z})\cdots f_{n}(\textbf{z})}\ \ \
(g\in\mathcal{O}(\overline{U}))$

is defined ([11]) by

(1.1) $\mathrm{Res}_{\textbf{0}}\ \omega=\frac{1}{(2\pi
i)^{n}}\int_{\Gamma}\omega,$

where $\Gamma$ is the real $n$-cycle defined by
$\Gamma=\\{\textbf{z}:|f_{i}(\textbf{z})|<\epsilon\\}$. Multiple residue also
has nice topological applications. Let $X$ be an oriented closed manifold such
that $H^{*}(X,\mathbb{C})\cong\mathbb{C}[z_{1},\cdots,z_{n}]/\langle
f_{1},\cdots,f_{n}\rangle$. Then for any cohomology class $g$ in
$H^{*}(X,\mathbb{C})$, one has

(1.2) $\langle g,\,[X]\rangle=\mathrm{Res}_{\textbf{0}}\
\frac{g(\textbf{z})dz_{1}\wedge\cdots\wedge dz_{n}}{f_{1}(\textbf{z})\cdots
f_{n}(\textbf{z})}.$

Typical examples of such $X$ are Grassmannian manifolds (where the ideal
arises from the Landau-Ginzburg potential, c.f. [21]).

Nevertheless when $\\{f_{i}\\}$’s are degenerated, namely their Jacobian
determinant vanishes, the form $\omega$ does not admit a well defined residue
as (1.1). In this paper, we look at the “iterated residue” (Definition 3)
under this degenerate situation,

$\mathrm{Res}_{0}\left\\{\cdots\mathrm{Res}_{0}\left\\{\frac{g(\textbf{z})}{f_{1}(\textbf{z})\cdots
f_{n}(\textbf{z})}dz_{1}\right\\}\cdots\right\\}dz_{n},$

and apply it to compute the Borisov-Gunnells toric form [4] and the Witten
genus.

######  Remark 1.

“Iterated residue” is similar to iterated integral in multiple variable
calculus and its value usually depends on the order of variables. The
“iterated residue” coincides with usual residue when $\\{f_{i}\\}$’s are non-
degenerated.

Our motivation for looking at the iterated residue arises from generalized
Bott manifolds, which are analogous to the relation between the multiple
residue (1.1) and Grassmannians.

A generalized Bott manifold $B_{n}$ (the total spaces of iterated projective
bundle over projective space, which will be reviewed in Section 2) has
cohomology ring

$H^{*}(B_{n})\cong{\mathbb{Z}}[u_{1},\cdots,u_{n}]/\langle
f_{i}(u_{1},\cdots,u_{n}):\ i=1,\cdots,n\rangle,$

where $f_{i}(u_{1},\cdots,u_{n})=u_{i}\prod_{j=1}^{n_{i}}(u_{i}+x_{ij})$ with
$\\{x_{ij}\\}$ being the formal Chern roots. We will prove that

######  Theorem 1.1 (Theorem 2.3).

For any top cohomology class $g$, one has

$\langle
g,\,[B_{n}]\rangle=\mathrm{Res}_{0}\left\\{\cdots\mathrm{Res}_{0}\left\\{\frac{g(\textbf{u})}{f_{1}(\textbf{u})\cdots
f_{n}(\textbf{u})}du_{1}\right\\}\cdots\right\\}du_{n},$

where the order of $u_{i}$ coincides with their position in the generalized
Bott tower.

######  Remark 2.

When the iterated projective bundles are trivial, generalized Bott manifolds
$B_{n}$ are simply products of projective spaces, and iterated residue reduces
to multiple residue.

For the genera of complete intersections in generalized Bott manifolds, we
have

######  Theorem 1.2 (Theorem 2.4).

For a generalized Bott manifold $B_{n}$

$B_{n}\overset{p_{n}}{\longrightarrow}B_{n-1}\overset{p_{n-1}}{\longrightarrow}\cdots\longrightarrow
B_{2}\overset{p_{2}}{\longrightarrow}B_{1}\longrightarrow\text{pt},$

let $-u_{k}$ be the first Chern class of the tautological line bundle over
$B_{k}={\mathbb{C}}P(\xi_{k}\oplus\underline{{\mathbb{C}}})$ and
$\\{x_{k1},x_{k2},\cdots,x_{kn_{k}}\\}$ the Chern roots of $\xi_{k}$. Denote
by $X$ the submanifold of $B_{n}$ Poincaré dual to
$a_{1}u_{1}+\cdots+a_{n}u_{n}\in H^{2}(B_{n};{\mathbb{Z}})$. For any genus
$\psi$ with the characteristic power series $Q(x)=x/f(x)$, we have

$\psi(X)=\mathrm{Res}_{0}\left\\{\cdots\left\\{\mathrm{Res}_{0}\frac{f(a_{1}u_{1}+\cdots+a_{n}u_{n})}{f(u_{1})^{n_{1}+1}}\cdot\prod_{i=1}^{n}\frac{1}{f(u_{i})}\cdot\prod_{j=1}^{n_{i}}\frac{1}{f(u_{i}+x_{ij})}du_{1}\right\\}\cdots\right\\}du_{n}.$

We will present two applications of above formula in this paper. One concerns
the Borisov-Gunnells toric form ([4]) and the other one concerns the Witten
genus [21].

Toric varieties are very important objects in both algebraic geometry and
geometric topology. Generalized Bott manifolds are examples of toric
varieties. In [4], given a toric variery $X$ and a “degree function”
$\mathrm{deg}$, Borisov and Gunnells introduced a function
$f_{N,\mathrm{deg}}$ on the upper half plane ${\mathbb{H}}$, called toric
form, and proved that it is a modular form when $\mathrm{deg}$ satisfies
certain natural condition. In Section 3, applying our iterated residue formula
to the toric forms of some concrete generalized Bott manifolds (Hirzebruch
surfaces and certain 4-folds), we obtain interesting theta function
identities. Let $\theta(z,\tau)$ denote the classical Jacobi theta function
(c.f. [7])

$\theta(z,\tau)=2q^{1/8}\sin(\pi z)\prod_{j=1}^{\infty}[(1-q^{j})(1-e^{2\pi
iz}q^{j})(1-e^{-2\pi iz}q^{j})].$

######  Theorem 1.3 (Proposition (3.1)).

When $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\notin\mathbb{Z}$ are complex
numbers, one has

(1.3) $\begin{split}&\sum_{a,b\in\mathbb{Z}}\frac{(1-e^{2\pi
i(\alpha_{3}+\alpha_{4})})(1-e^{2\pi
i(\alpha_{1}+\alpha_{2})}q^{ka})}{(1-e^{2\pi i\alpha_{4}}q^{a})(1-e^{2\pi
i\alpha_{1}}q^{b})(1-e^{2\pi i\alpha_{3}}q^{-a})(1-e^{2\pi
i\alpha_{2}}q^{ka-b})}\\\ =&\frac{1}{(2\pi
i)^{2}}\left\\{(\frac{\theta^{\prime}(-\alpha_{3})}{\theta(-\alpha_{3})}+\frac{\theta^{\prime}(-\alpha_{4})}{\theta(-\alpha_{4})})(\frac{\theta^{\prime}(-\alpha_{1})}{\theta(-\alpha_{1})}+\frac{\theta^{\prime}(-\alpha_{2})}{\theta(-\alpha_{2})})+\frac{k}{2}(\frac{\theta^{\prime\prime}(-\alpha_{3})}{\theta(-\alpha_{3})}-\frac{\theta^{\prime\prime}(-\alpha_{4})}{\theta(-\alpha_{4})})\right\\}.\end{split}$

######  Theorem 1.4 (Proposition (3.2)).

When $j,k\in\mathbb{Z}$, one has

(1.4) $\begin{split}&(2\pi
i)^{4}\cdot\sum_{a,b,c,d\in\mathbb{Z}}\frac{2(1+q^{jc+kd})}{(1+q^{a})(1+q^{b})(1+q^{c})(1+q^{d})(1+q^{jc+kd-
a-b})(1+q^{-c-d})}\\\
=&\left\\{(\frac{3\theta^{\prime\prime}(\frac{1}{2})}{2\theta(\frac{1}{2})}-\frac{\theta^{(3)}(0)}{2\theta^{\prime}(0)})^{2}+(j^{2}+k^{2}-jk)(\frac{\theta^{\prime\prime
2}(\frac{1}{2})}{4\theta^{2}(\frac{1}{2})}+\frac{5\theta^{(4)}(\frac{1}{2})}{24\theta(\frac{1}{2})}-\frac{7\theta^{(3)}(0)\theta^{\prime\prime}(\frac{1}{2})}{12\theta^{\prime}(0)\theta(\frac{1}{2})}-\frac{\theta^{(5)}(0)}{24\theta^{\prime}(0)}+\frac{(\theta^{(3)}(0))^{2}}{6\theta^{\prime
2}(0)})\right\\}.\end{split}$

In the special case $j=k=0$, this formula actually gives the classical
Rogers–Ramanujan type formula (c.f. [2, 3])

(1.5) $\sum_{m,n\geq
1}\frac{q^{m+n}}{(1+q^{m})(1+q^{n})(1+q^{m+n})}=\sum_{r\geq
1}\sigma_{1}(r)q^{2r},$

which was obtained by Borisov and Libgober using toric form [5]. In Section
3.1, we will review the proof. Our formula (1.4) can be viewed as a twisted
version of the classical formula (1.5), see Section 3.3 for details.

Concerning toric forms, using iterated residue formula, we are also able to
obtain the following vanishing result.

######  Theorem 1.5 (Theorem 3.3).

Consider toric variety $V:={\mathbb{C}}P(\eta^{\otimes
i_{1}}\oplus\eta^{\otimes i_{2}}\oplus\eta^{\otimes
i_{3}}\oplus\underline{\mathbb{C}})$ over ${\mathbb{C}}P^{n_{1}}$, where
$\eta$ denote the tautological bundle of ${\mathbb{C}}P^{n_{1}}$. Assume
$(i_{1},i_{2},i_{3})$ are coprime, $\ (i_{1},\ i_{1}-i_{2},\ i_{1}-i_{3}),\
(i_{2},\ i_{2}-i_{1},\ i_{2}-i_{3}),\ (i_{3},\ i_{3}-i_{1},i\ _{3}-i_{2})$ are
coprime respectively, $\sum_{j=0}^{3}\alpha_{n_{1}+2+j}\in\mathbb{Z}$,
$\sum_{i=1}^{n_{1}+1}\alpha_{i}-\sum_{j=1}^{3}i_{j}\alpha_{n_{1}+2+j}\in\mathbb{Z}$,
then the toric form $f_{N,\deg}(q)=0$.

######  Remark 3.

The coprime condition seems to be a bit strict. We will provide an example to
explain it. Let $(i_{1},i_{2},i_{3})=(1,3,4)$, we can check that $(1,3,4),\
(1,2,3),\ (3,2,1)$ and $(4,3,1)$ are coprime respectively. It is possible to
transfer iterated residue into residues at simple poles by residue theorem
when coprime condition hold.

Another application is about Witten genus, which plays an important role in
index theory. Let $M$ be a $4m$ dimensional compact oriented smooth manifold.
Let $\\{\pm 2\pi\sqrt{-1}z_{j},1\leq j\leq 2m\\}$ denote the formal Chern
roots of $T_{\mathbb{C}}M$, the complexification of the tangent vector bundle
$TM$ of $M$. Then the famous Witten genus of $M$ can be written as (c.f. [17])

$\varphi_{W}(M)=\left\langle\left(\prod_{j=1}^{2m}z_{j}\frac{\theta^{\prime}(0,\tau)}{\theta(z_{j},\tau)}\right),[M]\right\rangle\in\mathbb{Q}[[q]],$

with $\tau\in\mathbb{H}$, the upper half-plane, and $q=e^{\pi\sqrt{-1}\tau}$.
The Witten genus was first introduced in [21] and can be viewed as the loop
space analogue of the $\widehat{A}$-genus. It can be expressed as a
$q$-deformed $\widehat{A}$-genus as

$\varphi_{W}(M)=\left\langle\widehat{A}(TM)\mathrm{ch}\left(\Theta\left(T_{\mathbb{C}}M\right)\right),[M]\right\rangle,$

where

$\Theta(T_{\mathbb{C}}M)=\overset{\infty}{\underset{n=1}{\otimes}}S_{q^{2n}}(\widetilde{T_{\mathbb{C}}M}),\
\ {\rm with}\ \
\widetilde{T_{\mathbb{C}}M}=T_{\mathbb{C}}M-{\mathbb{C}}^{4m},$

is the Witten bundle defined in [21]. When the manifold $M$ is Spin, according
to the Atiyah-Singer index theorem [1], the Witten genus can be expressed
analytically as the index of twisted Dirac operators,
$\varphi_{W}(M)=\mathrm{ind}(D\otimes\Theta\left(T_{\mathbb{C}}M\right))\in\mathbb{Z}[[q]]$,
where $D$ is the Atiyah-Singer Spin Dirac operator on $M$ (c.f. [13]).
Moreover, if $M$ is String, i.e. $\frac{1}{2}p_{1}(TM)=0$, or even weaker, if
$M$ is Spin and the first rational Pontryagin class of $M$ vanishes, then
$\varphi_{W}(M)$ is a modular form of weight $2k$ over $SL(2,\mathbb{Z})$ with
integral Fourier expansion (c.f. [22]). The homotopy theoretical refinements
of the Witten genus on String manifolds leads to the theory of tmf
(topological modular form) developed by Hopkins and Miller [14]. The String
condition is the orientablity condition for this generalized cohomology
theory.

The Lichnerowicz theorem[16] asserts that if a closed spin manifold carries a
Riemannian metric of positive scalar curvature, then its $\widehat{A}$ genus
vanishes. Along this line, Stolz conjectured [19] that the Witten genus of a
string manifold carrying positive Ricci curvature metric vanishes. There are
two kinds of vanishing results to support the Stolz’s conjecture. One is the
theorem asserting that every string manifold carrying a nontrivial action of a
semi-simple Lie group $G$ has vanishing Witten genus ([10], [18]). The other
is the Landweber-Stong vanishing theorem (c.f. page 89 in [13]) asserting that
a string complete intersection in complex projective spaces has vanishing
Witten genus. The proof uses the calculation of residues. The Landweber-Stong
type vanishing results were also obtained for the following objects: string
complete intersections in products of complex projective spaces ([8], [9]),
string complete intersections in products of Grassmannians and flag manifolds
([25, 26]).

In this paper, by applying the iterated residue, we can prove a Landweber-
Stong type vanishing theorem for the Witten genus complete intersections in
two-staged generalized Bott manifolds.

######  Theorem 1.6 (Theorem 4.1).

For a string complete intersection $H^{\bf I}_{n_{1},\
3}(d_{1},d_{2};d_{3},d_{4})$, if ${\bf I}=(i,\ j,\ k)$ are coprime, $\ (i,\
i-j,\ i-k),\ (j,\ j-i,\ j-k),\ (k,\ k-i,\ k-j)$ are coprime respectively, then
the Witten genus

$\varphi_{W}(H^{\bf I}_{n_{1},\ 3}(d_{1},d_{2};d_{3},d_{4}))=0.$

Furthermore, it is also possible to calculate the ${\mathrm{mod}}$ $2$ Witten
genus of generalized Bott manifolds by making use of Rokhlin congruence
formula in [23, 24]. We will study this in a forthcoming project.

The paper is organised as follow. In Section 2, we introduce the concept of
“iterated residue” and give an explicit expression of the genus of complete
intersections in generalized Bott manifolds. In Section 3, we apply “iterated
residue” to toric forms and get interesting theta function identities. In
Section 4, we will use “iterated residue” to discuss when Witten genus of
complete intersections in generalized Bott manifolds vanishes and give the
proof of Theorem 1.6.

##  2\. Iterated residue in generalized Bott manifolds

In the study of symmetric spaces, Bott and Samelson [6] first introduced a
family of toric manifolds obtained as the total spaces of iterated bundles
over ${\mathbb{C}}P^{1}$ with fibre ${\mathbb{C}}P^{1}$. Grossberg and Karshon
[12] showed that these manifolds form an important family of smooth projective
toric varieties, and called them _Bott manifolds_.

######  Definition 1.

A generalized Bott tower of height $n$ is a tower of projective bundles

$B_{n}\overset{p_{n}}{\longrightarrow}B_{n-1}\overset{p_{n-1}}{\longrightarrow}\cdots\longrightarrow
B_{2}\overset{p_{2}}{\longrightarrow}B_{1}\longrightarrow\text{pt},$

where $B_{1}={\mathbb{C}}P^{n_{1}}$ and each $B_{k}$ is the complex
projectivisation of sum of $n_{k}$ complex line bundles and one trivial line
bundle over $B_{k-1}$. The fibre of the bundle $p_{k}:B_{k}\longrightarrow
B_{k-1}$ is ${\mathbb{C}}P^{n_{k}}$.

The last stage $B_{n}$ in a generalized Bott tower is called generalized Bott
manifold.

The topology of generalized Bott manifolds is very clear. Its cohomology ring
relies on the following result:

######  Theorem 2.1.

(c.f. [20, Chapter V]) Let $\xi$ be a complex $n$-dimensional vector bundle
over a finite cell complex $X$, the complex projectivisation of $\xi$ is
${\mathbb{C}}P(\xi)$. Let $-u\in H^{2}({\mathbb{C}}P(\xi))$ be the first Chern
class of the tautological line bundle over ${\mathbb{C}}P(\xi)$.
$H^{*}({\mathbb{C}}P(\xi);\mathbb{Z})$ is the quotient of polynomial ring
$H^{*}(X)[u]$ by single relation

$u^{n}+c_{1}(\xi)u^{n-1}+\cdots+c_{n}(\xi)=0.$

######  Corollary 1.

Let $-u_{k}$ be the first Chern class of tautological line bundle $\eta_{k}$
over $B_{k}={\mathbb{C}}P(\xi_{k}\oplus\underline{{\mathbb{C}}})$, suppose the
formal Chern roots of $\xi_{k}$ are $\\{x_{k1},x_{k2},\cdots,x_{kn_{k}}\\}$.
Then

$H^{*}(B_{n})\cong{\mathbb{Z}}[u_{1},\cdots,u_{n}]/\langle
f_{i}(u_{1},\cdots,u_{n}):\ i=1,\cdots,n\rangle,$

where $f_{i}(u_{1},\cdots,u_{n})=u_{i}\prod_{j=1}^{n_{i}}(u_{i}+x_{ij})$.

Moreover, its tangential bundle is clear,

$TB_{k+1}\oplus\underline{{\mathbb{C}}}\cong
p^{*}(TB_{k})\oplus(\bar{\eta}_{k}\otimes
p^{*}(\xi_{k}\oplus\underline{{\mathbb{C}}})).$

The generalized Bott manifolds provide a rich family of toric varieties on
which we can calculate all kinds of characteristic numbers and study their
geometry and topology. In [15], the authors introduced the concept of twisted
Milnor hypersurfaces, which is a generalization of Milnor hypersurfaces
(Milnor hypersurfaces can be used as the representatives of unitary bordism).
Consider the projective bundle over $\mathbb{C}P^{n_{1}}$ with fiber
$\mathbb{C}P^{n_{2}}$, i.e.

$V=\mathbb{C}P(\eta^{\otimes i_{1}}\oplus\cdots\oplus\eta^{\otimes
i_{n_{2}}}\oplus\underline{\mathbb{C}})\rightarrow\mathbb{C}P^{n_{1}},$

where $\eta$ is the tautological line bundle over $\mathbb{C}P^{n_{1}}$ and
$\underline{\mathbb{C}}$ is the trivial line bundle.

Let $\gamma$ be the vertical tautological line bundle over $V$,
$u=c_{1}(\overline{\eta}),v=c_{1}(\overline{\gamma})\in
H^{2}(V;{\mathbb{Z}})$. Denote $\mathbf{I}=(i_{1},\cdots,i_{n_{2}})$ be the
index.

######  Definition 2.

We call the smooth hypersurface Poincaré dual to $d_{1}u+d_{2}v$ in $V$
twisted Milnor hypersurface, denoted by $H_{n_{1},n_{2}}^{\bf
I}(d_{1},d_{2})$.

######  Remark 4.

We can also consider hypersurfaces or complete intersections on generalized
Bott manifolds, such as $H_{n_{1},n_{2}}^{\bf I}(d_{1},d_{2};d_{3},d_{4})$
denote the complete intersection Poincaré dual to
$(d_{1}u+d_{2}v)\cdot(d_{3}u+d_{4}v)$ in $V$.

As we all know, generalized Bott manifolds are toric varieties, but their
complete intersections are not necessarily algebraic.

Denote $X$ be the submanifold of $B_{n}$ Poincaré dual to
$a_{1}u_{1}+\cdots+a_{n}u_{n}\in H^{2}(B_{n};{\mathbb{Z}})$. Suppose $\nu$
denotes the normal bundle of inclusion $i:X\hookrightarrow B_{n}$. Then
$c_{1}(\nu)=i^{*}(a_{1}u_{1}+\cdots+a_{n}u_{n})$, while $i^{*}(TB_{n})\cong
TX\oplus\nu$.

For any genus $\psi$ with the charateristic power series $Q(x)=x/f(x)$, we
have

(2.1) $\psi(X)=\langle(\frac{u_{1}}{f(u_{1})})^{n_{1}+1}\cdot
f(a_{1}u_{1}+\cdots+a_{n}u_{n})\cdot\prod_{i=1}^{n}\frac{u_{i}}{f(u_{i})}\cdot\prod_{j=1}^{n_{i}}\frac{u_{i}+x_{ij}}{f(u_{i}+x_{ij})},[B_{n}]\rangle.$

If $B_{n}$ is a product of projective spaces, $\psi(X)$ can be simplified into
a very neat expression. As to general $B_{n}$, the idea $f_{i}$’s in
$H^{*}(B_{n};{\mathbb{Z}})$ will be the biggest obstacle in calculating genus,
so we have to cope with the relation between $u_{i}$’s carefully.

Motivated by the idea of Witten [21], we try to reduce $\psi(X)$ to multiple
residue. Recall the global residue theorem

######  Theorem 2.2.

$($Global Residue Theorem [11, Chapter 5]$)$ Let $M$ be a compact complex
$n$-manifold, $D=D_{1}+\cdots+D_{n}$ be a divisor on $M$ such that
$D_{1},\cdots,D_{n}$ are effective divisors on $M$ and the intersection
$D_{1}\cap\cdots\cap D_{n}$ is discrete, hence finite-set of points in $M$.
Then for any $\omega\in H^{0}(M,\Omega^{n}(D))$,

$\sum_{P\in\\{D_{1}\cap\cdots\cap D_{n}\\}}\mathrm{Res}_{P}\ \omega=0.$

Global Residue Theorem does not apply to generalized Bott manifolds, since the
ideas $f_{i}$’s are not necessarily non-degenerated, thus multiple residue at
point $(0,\cdots,0)$ can not be well defined.

To overcome this obstacle, we introduce the concept “iterated residue”.

######  Definition 3.

Let $U$ be the ball $\\{\textbf{z}\in\mathbb{C}^{n}:|\textbf{z}|<\epsilon\\}$
and $f_{1},\cdots,f_{m}\in\mathcal{O}(\overline{U})$ be functions holomorphic
in a neighborhood of the closure $\overline{U}$ of $U$. Then “iterated
residue” of a meromorphic $n$-form

$\omega=\frac{g(\textbf{z})dz_{1}\wedge\cdots\wedge
dz_{n}}{f_{1}(\textbf{z})\cdots f_{m}(\textbf{z})}\ \ \
(g\in\mathcal{O}(\overline{U}))$

is defined by

$\mathrm{Res}_{0}\left\\{\cdots\mathrm{Res}_{0}\left\\{\frac{g(\textbf{z})}{f_{1}(\textbf{z})\cdots
f_{m}(\textbf{z})}dz_{1}\right\\}\cdots\right\\}dz_{n}.$

######  Remark 5.

“Iterated residue” is similar to iterated integral in multiple calculus, its
value depends on the order of variables. Of course, “iterated residue”
coincides with usual residue when $f_{i}(z)$’s are non-degenerated.

“Iterated residue” admits a topological interpretation via characteristics of
generalized Bott manifolds.

######  Theorem 2.3.

For any $F\in H^{n_{1}+\cdots+n_{n}}(B_{n};{\mathbb{Z}})$, we have

$\langle\ F,\ [B_{n}]\
\rangle=\mathrm{Res}_{0}\left\\{\mathrm{Res}_{0}\cdots\left\\{\mathrm{Res}_{0}\frac{F}{\prod_{j=1}^{n_{i}}f_{j}(u_{1},\cdots,u_{n})}du_{1}\right\\}\cdots
du_{n-1}\right\\}du_{n},$

where the order of $u_{i}$ coincides with their position in the generalized
Bott tower.

###### Proof.

Since $\langle F,[B_{n}]\rangle$ and
$\mathrm{Res}_{0}\left\\{\mathrm{Res}_{0}\cdots\left\\{\mathrm{Res}_{0}\frac{F}{\prod_{j=1}^{n_{i}}f_{j}(u_{1},\cdots,u_{n})}du_{1}\right\\}\cdots
du_{n-1}\right\\}du_{n}$ are both linear forms,
$H^{n_{1}+\cdots+n_{n}}(B_{n};{\mathbb{Z}})\cong{\mathbb{Z}}\langle
u_{1}^{n_{1}}u_{2}^{n_{2}}\cdots u_{n}^{n_{n}}\rangle$, it remains to prove

$\langle u_{1}^{n_{1}}u_{2}^{n_{2}}\cdots
u_{n}^{n_{n}},[B_{n}]\rangle=\mathrm{Res}_{0}\left\\{\mathrm{Res}_{0}\cdots\left\\{\mathrm{Res}_{0}\frac{u_{1}^{n_{1}}u_{2}^{n_{2}}\cdots
u_{n}^{n_{n}}}{\prod_{j=1}^{n_{i}}f_{j}(u_{1},\cdots,u_{n})}du_{1}\right\\}\cdots
du_{n-1}\right\\}du_{n}.$

Obviously, $\langle u_{1}^{n_{1}}u_{2}^{n_{2}}\cdots
u_{n}^{n_{n}},[B_{n}]\rangle=1$, while

$\displaystyle\mathrm{Res}_{0}\left\\{\mathrm{Res}_{0}\cdots\left\\{\mathrm{Res}_{0}\frac{u_{1}^{n_{1}}u_{2}^{n_{2}}\cdots
u_{n}^{n_{n}}}{\prod_{j=1}^{n_{i}}f_{j}(u_{1},\cdots,u_{n})}du_{1}\right\\}\cdots
du_{n-1}\right\\}du_{n}$ $\displaystyle=$
$\displaystyle\mathrm{Res}_{0}\left\\{\cdots\left\\{\mathrm{Res}_{0}\frac{u_{2}^{n_{2}}\cdots
u_{n}^{n_{n}}}{\prod_{j=2}^{n_{i}}f_{j}(u_{2},\cdots,u_{n})}du_{2}\right\\}\cdots\right\\}du_{n}$
$\displaystyle=$ $\displaystyle 1.$

$\square$

For a generalized Bott manifold $B_{n}$

$B_{n}\overset{p_{n}}{\longrightarrow}B_{n-1}\overset{p_{n-1}}{\longrightarrow}\cdots\longrightarrow
B_{2}\overset{p_{2}}{\longrightarrow}B_{1}\longrightarrow\text{pt}.$

Let $-u_{k}$ be the first Chern class of the tautological line bundle over
$B_{k}={\mathbb{C}}P(\xi_{k}\oplus\underline{{\mathbb{C}}})$ over $B_{k-1}$,
the Chern roots of $\xi_{k}$ be $\\{x_{k1},x_{k2},\cdots,x_{kn_{k}}\\}$.
Denote $X$ be the submanifold of $B_{n}$ Poincaré dual to
$a_{1}u_{1}+\cdots+a_{n}u_{n}\in H^{2}(B_{n};{\mathbb{Z}})$.

Applying Theorem 2.3 to formula (2.1), we get our main result

######  Theorem 2.4.

For any genus $\psi$ with the charateristic power series $Q(x)=x/f(x)$, we
have

(2.2)
$\psi(X)=\mathrm{Res}_{0}\left\\{\cdots\left\\{\mathrm{Res}_{0}\frac{f(a_{1}u_{1}+\cdots+a_{n}u_{n})}{f(u_{1})^{n_{1}+1}}\cdot\prod_{i=1}^{n}\frac{1}{f(u_{i})}\cdot\prod_{j=1}^{n_{i}}\frac{1}{f(u_{i}+x_{ij})}du_{1}\right\\}\cdots\right\\}du_{n}.$

Formula (2.2) kills the ideals in $H^{*}(B_{n};\mathbb{Z})$ and only $f$ is
left. If power series $f$ admits good property, it is possible to give
explicit formula of $\psi(X)$.

##  3\. Borisov-Gunnellls Toric forms: theta functions identities and
vanishing results

Toric variety is a very important object in both algebraic geometry and
geometric topology. Borisov and Libgober[5] gave the explicit formula of
elliptic genus of a toric variety by its combinatorial data, and showed that
elliptic genera of a Calabi-Yau hypersurface in a toric variety and its mirror
coincide up to sign. In [4], Borisov and Gunnells defined algebraic _toric
form_ which is motivated from the expression of normalized ellptic genus of
toric variety[5]. Let $N\in\mathbb{R}^{r}$ be a lattice, $M$ be its dual
lattice. For a complete rational polyhedral fan $\Sigma\subset
N\otimes\mathbb{R}$. A _degree function_ $\deg:N\longrightarrow\mathbb{C}$ is
a piecewise linear function on the cones of $\Sigma$. For every cone
$C\in\Sigma$, they defined a map

$f_{C}:\mathbb{H}\times M_{\mathbb{C}}\longrightarrow\mathbb{C}$

as follows. Write $q=e^{2\pi i\tau}$, $\tau\in\mathbb{H}$, the upper
halfplane. If $m\in M_{\mathbb{C}}$ satisfies

$m\cdot(C\setminus\\{0\\})>0,$

for all $\tau$ with sufficiently large imaginart part, they set

$f_{C}(q,m)\coloneqq\sum_{n\in C\cap N}q^{m\cdot n}e^{2\pi i\deg(n)}.$

The _toric form_ associated to $(N,\deg)$ is the function
$f_{N,\deg}:\mathbb{H}\longrightarrow\mathbb{C}$ defined by

$f_{N,\deg}(q)\coloneqq\sum_{m\in M}(\sum_{C\in\Sigma}(-1)^{codim\
C}a.c.(\sum_{n\in C}q^{m\cdot n}e^{2\pi i\deg(n)})),$

here “$a.c.$” denotes analytic continuation of $f_{C}$.

The definition of $f_{N,\deg}$ is well-defined. More precisely, there exists
$\epsilon>0$ such that the sum over $M$ converges absolutely and uniformly for
all $|q|<\epsilon$.

######  Remark 6.

Suppose that $\deg(d)=1/2$ for all generators $d$ of one-dimensional cones of
$\Sigma$, and toric variety $X$ associated to $\Sigma$ is nonsingular. Then
the function $f_{N,\deg}(q)$ is the normalized elliptic genus of $X$. This
example is the main motivation of toric forms [5].

Furthermore they proved that the toric forms are modular forms under certain
conditions on $\deg$.

######  Theorem 3.1 (Borisov, Gunnells).

Suppose $\deg$ fucntion takes values in $\frac{1}{l}\mathbb{Z}$, and $\deg$ is
not integral valued on the primitive generator of any 1-cone of $\Sigma$. Then
toric form $f_{N,\deg}(q)$ is holomorphic modular form of weight $r$ for the
congruence subgroup $\Gamma_{1}(l)$.

Borisov and Gunnells gave a topological interpretation of toric forms by
Hirzebruch-Riemann-Roch theorem. Let $\\{d_{i}\\}$ be the set of primitive
generator of 1-cone of $\Sigma$, $X$ be toric variety associated to $\Sigma$,
and for each $d_{i}$, $D_{i}\subset X$ be the corresponding toric divisor. In
the following, we abuse $D_{i}$ to mean either the divisor or its cohomology
class. Recall that

$\theta(z,\tau)=2q^{1/8}\sin(\pi z)\prod_{j=1}^{\infty}[(1-q^{j})(1-e^{2\pi
iz}q^{j})(1-e^{-2\pi iz}q^{j})].$

######  Theorem 3.2 ( Borisov, Gunnells).

Assume that the toric variety $X$ is nonsingular, and that
$\alpha_{i}\notin\mathbb{Z}$ for all primitive generator of 1-cone of
$\Sigma$. Then

$f_{N,\deg}(q)=\int_{X}\prod_{i}\frac{(D_{i}/2\pi i)\theta(D_{i}/2\pi
i-\alpha_{i})\theta^{\prime}(0)}{\theta(D_{i}/2\pi i)\theta(-\alpha_{i})}.$

Our “iterated residue” can be applied to calculate above Euler characteristic
for some generalized Bott manifolds, thus getting interesting theta function
identities regarding to toric forms.

###  3.1. Projective plane $\mathbb{C}P^{2}$

Let $N=\mathbb{Z}^{2}$, and $\Sigma$ be the fan in Figure 1. Then the
corresponding toric variety is projective plane $\mathbb{C}P^{2}$.

$\textbf{e}_{1}$$\textbf{e}_{2}$$-\textbf{e}_{1}-\textbf{e}_{2}$Figure 1

Assume that $\deg$ takes $\alpha,\beta,\gamma$ on the generators
$\textbf{e}_{1},\ \textbf{e}_{2},\ -\textbf{e}_{1}-\textbf{e}_{2}$
respectively. Then the toric form is

$\displaystyle f_{N,\deg}(q)$ $\displaystyle=$
$\displaystyle\sum_{a,b\in\mathbb{Z}}\frac{1}{(1-e^{2\pi
i\alpha}q^{a})(1-e^{2\pi i\beta}q^{b})}+\frac{1}{(1-e^{2\pi
i\beta}q^{b})(1-e^{2\pi i\gamma}q^{-a-b})}$ $\displaystyle+\frac{1}{(1-e^{2\pi
i\alpha}q^{a})(1-e^{2\pi i\gamma}q^{-a-b})}-\frac{1}{1-e^{2\pi
i\alpha}q^{a}}-\frac{1}{1-e^{2\pi i\beta}q^{b}}-\frac{1}{1-e^{2\pi
i\gamma}q^{-a-b}}+1$ $\displaystyle=$
$\displaystyle\sum_{a,b\in\mathbb{Z}}\frac{1-e^{2\pi
i(\alpha+\beta+\gamma)}}{(1-e^{2\pi i\alpha}q^{a})(1-e^{2\pi
i\beta}q^{b})(1-e^{2\pi i\gamma}q^{-a-b})}\ .$

In the case $\alpha=\beta=\gamma=\frac{1}{2}$, Borisov and Libgober [5]
deduced the famous identity

$\sum_{m,n\geq 1}\frac{q^{m+n}}{(1+q^{m})(1+q^{n})(1+q^{m+n})}=\sum_{r\geq
1}q^{2r}\sum_{k|r}k=\sum_{r\geq 1}\sigma_{1}(r)q^{2r}.$

The deduction is the following. By Theorem 3.2,

$\displaystyle\sum_{m,n\in\mathbb{Z}}\frac{2q^{m+n}}{(1+q^{m})(1+q^{n})(1+q^{m+n})}$
$\displaystyle=\int_{X}\prod_{i}\frac{(D_{i}/2\pi i)\theta(D_{i}/2\pi
i-\frac{1}{2})\theta^{\prime}(0)}{\theta(D_{i}/2\pi i)\theta(-\frac{1}{2})}$
$\displaystyle=\frac{1}{(2\pi i)^{2}}\cdot\frac{\theta^{\prime
3}(0)}{\theta^{2}(\frac{1}{2})}\cdot\mathrm{Res}_{0}\frac{1}{\theta^{3}(u)}du$
$\displaystyle=\frac{1}{(2\pi
i)^{2}}\left\\{\frac{3\theta^{\prime\prime}(\frac{1}{2})}{2\theta(\frac{1}{2})}-\frac{\theta^{(3)}(0)}{2\theta^{\prime}(0)}\right\\}.$

Then we see

$\displaystyle LHS$ $\displaystyle=$ $\displaystyle 4\sum_{m,n\geq
1}\left\\{\frac{q^{m+n}}{(1+q^{m})(1+q^{n})(1+q^{m+n})}+\frac{q^{m+n}}{(1+q^{m})(1+q^{n})(q^{m}+q^{n})}\right\\}+4\sum_{m=1}^{\infty}\frac{q^{m}}{(1+q^{m})^{2}}+\frac{1}{4}$
$\displaystyle=$ $\displaystyle 12\sum_{m,n\geq
1}\frac{q^{m+n}}{(1+q^{m})(1+q^{n})(1+q^{m+n})}+6\sum_{m=1}^{\infty}\frac{q^{m}}{(1+q^{m})^{2}}+\frac{1}{4}.$

On the other side, we list the derivatives of theta function first.
$\theta^{\prime}(z)=2q^{1/8}\prod_{j=1}^{\infty}[(1-q^{j})(1-e^{2\pi
iz}q^{j})(1-e^{-2\pi iz}q^{j})]\cdot\bigg{\\{}\pi\cos(\pi z)-\sin(\pi
z)\sum_{j=1}^{\infty}\frac{2\pi iq^{j}(e^{2\pi iz}-e^{-2\pi iz})}{(1-e^{2\pi
iz}q^{j})(1-e^{-2\pi iz}q^{j})}\bigg{\\}}$,

$\displaystyle\theta^{\prime\prime}(z)=$ $\displaystyle
2q^{1/8}\prod_{j=1}^{\infty}[(1-q^{j})(1-e^{2\pi iz}q^{j})(1-e^{-2\pi
iz}q^{j})]\cdot\bigg{\\{}-2\pi\cos(\pi z)\sum_{j=1}^{\infty}\frac{2\pi
iq^{j}(e^{2\pi iz}-e^{-2\pi iz})}{(1-e^{2\pi iz}q^{j})(1-e^{-2\pi iz}q^{j})}$
$\displaystyle-\pi^{2}\sin(\pi z)+\sin(\pi z)(\sum_{j=1}^{\infty}\frac{2\pi
iq^{j}(e^{2\pi iz}-e^{-2\pi iz})}{(1-e^{2\pi iz}q^{j})(1-e^{-2\pi
iz}q^{j})})^{2}$ $\displaystyle-\sin(\pi z)\sum_{j=1}^{\infty}\frac{(2\pi
i)^{2}q^{j}(1-e^{2\pi iz}q^{j})(1-e^{-2\pi iz}q^{j})(e^{2\pi iz}+e^{-2\pi
iz})-(2\pi iq^{j})^{2}(e^{2\pi iz}-e^{-2\pi iz})^{2}}{(1-e^{2\pi
iz}q^{j})^{2}(1-e^{-2\pi iz}q^{j})^{2}}\bigg{\\}}.$

Then

$\displaystyle RHS$ $\displaystyle=$ $\displaystyle\frac{1}{(2\pi
i)^{2}}\left\\{\frac{3}{2}(-\pi^{2}+\sum_{j=1}^{\infty}\frac{(2\pi
i)^{2}2q^{j}}{(1+q^{j})^{2}})-\frac{1}{2}(-\pi^{2}-3\sum_{j=1}^{\infty}\frac{(2\pi
i)^{2}2q^{j}}{(1-q^{j})^{2}})\right\\}$ $\displaystyle=$
$\displaystyle\frac{1}{4}+3\sum_{j=1}^{\infty}\left\\{\frac{q^{j}}{(1+q^{j})^{2}}+\frac{q^{j}}{(1-q^{j})^{2}}\right\\}.$

Thus

$\displaystyle\sum_{m,n\geq 1}\frac{q^{m+n}}{(1+q^{m})(1+q^{n})(1+q^{m+n})}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{j=1}^{\infty}\left\\{\frac{q^{j}}{(1-q^{j})^{2}}-\frac{q^{j}}{(1+q^{j})^{2}}\right\\}$
$\displaystyle=$ $\displaystyle\sum_{j=1}^{\infty}\ \ \sum_{a>b,\ a+b\ odd}\ \
q^{j}\cdot q^{aj}\cdot q^{bj}$ $\displaystyle=$ $\displaystyle\sum_{r\geq
1}q^{2r}\sum_{k|r}k$ $\displaystyle=$ $\displaystyle\sum_{r\geq
1}\sigma_{1}(r)q^{2r}.$

###  3.2. Hirzebruch surface

Hirzebruch surface
$F_{k}=\mathbb{C}P(\underline{\mathbb{C}}\oplus\mathcal{O}(k))$ is a
$\mathbb{C}P^{1}$ bundle over $\mathbb{C}P^{1}$. Its corresponding fan is
spanned by four primitive vectors $\textbf{e}_{1},\ \textbf{e}_{2},\
-\textbf{e}_{1}+k\textbf{e}_{2},\ -\textbf{e}_{2}$, see Figure $2$.

$\textbf{e}_{1}$$\textbf{e}_{2}$$-\textbf{e}_{2}$$-\textbf{e}_{1}+k\textbf{e}_{2}$Figure
2

Assume that $\deg$ takes $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$ on the
generators $\textbf{e}_{1},\ -\textbf{e}_{1}+k\textbf{e}_{2},\
-\textbf{e}_{2},\ \textbf{e}_{2},\ $. Then the toric form is

$\displaystyle f_{N,\deg}(q)$ $\displaystyle=$
$\displaystyle\sum_{a,b\in\mathbb{Z}}\frac{1}{(1-e^{2\pi
i\alpha_{1}}q^{b})(1-e^{2\pi i\alpha_{4}}q^{a})}+\frac{1}{(1-e^{2\pi
i\alpha_{1}}q^{b})(1-e^{2\pi i\alpha_{3}}q^{-a})}$
$\displaystyle+\frac{1}{(1-e^{2\pi i\alpha_{3}}q^{-a})(1-e^{2\pi
i\alpha_{2}}q^{ka-b})}+\frac{1}{(1-e^{2\pi i\alpha_{4}}q^{a})(1-e^{2\pi
i\alpha_{2}}q^{ka-b})}$ $\displaystyle-\frac{1}{1-e^{2\pi
i\alpha_{4}}q^{a}}-\frac{1}{1-e^{2\pi i\alpha_{1}}q^{b}}-\frac{1}{1-e^{2\pi
i\alpha_{2}}q^{ka-b}}-\frac{1}{1-e^{2\pi i\alpha_{3}}q^{-a}}+1$
$\displaystyle=$ $\displaystyle\sum_{a,b\in\mathbb{Z}}\frac{(1-e^{2\pi
i(\alpha_{3}+\alpha_{4})})(1-e^{2\pi
i(\alpha_{1}+\alpha_{2})}q^{ka})}{(1-e^{2\pi i\alpha_{4}}q^{a})(1-e^{2\pi
i\alpha_{1}}q^{b})(1-e^{2\pi i\alpha_{3}}q^{-a})(1-e^{2\pi
i\alpha_{2}}q^{ka-b})}.$

On the other hand,

$\displaystyle f_{N,\deg}(q)$ $\displaystyle=$
$\displaystyle\int_{X}\prod_{i=1}^{4}\frac{(D_{i}/2\pi i)\theta(D_{i}/2\pi
i-\alpha_{i})\theta^{\prime}(0)}{\theta(D_{i}/2\pi i)\theta(-\alpha_{i})}$
$\displaystyle=$ $\displaystyle\frac{\theta^{\prime}(0)^{4}}{(2\pi
i)^{4}\prod_{i=1}^{4}\theta(-\alpha_{i})}\textrm{Res}_{0}\left\\{\textrm{Res}_{0}\frac{\theta(\frac{u}{2\pi
i}-\alpha_{1})\theta(\frac{u}{2\pi i}-\alpha_{2})\theta(\frac{v}{2\pi
i}-\alpha_{3})\theta(\frac{v-ku}{2\pi i}-\alpha_{4})}{\theta^{2}(\frac{u}{2\pi
i})\theta(\frac{v}{2\pi i})\theta(\frac{v-ku}{2\pi i})}du\right\\}dv$
$\displaystyle=$ $\displaystyle\frac{\theta^{\prime}(0)^{4}}{(2\pi
i)^{2}\prod_{i=1}^{4}\theta(-\alpha_{i})}\textrm{Res}_{0}\frac{\theta(v-\alpha_{3})}{\theta(v)}\left\\{\textrm{Res}_{0}\frac{\theta(u-\alpha_{1})\theta(u-\alpha_{2})\theta(v-ku-\alpha_{4})}{\theta^{2}(u)\theta(v-ku)}du\right\\}dv$
$\displaystyle=$ $\displaystyle\frac{\theta^{\prime}(0)^{2}}{(2\pi
i)^{2}\theta(-\alpha_{3})\theta(-\alpha_{4})}\textrm{Res}_{0}\frac{\theta(v-\alpha_{3})\theta(v-\alpha_{4})}{\theta^{2}(v)}\left\\{\frac{\theta^{\prime}(-\alpha_{1})}{\theta(-\alpha_{1})}+\frac{\theta^{\prime}(-\alpha_{2})}{\theta(-\alpha_{2})}-k\frac{\theta^{\prime}(v-\alpha_{4})}{\theta(v-\alpha_{4})}+k\frac{\theta^{\prime}(v)}{\theta(v)}\right\\}dv.$

Similarly, we have

$\textrm{Res}_{0}\
\frac{\theta(v-\alpha_{3})\theta(v-\alpha_{4})}{\theta^{2}(v)}dv=\frac{\theta(-\alpha_{3})\theta(-\alpha_{4})}{\theta^{\prime
2}(0)}\left\\{\frac{\theta^{\prime}(-\alpha_{3})}{\theta(-\alpha_{3})}+\frac{\theta^{\prime}(-\alpha_{4})}{\theta(-\alpha_{4})}\right\\},$
$\textrm{Res}_{0}\
\frac{\theta(v-\alpha_{3})\theta^{\prime}(v-\alpha_{4})}{\theta^{2}(v)}dv=\frac{\theta(-\alpha_{3})\theta^{\prime}(-\alpha_{4})}{\theta^{\prime
2}(0)}\left\\{\frac{\theta^{\prime}(-\alpha_{3})}{\theta(-\alpha_{3})}+\frac{\theta^{\prime\prime}(-\alpha_{4})}{\theta^{\prime}(-\alpha_{4})}\right\\}.$

and

$\displaystyle\textrm{Res}_{0}\
\frac{\theta(v-\alpha_{3})\theta(v-\alpha_{4})\theta^{\prime}(v)}{\theta^{3}(v)}dv$
$\displaystyle=$ $\displaystyle\textrm{Res}_{0}\
\frac{(\theta(-\alpha_{3})+\theta^{\prime}(-\alpha_{3})v+\frac{\theta^{\prime\prime}(-\alpha_{3})v^{2}}{2})(\theta(-\alpha_{4})+\theta^{\prime}(-\alpha_{4})v+\frac{\theta^{\prime\prime}(-\alpha_{4})v^{2}}{2})(\theta^{\prime}(0)+\frac{\theta^{(3)}(0)v^{2}}{2})}{(\theta^{\prime}(0)v+\theta^{(3)}(0)v^{3}/6)^{3}}dv$
$\displaystyle=$ $\displaystyle\textrm{Res}_{0}\
\frac{(\theta(-\alpha_{3})+\theta^{\prime}(-\alpha_{3})v+\frac{\theta^{\prime\prime}(-\alpha_{3})v^{2}}{2})(\theta(-\alpha_{4})+\theta^{\prime}(-\alpha_{4})v+\frac{\theta^{\prime\prime}(-\alpha_{4})v^{2}}{2})}{\theta^{\prime
2}(0)v^{3}}dv$ $\displaystyle=$
$\displaystyle\frac{\theta(-\alpha_{3})\theta(-\alpha_{4})}{2\theta^{\prime
2}(0)}\left\\{(\frac{\theta^{\prime\prime}(-\alpha_{3})}{\theta(-\alpha_{3})}+\frac{\theta^{\prime\prime}(-\alpha_{4})}{\theta(-\alpha_{4})}+2\frac{\theta^{\prime}(-\alpha_{3})\theta^{\prime}(-\alpha_{4})}{\theta(-\alpha_{3})\theta(-\alpha_{4})})\right\\}.$

######  Proposition 3.1.

For Hirzebruch surface
$F_{k}=\mathbb{C}P(\underline{\mathbb{C}}\oplus\mathcal{O}(k))$, we have theta
function identity

$\displaystyle\sum_{a,b\in\mathbb{Z}}\frac{(1-e^{2\pi
i(\alpha_{3}+\alpha_{4})})(1-e^{2\pi
i(\alpha_{1}+\alpha_{2})}q^{ka})}{(1-e^{2\pi i\alpha_{4}}q^{a})(1-e^{2\pi
i\alpha_{1}}q^{b})(1-e^{2\pi i\alpha_{3}}q^{-a})(1-e^{2\pi
i\alpha_{2}}q^{ka-b})}$ $\displaystyle=$ $\displaystyle\frac{1}{(2\pi
i)^{2}}\left\\{(\frac{\theta^{\prime}(-\alpha_{3})}{\theta(-\alpha_{3})}+\frac{\theta^{\prime}(-\alpha_{4})}{\theta(-\alpha_{4})})(\frac{\theta^{\prime}(-\alpha_{1})}{\theta(-\alpha_{1})}+\frac{\theta^{\prime}(-\alpha_{2})}{\theta(-\alpha_{2})})+\frac{k}{2}(\frac{\theta^{\prime\prime}(-\alpha_{3})}{\theta(-\alpha_{3})}-\frac{\theta^{\prime\prime}(-\alpha_{4})}{\theta(-\alpha_{4})})\right\\}.$

It is easy to check that if $\alpha_{3}+\alpha_{4}\in\mathbb{Z}$, both sides
equal zero.

###  3.3. $\mathbb{C}P^{2}$ bundle over $\mathbb{C}P^{2}$

Consider a $\mathbb{C}P^{2}$ bundle over $\mathbb{C}P^{2}$, its corresponding
fan is spanned by five primitive vectors $\textbf{e}_{1},\ \textbf{e}_{2},\
\textbf{e}_{3},\ \textbf{e}_{4},\
-\textbf{e}_{1}-\textbf{e}_{2}+j\textbf{e}_{3}+k\textbf{e}_{4},\
-\textbf{e}_{3}-\textbf{e}_{4}$ in $\mathbb{Z}^{4}$.

Assume that $\deg$ takes $\frac{1}{2}$ on all generators $\textbf{e}_{1},\
\textbf{e}_{2},\ \textbf{e}_{3},\ \textbf{e}_{4},\
-\textbf{e}_{1}-\textbf{e}_{2}+j\textbf{e}_{3}+k\textbf{e}_{4},\
-\textbf{e}_{3}-\textbf{e}_{4}$ in $\mathbb{Z}^{4}$. Then the toric form is

$f_{N,\deg}(q)=\sum_{a,b,c,d\in\mathbb{Z}}\frac{2(1+q^{jc+kd})}{(1+q^{a})(1+q^{b})(1+q^{c})(1+q^{d})(1+q^{jc+kd-
a-b})(1+q^{-c-d})}.$

On the other hand,

$\displaystyle f_{N,\deg}(q)$ $\displaystyle=$
$\displaystyle\int_{X}\prod_{i=1}^{6}\frac{(D_{i}/2\pi i)\theta(D_{i}/2\pi
i-\alpha_{i})\theta^{\prime}(0)}{\theta(D_{i}/2\pi i)\theta(-\alpha_{i})}$
$\displaystyle=$ $\displaystyle\frac{\theta^{\prime}(0)^{6}}{(2\pi
i)^{6}\theta^{6}(\frac{1}{2})}\textrm{Res}_{0}\left\\{\textrm{Res}_{0}\frac{\theta^{3}(\frac{u}{2\pi
i}-\frac{1}{2})\theta(\frac{v}{2\pi i}-\frac{1}{2})\theta(\frac{v-ju}{2\pi
i}-\frac{1}{2})\theta(\frac{v-ku}{2\pi
i}-\frac{1}{2})}{\theta^{3}(\frac{u}{2\pi i})\theta(\frac{v}{2\pi
i})\theta(\frac{v-ju}{2\pi i})\theta(\frac{v-ku}{2\pi i})}du\right\\}dv$
$\displaystyle=$ $\displaystyle\frac{\theta^{\prime}(0)^{6}}{(2\pi
i)^{4}\theta^{6}(\frac{1}{2})}\textrm{Res}_{0}\frac{\theta(v-\frac{1}{2})}{\theta(v)}\left\\{\textrm{Res}_{0}\frac{\theta^{3}(u-\frac{1}{2})\theta(v-ju-\frac{1}{2})\theta(v-ku-\frac{1}{2})}{\theta^{3}(u)\theta(v-ju)\theta(v-ku)}du\right\\}dv$
$\displaystyle=$ $\displaystyle\frac{\theta^{\prime}(0)^{3}}{(2\pi
i)^{4}\theta^{3}(\frac{1}{2})}\textrm{Res}_{0}\frac{\theta^{3}(v+\frac{1}{2})}{\theta^{3}(v)}\bigg{\\{}\frac{3\theta^{\prime\prime}(\frac{1}{2})}{2\theta(\frac{1}{2})}-\frac{\theta^{(3)}(0)}{2\theta^{\prime}(0)}+\frac{j^{2}+k^{2}}{2}(\frac{\theta^{\prime\prime}(v+\frac{1}{2})}{\theta(v+\frac{1}{2})}-\frac{\theta^{\prime\prime}(v)}{\theta(v)})$
$\displaystyle+jk(\frac{\theta^{\prime
2}(v+\frac{1}{2})}{\theta^{2}(v+\frac{1}{2})}-\frac{\theta^{\prime
2}(v)}{\theta^{2}(v)})+(j+k)^{2}\frac{\theta^{\prime
2}(v)}{\theta^{2}(v)}-(j+k)^{2}\frac{\theta^{\prime}(v)\theta^{\prime}(v+\frac{1}{2})}{\theta(v)\theta(v+\frac{1}{2})}\bigg{\\}}dv$
$\displaystyle=$ $\displaystyle\frac{1}{(2\pi
i)^{4}}\bigg{\\{}(\frac{3\theta^{\prime\prime}(\frac{1}{2})}{2\theta(\frac{1}{2})}-\frac{\theta^{(3)}(0)}{2\theta^{\prime}(0)})^{2}+(j^{2}+k^{2}-jk)(\frac{\theta^{\prime\prime
2}(\frac{1}{2})}{4\theta^{2}(\frac{1}{2})}+\frac{5\theta^{(4)}(\frac{1}{2})}{24\theta(\frac{1}{2})}-\frac{7\theta^{(3)}(0)\theta^{\prime\prime}(\frac{1}{2})}{12\theta^{\prime}(0)\theta(\frac{1}{2})}-\frac{\theta^{(5)}(0)}{24\theta^{\prime}(0)}+\frac{(\theta^{(3)}(0))^{2}}{6\theta^{\prime
2}(0)})\bigg{\\}}.$

######  Proposition 3.2.

We have the following theta function identity

$\displaystyle\sum_{a,b,c,d\in\mathbb{Z}}\frac{2(1+q^{jc+kd})}{(1+q^{a})(1+q^{b})(1+q^{c})(1+q^{d})(1+q^{jc+kd-
a-b})(1+q^{-c-d})}$ $\displaystyle=$ $\displaystyle\frac{1}{(2\pi
i)^{4}}\left\\{(\frac{3\theta^{\prime\prime}(\frac{1}{2})}{2\theta(\frac{1}{2})}-\frac{\theta^{(3)}(0)}{2\theta^{\prime}(0)})^{2}+(j^{2}+k^{2}-jk)(\frac{\theta^{\prime\prime
2}(\frac{1}{2})}{4\theta^{2}(\frac{1}{2})}+\frac{5\theta^{(4)}(\frac{1}{2})}{24\theta(\frac{1}{2})}-\frac{7\theta^{(3)}(0)\theta^{\prime\prime}(\frac{1}{2})}{12\theta^{\prime}(0)\theta(\frac{1}{2})}-\frac{\theta^{(5)}(0)}{24\theta^{\prime}(0)}+\frac{(\theta^{(3)}(0))^{2}}{6\theta^{\prime
2}(0)})\right\\}.$

In special case $j=k=0$, $f_{N,\deg}(q)$ is exactly the square of normalized
elliptic genus of $\mathbb{C}P^{2}$ which we have already discussed in Example
3.1.

###  3.4. Vanishing result

Besides the accurate calculation in low dimensions, we can also get a
Landweber-Stong type vanishing result for a family of generalized Bott
manifolds.

######  Theorem 3.3.

Consider toric variety $V\coloneqq{\mathbb{C}}P(\eta^{\otimes
i_{1}}\oplus\eta^{\otimes i_{2}}\oplus\eta^{\otimes
i_{3}}\oplus\underline{\mathbb{C}})$ over ${\mathbb{C}}P^{n_{1}}$, where
$\eta$ denote the tautological bundle of ${\mathbb{C}}P^{n_{1}}$.

If $(i_{1},\ i_{2},\ i_{3})$ are coprime, $\ (i_{1},\ i_{1}-i_{2},\
i_{1}-i_{3}),\ (i_{2},\ i_{2}-i_{1},\ i_{2}-i_{3}),\ (i_{3},\ i_{3}-i_{1},\
i_{3}-i_{2})$ are coprime respectively,
$\sum_{j=0}^{3}\alpha_{n_{1}+2+j}\in\mathbb{Z}$,
$\sum_{i=1}^{n_{1}+1}\alpha_{i}-\sum_{j=1}^{3}i_{j}\alpha_{n_{1}+2+j}\in\mathbb{Z}$,
then the toric form $f_{N,\deg}(q)=0$.

The proof of Theorem 3.3 is similar to that of Theorem 4.1.

##  4\. Witten genus: a vanishing result

Let $M$ be a $4k$ dimensional closed oriented smooth manifold, $E$ be a
complex vector bundle over $M$. For any complex number $t$, set

$S_{t}(E)=\underline{{\mathbb{C}}}+tE+t^{2}S^{2}(E)+\cdots,$

where $S^{j}(E)$ is the $j$-th symmetric power of $E$.

Let $q=e^{2\pi i\tau}$ with $\tau\in\mathbb{H}$, the upper half plane. Witten
defined

$\Theta_{q}(E)=\underset{n\geq 1}{\bigotimes}\ S_{q^{n}}(E),$

then defined the Witten genus

$\varphi_{W}(M)=\left\langle\widehat{A}(M)ch(\Theta_{q}(TM\otimes{\mathbb{C}}-\underline{{\mathbb{C}}}^{4k})),[M]\right\rangle.$

Let ${\pm 2\pi ix_{j}\ (\ 1\leq j\leq 2k)}$ be the formal Chern roots of
$TM\otimes{\mathbb{C}}$. The Witten genus can be rewritten as (c.f. [17])

$\varphi_{W}(M)=\left\langle\left(\prod_{j=1}^{2m}z_{j}\frac{\theta^{\prime}(0,\tau)}{\theta(z_{j},\tau)}\right),[M]\right\rangle,$

where $\theta(z,\tau)$ is the Jacobi theta function

$\theta(z,\tau)=2q^{1/8}sin(\pi z)\prod_{j=1}^{\infty}[(1-q^{j})(1-e^{2\pi
iz}q^{j})(1-e^{-2\pi iz}q^{j})],\ q=e^{2\pi i\tau}.$

The manifold $M$ is called spin if $\omega_{1}(M)=0,\ \omega_{2}(M)=0,$ where
$\omega_{1}(M),\ \omega_{2}(M)$ are the first and second Stiefel-Whitney
classes of $M$. According to Atiyah-Singer index theorem, when $M$ is spin,
$\varphi_{W}(M)\in{\mathbb{Z}}[[q]]$.

A spin manifold $M$ is called string if $\frac{1}{2}p_{1}(M)=0$, where
$p_{1}(M)$ is the first Pontrjagin class of $M$. It is well known that if $M$
is string, $\varphi_{W}(M)$ is a modular form of weight $2k$ over
$SL(2,\mathbb{Z})$, see [13, Hirzebruch]. In this section, we mainly discuss
the Witten genus of string manifolds.

###  4.1. String complete intersection in generalized Bott manifolds

Consider a two staged generalized Bott manifold
$V\coloneqq{\mathbb{C}}P(\eta^{\otimes i_{1}}\oplus\cdots\eta^{\otimes
i_{n_{2}}}\oplus\underline{{\mathbb{C}}})$ over ${\mathbb{C}}P^{n_{1}}$, where
$\eta$ denotes the tautological bundle of ${\mathbb{C}}P^{n_{1}}$.

Let $\textbf{I}=(i_{1},\cdots,i_{n_{2}})$ denote the index of projective
bundle and $H^{\textbf{I}}_{n_{1},n_{2}}(d_{1},d_{2})$ denote the submanifold
Poincaré dual to $d_{1}u+d_{2}v\in H^{2}(V;{\mathbb{Z}})$. Let
$i:H^{\textbf{I}}_{n_{1},n_{2}}(d_{1},d_{2})\longrightarrow V$ be the natural
embedding, $\nu$ denote the normal bundle of this embedding. We have

$i^{*}(TV)\cong i^{*}(\nu)\oplus TH^{\textbf{I}}_{n_{1},n_{2}}(d_{1},d_{2}).$

Thus

$c_{1}(H^{\textbf{I}}_{n_{1},n_{2}}(d_{1},d_{2}))=i^{*}((n_{1}+1)u+v+\sum_{j=1}^{n_{2}}(v-i_{j}u)-(d_{1}u+d_{2}v)).$

and

$p_{1}(H^{\bf
I}_{n_{1},n_{2}}(d_{1},d_{2}))=i^{*}\left\\{(n_{1}+1+\sum_{j=1}^{n_{2}}i_{j}^{2}-d_{1}^{2})u^{2}+(1+n_{2}-d_{2}^{2})v^{2}-2(d_{1}d_{2}+\sum_{j=1}^{n_{2}}i_{j})uv\right\\}.$

######  Lemma 4.1.

Twisted Milnor hypersurfaces $H^{\bf I}_{n_{1},n_{2}}(d_{1},d_{2})$ can’t be
string for $n_{2}\geq 3$.

###### Proof.

Let $i_{!}:H^{*}(H^{\bf I}_{n_{1},n_{2}}(d_{1},d_{2}))\longrightarrow
H^{*}(V)$ be the pushforward map.

First we assume

$\left\\{\begin{array}[]{lr}n_{1}+1+\sum\limits_{j=1}^{n_{2}}i_{j}^{2}=d_{1}^{2}\\\
1+n_{2}=d_{2}^{2}\\\ d_{1}d_{2}+\sum_{j=1}^{n_{2}}i_{j}=0.\end{array}\right.$

Thus
$(n_{1}+1+\sum_{j=1}^{n_{2}}i_{j}^{2})(1+n_{2})=(d_{1}d_{2})^{2}=(\sum_{j=1}^{n_{2}}i_{j})^{2}$,
which is impossible. Thus
$(n_{1}+1+\sum_{j=1}^{n_{2}}i_{j}^{2}-d_{1}^{2})u^{2}+(1+n_{2}-d_{2}^{2})v^{2}+(-2\sigma_{1}-2d_{1}d_{2})uv\neq
0$.

On the other hand, if $p_{1}(H^{\bf I}_{n_{1},n_{2}}(d_{1},d_{2}))=0$, then

$\displaystyle i_{!}p_{1}(H^{\bf I}_{n_{1},n_{2}}(d_{1},d_{2}))$
$\displaystyle=$ $\displaystyle
i_{!}i^{*}\left\\{(n_{1}+1+\sum_{j=1}^{n_{2}}i_{j}^{2}-d_{1}^{2})u^{2}+(1+n_{2}-d_{2}^{2})v^{2}+(-2\sigma_{1}-2d_{1}d_{2})uv\right\\}$
$\displaystyle=$
$\displaystyle(d_{1}u+d_{2}v)\left\\{(n_{1}+1+\sum_{j=1}^{n_{2}}i_{j}^{2}-d_{1}^{2})u^{2}+(1+n_{2}-d_{2}^{2})v^{2}+(-2\sigma_{1}-2d_{1}d_{2})uv\right\\}$
$\displaystyle=$ $\displaystyle 0.$

Since $d_{1}u+d_{2}v\neq 0$, as long as $n_{2}\geq 3$, it is impossible for
$p_{1}=0$.

$\square$

######  Remark 7.

It is possible for twisted Milnor hypersurface $H^{\bf
I}_{n_{1},n_{2}}(d_{1},d_{2})$ to be string when $n_{2}<3$. For example,
choose $n_{2}=1,d_{2}=0$, we have

$p_{1}=i^{*}\bigg{\\{}(n_{1}+1+i_{1}^{2}-d_{1}^{2})u^{2}+2v^{2}-2i_{1}uv\bigg{\\}},$

where $2v^{2}-2i_{1}uv$ is killed by the relation in the cohomology ring of 2
staged generalized Bott manifold. Thus if $n_{1}+1+i_{1}^{2}-d_{1}^{2}=0$, we
have $p_{1}=0$. For example, twisted Milnor hypersurface $H^{\pm 6}_{12,1}(\pm
7,0)$ is the required 24-dim string manifold.

Due to the restriction of Lemma 4.1, we proceed to investigate string complete
intersections. The submanifold $H^{\bf
I}_{n_{1},n_{2}}(d_{1},d_{2};d_{3},d_{4})$ Poincaré dual to cohomology class
$(d_{1}u+d_{2}v)\cdot(d_{3}u+d_{4}v)\in H^{4}(V;{\mathbb{Z}})$ is string as
long as the equation set
$\Bigg{\\{}\begin{array}[]{lr}n_{1}+1+\sum\limits_{j=1}^{n_{2}}i_{j}^{2}=d_{1}^{2}+d_{3}^{2}\\\
1+n_{2}=d_{2}^{2}+d_{4}^{2}\\\
d_{1}d_{2}+d_{3}d_{4}+\sum\limits_{j=1}^{n_{2}}i_{j}=0\end{array}$ has integer
solutions. Actually, there exist plenty of string complete intersections.

###  4.2. Witten genus

Consider the Jacobi theta function

$\theta(z,\tau)=2q^{1/8}sin(\pi z)\prod_{j=1}^{\infty}[(1-q^{j})(1-e^{2\pi
iz}q^{j})(1-e^{-2\pi iz}q^{j})],$

where $\theta$ admits periodicity, $\theta(z+1,\tau)=-\theta(z,\tau),\
\theta(z+b\tau,\tau)=(-1)^{b}e^{-\pi i(2bz+b^{2}\tau)}\theta(z,\tau)$. Clearly
$\theta$ has simple zeros on lattice ${\mathbb{Z}}+{\mathbb{Z}}\tau$ and no
pole. Applying our iterated residue, the Witten genus can be reformulated as

$\displaystyle\varphi_{W}(H^{\bf I}_{n_{1},n_{2}})$ $\displaystyle=$
$\displaystyle\theta^{\prime
n_{1}+n_{2}}(0)\langle(\frac{u}{\theta(u)})^{n_{1}+1}\frac{v}{\theta(v)}\prod_{j=1}^{n_{2}}\frac{v-i_{j}u}{\theta(v-i_{j}u)}\frac{\theta(d_{1}u+d_{2}v)}{d_{1}u+d_{2}v}\frac{\theta(d_{3}u+d_{4}v)}{d_{3}u+d_{4}v},[H^{\bf
I}_{n_{1},n_{2}}]\rangle$ $\displaystyle=$ $\displaystyle\frac{\theta^{\prime
n_{1}+n_{2}}(0)}{(2\pi
i)^{-2}}\langle(\frac{u}{\theta(u)})^{n_{1}+1}\frac{v}{\theta(v)}\prod_{j=1}^{n_{2}}\frac{v-i_{j}u}{\theta(v-i_{j}u)}\theta(d_{1}u+d_{2}v)\cdot\theta(d_{3}u+d_{4}v),[V]\rangle$
$\displaystyle=$ $\displaystyle\frac{\theta^{\prime n_{1}+n_{2}}(0)}{(2\pi
i)^{n_{1}+n_{2}-2}}\mathrm{Res}_{0}\left\\{\mathrm{Res}_{0}\frac{\theta(d_{1}u+d_{2}v)\cdot\theta(d_{3}u+d_{4}v)}{\theta^{n_{1}+1}(u)\theta(v)\prod_{j=1}^{n_{2}}\theta(v-i_{j}u)}du\right\\}dv.$

######  Remark 8.

If meromorphic function $g$ has pole of order $n$ at point $c$, then

$\mathrm{Res}_{c}\,g=\frac{1}{(n-1)!}\cdot\underset{z\rightarrow c}{\lim}\
\frac{d^{n-1}}{dz^{n-1}}((z-c)^{n}g(z)).$

It is usually difficult to calculate the residue at high order poles for both
variables $u$ and $v$ directly, thus we shall make use of residue theorem to
reduce iterated residue at high order poles to simple poles. This is the
common operation for Landweber-Stong type vanishing results.

Under certain condition, we obtain a Landweber-Stong type vanishing result for
string complete intersections in generalized Bott manifolds. We add the
coprime condition to make sure that the poles except for $0$ are simple.

######  Theorem 4.1.

For a string complete intersection $H^{\bf I}_{n_{1},\
3}(d_{1},d_{2};d_{3},d_{4})$, if ${\bf I}=(i,\ j,\ k)$ are coprime, $\ (i,\
i-j,\ i-k),\ (j,\ j-i,\ j-k),\ (k,\ k-i,\ k-j)$ are coprime respectively, then
the Witten genus

$\varphi_{W}(H^{\bf I}_{n_{1},\ 3}(d_{1},d_{2};d_{3},d_{4}))=0.$

###### Proof.

$\varphi_{W}(H^{\bf I})=\frac{\theta^{\prime n_{1}+n_{2}}(0)}{(2\pi
i)^{n_{1}+n_{2}-2}}\mathrm{Res}_{0}\left\\{\mathrm{Res}_{0}\frac{\theta(d_{1}u+d_{2}v)\theta(d_{3}u+d_{4}v)}{\theta^{n_{1}+1}(u)\theta(v)\theta(v+iu)\theta(v+ju)\theta(v+ku)}du\right\\}dv.$

Denote
$\omega(u)=\frac{\theta(d_{1}u+d_{2}v)\cdot\theta(d_{3}u+d_{4}v)}{\theta^{n_{1}+1}(u)\theta(v)\prod_{j=1}^{n_{2}}\theta(v-i_{j}u)}du$,
we check its periodicity

$\omega(u+1)=(-1)^{d_{1}+d_{3}-n_{1}-1-\sum i_{j}}\omega(u)=\omega(u);$
$\omega(u+\tau)=\frac{(-1)^{d_{1}+d_{3}}e^{-\pi
i(2d_{1}(d_{1}u+d_{2}v)+2d_{3}(d_{3}u+d_{4}v)+(d_{1}^{2}+d_{3}^{2})\tau)}\theta(d_{1}u+d_{2}v)\cdot\theta(d_{3}u+d_{4}v)}{(-1)^{n_{1}+1+\sum
i_{j}}e^{-\pi
i((n_{1}+1)(2u+\tau)+\sum(-2i_{j}(v-i_{j}u)+i_{j}^{2}\tau))}\theta^{n_{1}+1}(u)\theta(v)\prod\theta(v-i_{j}u)}du.$

Since $H^{\bf I}_{n_{1},n_{2}}(d_{1},d_{2};d_{3},d_{4})$ is string, we have
$\omega(u+\tau)=\omega(u)$. We can claim that $\omega(u)$ admits double
periodicity on lattice $L\coloneqq{\mathbb{Z}}+{\mathbb{Z}}\tau$. Let
$T^{2}\cong{\mathbb{C}}/L$, thus $\omega(u)$ can be defined on closed surface
$T^{2}$. Then we can apply residue theorem on $\omega(u)$. Since $\theta$ has
simple zeros on ${\mathbb{Z}}+{\mathbb{Z}}\tau$ and no poles, the possible
poles of $\omega$ are

$0,\ \frac{a+b\tau-v}{i}(1\leq a,\ b\leq i),\ \frac{c+d\tau-v}{j}(1\leq c,\
d\leq j),\ \frac{m+n\tau-v}{k}(1\leq m,\ n\leq k).$

Applying residue theorem on $u$, we get

$\\{\mathrm{Res}_{0}+\mathrm{Res}_{1+\tau-\frac{v}{i}}+\mathrm{Res}_{1+\tau-\frac{v}{j}}+\mathrm{Res}_{1+\tau-\frac{v}{k}}+\mathrm{Res}_{\frac{a+b\tau-v}{i}}+\mathrm{Res}_{\frac{c+d\tau-v}{j}}+\mathrm{Res}_{\frac{m+n\tau-v}{k}}\\}\omega(u)=0.$

Moreover

$\mathrm{Res}_{0}\\{\mathrm{Res}_{1+\tau-\frac{v}{i}}\
\omega(u)\\}dv=\frac{1}{\theta^{\prime}(0)}\cdot\mathrm{Res}_{0}\frac{\theta((d_{2}i-d_{1})v)\theta((d_{4}i-d_{3})v)}{\theta^{n_{1}+1}(-v)\theta(iv)\theta((i-j)v)\theta((i-k)v)}dv,$

$\mathrm{Res}_{0}\\{\mathrm{Res}_{1+\tau-\frac{v}{j}}\
\omega(u)\\}dv=\frac{1}{\theta^{\prime}(0)}\cdot\mathrm{Res}_{0}\frac{\theta((d_{2}j-d_{1})v)\theta((d_{4}j-d_{3})v)}{\theta^{n_{1}+1}(-v)\theta(jv)\theta((j-i)v)\theta((j-k)v)}dv,$

$\mathrm{Res}_{0}\\{\mathrm{Res}_{1+\tau-\frac{v}{k}}\
\omega(u)\\}dv=\frac{1}{\theta^{\prime}(0)}\cdot\mathrm{Res}_{0}\frac{\theta((d_{2}k-d_{1})v)\theta((d_{4}k-d_{3})v)}{\theta^{n_{1}+1}(-v)\theta(kv)\theta((k-i)v)\theta((k-j)v)}dv,$

$\mathrm{Res}_{0}\\{\mathrm{Res}_{\frac{a+b\tau-v}{i}}\
\omega(u)\\}dv=\frac{(-1)^{a+b+1}}{i\cdot\theta^{\prime 2}(0)}e^{2\pi
ib^{2}\tau}\cdot\frac{\theta(\frac{d_{1}(a+b\tau)}{i})\theta(\frac{d_{3}(a+b\tau)}{i})}{\theta^{n_{1}+1}(\frac{a+b\tau}{i})\theta(\frac{j(a+b\tau)}{i})\theta(\frac{k(a+b\tau)}{i})},$

$\mathrm{Res}_{0}\\{\mathrm{Res}_{\frac{c+d\tau-v}{j}}\
\omega(u)\\}dv=\frac{(-1)^{c+d+1}}{j\cdot\theta^{\prime 2}(0)}e^{2\pi
id^{2}\tau}\cdot\frac{\theta(\frac{d_{1}(c+d\tau)}{j})\theta(\frac{d_{3}(c+d\tau)}{j})}{\theta^{n_{1}+1}(\frac{c+d\tau}{j})\theta(\frac{i(c+d\tau)}{j})\theta(\frac{k(c+d\tau)}{j})},$

$\mathrm{Res}_{0}\\{\mathrm{Res}_{\frac{m+n\tau-v}{k}}\
\omega(u)\\}dv=\frac{(-1)^{m+n+1}}{k\cdot\theta^{\prime 2}(0)}e^{2\pi
in^{2}\tau}\cdot\frac{\theta(\frac{d_{1}(m+n\tau)}{k})\theta(\frac{d_{3}(m+n\tau)}{k})}{\theta^{n_{1}+1}(\frac{m+n\tau}{k})\theta(\frac{i(m+n\tau)}{k})\theta(\frac{j(m+n\tau)}{k})}.$

Apply residue theorem again on $v$ to the followings

$\mathrm{Res}_{0}\\{\mathrm{Res}_{1+\tau-\frac{v}{i}}\ \omega(u)\\}dv,\
\mathrm{Res}_{0}\\{\mathrm{Res}_{1+\tau-\frac{v}{j}}\ \omega(u)\\}dv,\
\mathrm{Res}_{0}\\{\mathrm{Res}_{1+\tau-\frac{v}{k}}\ \omega(u)\\}dv.$

Here we take $\mathrm{Res}_{0}\\{\mathrm{Res}_{1+\tau-\frac{v}{i}}\
\omega(u)\\}dv$ for example. Denote

$\omega(v)=\frac{\theta((d_{2}i-d_{1})v)\theta((d_{4}i-d_{3})v)}{\theta^{n_{1}+1}(-v)\theta(iv)\theta((i-j)v)\theta((i-k)v)}dv.$

Since $i,(i-j),i-k$ are coprime, it is easy to check $\omega(v)$ can also be
defined on $T^{2}\cong\mathbb{C}/L$. The possible poles of $\omega(v)$ include
$0,\frac{a+b\tau}{i},\ \frac{a+b\tau}{i-j},\ \frac{a+b\tau}{i-k}$. We have

$\mathrm{Res}_{\frac{a+b\tau}{i}}\
\omega(v)=\frac{(-1)^{a+b}}{i\cdot\theta^{\prime}(0)}e^{2\pi
ib^{2}\tau}\cdot\frac{\theta(\frac{(d_{2}i-d_{1})(a+b\tau)}{i})\theta(\frac{(d_{4}i-d_{3})(a+b\tau)}{i})}{\theta^{n_{1}+1}(-\frac{a+b\tau}{i})\theta(\frac{(i-j)(a+b\tau)}{i})\theta(\frac{(i-k)(a+b\tau)}{i})}.$

Since $\theta(z+b\tau)=(-1)^{b}e^{-2\pi ibz-\pi ib^{2}\tau}\theta(z)$,

$\displaystyle\frac{\mathrm{Res}_{\frac{a+b\tau}{i}}\
\omega(v)}{\mathrm{Res}_{0}\\{\mathrm{Res}_{\frac{a+b\tau-v}{i}}\
\omega(u)\\}dv}$ $\displaystyle=$
$\displaystyle(-1)^{n_{1}+1+(d_{2}+d_{4})(a+b)}\cdot e^{-2\pi
i(-d_{2}d_{1}-d_{4}d_{3}+j+k)b(\frac{a+b\tau}{i})-\pi
ib^{2}(d_{2}^{2}+d_{4}^{2}-2)\tau}$ $\displaystyle=$ $\displaystyle 1.$

Similarly,

$\mathrm{Res}_{\frac{c+d\tau}{j}}\\{\mathrm{Res}_{1+\tau-\frac{v}{j}}\
\omega(u)\\}dv=\mathrm{Res}_{0}\\{\mathrm{Res}_{\frac{c+d\tau-v}{j}}\
\omega(u)\\}dv,$

$\mathrm{Res}_{\frac{m+n\tau}{k}}\ \\{\mathrm{Res}_{1+\tau-\frac{v}{k}}\
\omega(u)\\}dv=\mathrm{Res}_{0}\\{\mathrm{Res}_{\frac{m+n\tau-v}{k}}\
\omega(u)\\}dv$

and

$\displaystyle\frac{\mathrm{Res}_{\frac{a+b\tau}{i-j}}\
\omega(v)}{\mathrm{Res}_{\frac{a+b\tau}{i-j}}\\{\mathrm{Res}_{1+\tau-\frac{v}{j}}\
\omega(u)\\}dv}$ $\displaystyle=$
$\displaystyle(-1)^{1+(d_{2}+d_{4})(a+b)}\cdot e^{-2\pi
i(d_{2}(d_{2}j-d_{1})+d_{4}(d_{4}j-d_{3})-2j+k)b\frac{a+b\tau}{i-j}-\pi
ib^{2}(d_{2}^{2}+d_{4}^{2}-2)\tau}$ $\displaystyle=$ $\displaystyle-1.$

Similarly,

$\mathrm{Res}_{\frac{a+b\tau}{i-k}}\
\omega(v)=-\mathrm{Res}_{\frac{a+b\tau}{i-k}}\\{\mathrm{Res}_{1+\tau-\frac{v}{k}}\
\omega(u)\\}dv$

and

$\mathrm{Res}_{\frac{a+b\tau}{j-k}}\\{\mathrm{Res}_{1+\tau-\frac{v}{j}}\
\omega(u)\\}dv=-\mathrm{Res}_{\frac{a+b\tau}{j-k}}\\{\mathrm{Res}_{1+\tau-\frac{v}{k}}\
\omega(u)\\}dv.$

Glue all the calculations together, we have
$\mathrm{Res}_{0}\\{\mathrm{Res}_{0}\ \omega(u)\\}dv=0$, i.e.,

$\varphi_{W}(H^{\bf I}_{n_{1},\ 3}(d_{1},d_{2};d_{3},d_{4}))=0.$

$\square$

## References

  * [1] M. F. Atiyah, I.M. Singer, The index of elliptic operators, III, Ann. Math. 1968 (87), 546–604.
  * [2] G. Andrews, A. Schilling, S. Warnaar, An $A_{2}$ Bailey lemma and Rogers-Ramanujan-type identities, Journal of the American Mathematical Society, 1999, 12(3): 677–702.
  * [3] W. N. Bailey, An Algebraic Identity, J. London Math. Soc. 1936 (11), 156–160
  * [4] L. A. Borisov, P. E. Gunnells, Toric varieties and modular forms, Inventiones mathematicae, 2001, 144(2): 297–325.
  * [5] L. A. Borisov, A. Libgober, Elliptic genera of toric varieties and applications to mirror symmetry, Inventiones mathematicae, 2000, 140(2): 453–485.
  * [6] R. Bott, H. Samelson, Application of the theory of Morse to symmetric spaces, Amer. J. of Math. 1958 (80), 964–1029.
  * [7] K. Chandrasekharan, Elliptic Functions, Springer-Verlag, 1985.
  * [8] Q. Chen, F. Han, Witten genus and string complete intersections, Pacific J. Math. 2008 234(2) : 249–259.
  * [9] Q. Chen, F. Han, and W. Zhang, Generalized Witten genus and vanishing theorems, J. Differential Geom. 2011 88(1) : 1–39.
  * [10] A. Dessai, The Witten Genus and $S^{3}$-actions on Manifolds, Fachbereich Mathematik, Universität Mainz, 1994.
  * [11] P. Griffiths, J. Harris, Principles of Algebraic Geometry, Vol. 52 (John Wiley and Sons, Springer, New York, 2011).
  * [12] M. Grossberg, Y. Karshon, Bott towers, complete integrability, and the extended character of representations, Duke Math. J. 1994 (76): 23–58.
  * [13] F. Hirzebruch, T. Berger, R. Jung, P. S. Landweber, Manifolds and modular forms, Braunschweig: Vieweg, 1992.
  * [14] M. Hopkins, Algebraic topology and modular forms, Plenary talk, ICM, Beijing, 2002, MR 1989190, Zbl 1031.55007.
  * [15] J. Lian, F. Han, H. Li and Z. Lü, Twisted Milnor hypersurface, Pure and Applied Mathematics Quarterly, 2022, Volume 18, Number 3, 923–970.
  * [16] A. Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris, Série A-B. 257 (1963), 7–9.
  * [17] K. Liu, On elliptic genera and theta-functions, Topology 1996 35(3): 617–640.
  * [18] K. Liu, On modular invariance and rigidity theorems, J. Differential Geom. 1995 (41): 343–396.
  * [19] S. Stolz, A conjecture concerning positive Ricci curvature and the Witten genus, Mathematische Annalen, 1996, 304(1): 785–800.
  * [20] R. E. Stong, Notes on cobordism theory, Princeton Univ. Press, Princeton, NJ, 1968.
  * [21] E. Witten, The verlinde algebra and the cohomology of the Grassmannian, Geometry, Topology, Physics (International Press, Cambridge, MA), 1995: 357–422.
  * [22] D. Zagier, Note on the Landweber-Stong elliptic genus, in P.S. Landweber, ed., Elliptic Curves and Modular Forms in Algebraic Topology (Proceedings, Princeton 1986), Lecture Notes in Math., 1326 Springer, 1988: 216–224.
  * [23] W. Zhang, Spinc-manifolds and Rokhlin congruences, C. R. Acad. Sci. Paris, Série I 1993 (317): 689–692, MR 1245100, Zbl 0804.57012.
  * [24] W. Zhang, Cobordism and Rokhlin congruences, Acta Math. Scientia, 2009 (29B): 609–612, MR 2514364, Zbl pre05732980.
  * [25] J. Zhou, X. Zhuang, Witten genus of generalized complete intersections in products of Grassmannians, International Journal of Mathematics, 2014 25(10) 1450095.
  * [26] X. Zhuang, Vanishing theorems of generalized Witten geneus for generalized complete intersections in flag manifolds, International Journal of Mathematics, 2016 27(9) 1650076.