# On equivariant index of a generalized Bott manifold

Yuki Sugiyama

Abstract. In this paper, we consider the equivariant index of a generalized
Bott manifold. We show the multiplicity function of the equivariant index is
given by the density function of a generalized twisted cube. In addition, we
give a Demazure-type character formula of this representation.

##  1 Introduction

A Bott tower of height $n$ is a sequence:

$M_{n}\stackrel{{\scriptstyle\pi_{n}}}{{\to}}M_{n-1}\stackrel{{\scriptstyle\pi_{n-1}}}{{\to}}\cdots\stackrel{{\scriptstyle\pi_{2}}}{{\to}}M_{1}\stackrel{{\scriptstyle\pi_{1}}}{{\to}}M_{0}=\\{\textup{a
point}\\}$

of complex manifolds $M_{j}=\mathbb{P}(\underline{\mathbb{C}}\oplus E_{j})$,
where $\underline{\mathbb{C}}$ is the trivial line bundle over $M_{j-1}$,
$E_{j}$ is a holomorphic line bundle over $M_{j-1}$, $\mathbb{P}(\cdot)$
denotes the projectivization, and $\pi_{j}:M_{j}\to M_{j-1}$ is the projection
of the $\mathbb{C}P^{1}$-bundle. We call $M_{j}$ a $j$-stage Bott manifold.
The notion of a Bott tower was introduced by Grossberg and Karshon ([6]).

A generalized Bott tower is a generalization of a Bott tower. A generalized
Bott tower of height $m$ is a sequence:

$B_{m}\stackrel{{\scriptstyle\pi_{m}}}{{\to}}B_{m-1}\stackrel{{\scriptstyle\pi_{m-1}}}{{\to}}\cdots\stackrel{{\scriptstyle\pi_{2}}}{{\to}}B_{1}\stackrel{{\scriptstyle\pi_{1}}}{{\to}}B_{0}=\\{\textup{a
point}\\},$

of complex manifolds $B_{j}=\mathbb{P}(\underline{\mathbb{C}}\oplus
E_{j}^{(1)}\oplus\cdots\oplus E_{j}^{(n_{j})})$, where
$\underline{\mathbb{C}}$ is the trivial line bundle over $B_{j-1}$,
$E_{j}^{(k)}$ is a holomorphic line bundle over $B_{j-1}$ for
$k=1,\dots,n_{j}$. We call $B_{j}$ a $j$-stage generalized Bott manifold. A
generalized Bott tower has been studied from various points of view (see,
e.g., [2, 3, 8]). A generalized Bott manifold is a certain class of toric
manifold, so it is interesting to investigate the properties of generalized
Bott towers.

In [6], Grossberg and Karshon showed the multiplicity function of the
equivariant index (see $\S 2.4$) for a holomorphic line bundle over a Bott
manifold is given by the density function of a twisted cube, which is
determined by the structure of the Bott manifold and the line bundle over it.
From this, they derived a Demazure-type character formula.

The purpose of this paper is to generalize the results in [6] to generalized
Bott manifolds. We generalize the twisted cube, and we call it the generalized
twisted cube. It is a special case of twisted polytope introduced by Karshon
and Tolman [9] for the presymplectic toric manifold, and it is a special case
of multi-polytope introduced by Hattori and Masuda [7] for the torus manifold.
We show the multiplicity function of the equivariant index for the holomorphic
line bundle over the generalized Bott manifold is given by the density
function of the generalized twisted cube. From this, we derive a Demazure-type
character formula. In order to state the main results, we give some notation.
Let $\mathbf{L}$ be a holomorphic line bundle over a generalized Bott manifold
$B_{m}$, which is constructed from integers $\\{\ell_{i}\\}$ and
$\\{c_{i,j}^{(k)}\\}$ (see $\S 2.1$). Let $N=\sum_{j=1}^{m}n_{j}$, and let
$T^{N}=S^{1}\times\cdots\times S^{1}$. We consider the action of $T^{N}$ on
$B_{m}$ as follows:

$(\mathbf{t}_{1},\dots,\mathbf{t}_{m})\cdot[\mathbf{z}_{1},\dots,\mathbf{z}_{m}]=[\mathbf{t}_{1}\mathbf{z}_{1},\dots,\mathbf{t}_{m}\mathbf{z}_{m}],$

where
$\mathbf{t}_{i}=(t_{i,1},\dots,t_{i,n_{i}}),\mathbf{z}_{i}=(z_{i,0},\dots,z_{i,n_{i}}),\mathbf{t}_{i}\mathbf{z}_{i}=(z_{i,0},t_{i,1}z_{i,1},\dots,t_{i,n_{i}}z_{i,n_{i}})$
for $i=1,\dots,m$. Also we consider the action of $T=T^{N}\times S^{1}$ on
$\mathbf{L}$ as follows:

$(\mathbf{t}_{1},\dots,\mathbf{t}_{m},t_{m+1})\cdot[\mathbf{z}_{1},\dots,\mathbf{z}_{m},v]=[\mathbf{t}_{1}\mathbf{z}_{1},\dots,\mathbf{t}_{m}\mathbf{z}_{m},t_{m+1}v].$
(1.1)

We define the generalized twisted cube as follows. It is defined to be the set
of $x=(x_{1,1},\dots,x_{m,n_{m}})\in\mathbb{R}^{N}$ which satisfies

$\displaystyle A_{i}(x)\leq\sum_{k=1}^{n_{i}}x_{i,k}\leq 0,\,\,x_{i,k}\leq
0\,\,\,\,(1\leq k\leq n_{i})$ $\displaystyle{\rm
or}\,\,0<\sum_{k=1}^{n_{i}}x_{i,k}<A_{i}(x),\,\,x_{i,k}>0\,\,\,\,(1\leq k\leq
n_{i}),$

for $1\leq i\leq m$, where

$A_{i}(x)=\begin{cases}-\ell_{m}&(i=m)\\\
-(\ell_{i}+\sum_{j=i+1}^{m}\sum_{k=1}^{n_{j}}c_{i,j}^{(k)}x_{j,k})&(1\leq
i\leq m-1).\end{cases}$

We denote the generalized twisted cube by $C$. We also define ${\rm
sgn}(x_{i,k})=1$ for $x_{i,k}>0$ and ${\rm sgn}(x_{i,k})=-1$ for $x_{i,k}\leq
0$. The density function of the generalized twisted cube is defined to be
$\rho(x)=(-1)^{N}\prod_{i,k}{\rm sgn}(x_{i,k})$ when $x\in C$ and $0$
elsewhere.

Let $\mathfrak{t}$ be the Lie algebra of $T$ and let $\mathfrak{t}^{\ast}$ be
its dual space. Let $\ell^{\ast}\subset i\mathfrak{t}^{\ast}$ be the integral
weight lattice and let ${\rm mult}:\ell^{\ast}\to\mathbb{Z}$ be the
multiplicity function of the equivariant index. The first main result of this
paper is the following:

###### Theorem 1.1

Fix integers $\\{c_{i,j}^{(k)}\\}$ and $\\{\ell_{j}\\}$. Let $\mathbf{L}\to
B_{m}$ be the corresponding line bundle over a generalized Bott manifold. Let
$\rho\,:\,\mathbb{R}^{N}\to\\{-1,0,1\\}$ be the density function of the
generalized twisted cube $C$ which is determined by these integers. Consider
the torus action of $T=T^{N}\times S^{1}$ as in (1.1). Then the multiplicity
function for $\ell^{\ast}\cong\mathbb{Z}^{N}\times\mathbb{Z}$ is given by

${\rm mult}(x,k)=\begin{cases}\rho(x)&(k=1)\\\ 0&(k\neq 1).\end{cases}$

Karshon and Tolman found a toric manifold for which the multiplicities of the
equivariant index are $0,-1$, or $-2$ ([9, Example 6.7]). A generalized Bott
manifold is different from this case by Theorem 1.1.

Next, we give our character formula. Let
$\\{e_{1,1},\dots,e_{m,n_{m}},e_{m+1}\\}$ be the standard basis in
$\mathbb{R}^{N+1}$, $x_{i}=(x_{i,1},\dots,x_{i,n_{i}})$, and
$e_{i}=(e_{i,1},\dots,e_{i,n_{i}})$. Let
$\Delta_{n,r}^{-}=\left\\{z=(z_{1},\dots,z_{n})\in\mathbb{Z}_{\leq
0}^{n}\,\middle|\,\right.$
$\left.z_{1}+\cdots+z_{n}=-r\right\\}$, and let
$\Delta_{n,r}^{+}=\left\\{z=(z_{1},\dots,z_{n})\in\mathbb{Z}_{>0}^{n}\,\middle|\,z_{1}+\cdots+z_{n}=r-1\right\\}$.
For every integral weight $\mu\in\ell^{\ast}$ we have a homomorphism
$\lambda^{\mu}:T\to S^{1}$. We denote the integral combinations of these
$\lambda^{\mu}$’s by $\mathbb{Z}[T]$. Then the operators
$D_{i}:\mathbb{Z}[T]\to\mathbb{Z}[T]$ are defined using $c_{i,j}^{(k)}$ and
$\ell_{j}$ in the following way:

$D_{i}(\lambda^{\mu})=\begin{cases}\displaystyle\sum_{0\leq r\leq
k_{i}}\sum_{x_{i}\in\Delta_{n_{i},r}^{-}}\lambda^{\mu+\langle
x_{i},e_{i}\rangle}&\text{if}\,\,k_{i}\geq 0\\\ 0&\text{if}\,\,-n_{i}\leq
k_{i}\leq-1\\\ \displaystyle\sum_{n_{i}+1\leq r\leq-
k_{i}}\sum_{x_{i}\in\Delta_{n_{i},r}^{+}}(-1)^{n_{i}}\lambda^{\mu+\langle
x_{i},e_{i}\rangle}&\text{if}\,\,k_{i}\leq-n_{i}-1,\end{cases}$

where the functions $k_{i}$ are defined as follows: if
$\mu=e_{m+1}+\sum_{j=i+1}^{m}\sum_{k=1}^{n_{j}}x_{j,k}e_{j,k}$, then
$k_{i}(\mu)=\ell_{i}+\sum_{j=i+1}^{m}\sum_{k=1}^{n_{j}}c_{i,j}^{(k)}x_{j,k}$.
From Theorem 1.1, we obtain the following theorem:

###### Theorem 1.2

Consider the action of the torus $T$ on $\mathbf{L}\to B_{m}$ as in (1.1).
Denote the $(N+1)$-th component of the standard basis in $\mathbb{R}^{N+1}$ by
$e_{m+1}$. Then the character is given by the following element of
$\mathbb{Z}[T]$:

$\chi=D_{1}\cdots D_{m}(\lambda^{e_{m+1}}).$

This is a Demazure-type character formula. On the other hand, the character is
also given by the localization formula with respect to the action of $T$ ([7,
Corollary 7.4]). We compare our formula and the localization formula (see
Remark 3.8).

This paper is organized as follows. In Section 2, we recall the equivariant
index and the generalized Bott towers, and we give the definition of
generalized twisted cubes. In Section 3, we prove the main theorems.

##  2 Preliminaries

In this section, we set up the tools to prove the main theorems.

###  2.1 Generalized Bott manifolds

######  Definition 2.1 ([2])

A generalized Bott tower of height $m$ is a sequence:

$B_{m}\stackrel{{\scriptstyle\pi_{m}}}{{\to}}B_{m-1}\stackrel{{\scriptstyle\pi_{m-1}}}{{\to}}\cdots\stackrel{{\scriptstyle\pi_{2}}}{{\to}}B_{1}\stackrel{{\scriptstyle\pi_{1}}}{{\to}}B_{0}=\\{\textup{a
point}\\},$

of manifolds $B_{j}=\mathbb{P}(\underline{\mathbb{C}}\oplus
E_{j}^{(1)}\oplus\cdots\oplus E_{j}^{(n_{j})})$, where
$\underline{\mathbb{C}}$ is the trivial line bundle over $B_{j-1}$,
$E_{j}^{(k)}$ is a holomorphic line bundle over $B_{j-1}$ for $k=1,\dots
n_{j}$, and $\mathbb{P}(\cdot)$ denotes the projectivization. We call $B_{j}$
a $j$-stage generalized Bott manifold.

The construction of the generalized Bott tower is as follows. A 1-step
generalized Bott tower can be written as
$B_{1}=\mathbb{C}P^{n_{1}}=(\mathbb{C}^{n_{1}+1})^{\times}/\mathbb{C}^{\times}$,
where $\mathbb{C}^{\times}$ acts diagonally. We construct a line bundle over
$B_{1}$ by
$E_{2}^{(k)}=(\mathbb{C}^{n_{1}+1})^{\times}\times_{\mathbb{C}^{\times}}\mathbb{C}$
for $k=1,\dots,n_{2}$, where $\mathbb{C}^{\times}$ acts on $\mathbb{C}$ by
$a:v\mapsto a^{-c_{k}}v$ for some integer $c_{k}$. In $E_{2}^{(k)}$ we have
$[z_{1,0},\dots,z_{1,n_{1}},v]=[z_{1,0}a,\dots,z_{1,n_{1}}a,a^{c_{k}}v]$ for
all $a\in\mathbb{C}^{\times}$. A 2-step generalized Bott tower
$B_{2}=\mathbb{P}(\underline{\mathbb{C}}\oplus E_{2}^{(1)}\oplus\cdots\oplus
E_{2}^{(n_{2})})$ can be written as
$B_{2}=((\mathbb{C}^{n_{1}+1})^{\times}\times(\mathbb{C}^{n_{2}+1})^{\times})/G$,
where the right action of $G=(\mathbb{C}^{\times})^{2}$ is given by

$(\mathbf{z}_{1},\mathbf{z}_{2})\cdot(a_{1},a_{2})=(z_{1,0}a_{1},z_{1,1}a_{1},\dots,z_{1,n_{1}}a_{1},z_{2,0}a_{2},a_{1}^{c_{1}}z_{2,1}a_{2},\dots,a_{1}^{c_{n_{2}}}z_{2,n_{2}}a_{2}),$

where $\mathbf{z}_{j}=(z_{j,0},z_{j,1},\dots,z_{j,n_{j}})$ for $j=1,2$.

We can construct higher generalized Bott tower in a similar way. In this way
we get an $m$-step generalized Bott manifold
$B_{m}=\mathbb{P}(\underline{\mathbb{C}}\oplus E_{m}^{(1)}\oplus\cdots\oplus
E_{m}^{(n_{m})})$ from any collection of integers $\\{c_{i,j}^{(k)}\\}$:

$B_{m}=((\mathbb{C}^{n_{1}+1})^{\times}\times\cdots\times(\mathbb{C}^{n_{m}+1})^{\times})/G,$

where the right action of $G=(\mathbb{C}^{\times})^{m}$ is given by

$(\mathbf{z}_{1},\dots,\mathbf{z}_{m})\cdot\mathbf{a}=(\mathbf{z}_{1}^{\prime},\mathbf{z}_{2}^{\prime},\dots,\mathbf{z}_{m}^{\prime}),$

where $\mathbf{z}_{i}=(z_{i,0},\dots,z_{i,n_{i}})$ for $i=1,\dots,m$,
$\mathbf{a}=(a_{1},\dots,a_{m})\in(\mathbb{C}^{\times})^{m}$,
$\mathbf{z}_{1}^{\prime}=(z_{1,0}a_{1},z_{1,1}a_{1},\dots,z_{1,n_{1}}a_{1})$
and $\mathbf{z}_{j}^{\prime}=(z_{j,0}a_{j},a_{1}^{c_{1,j}^{(1)}}\cdots
a_{j-1}^{c_{j-1,j}^{(1)}}z_{j,1}a_{j},\dots,a_{1}^{c_{1,j}^{(n_{j})}}\cdots
a_{j-1}^{c_{j-1,j}^{(n_{j})}}z_{j,n_{j}}a_{j})$ for $j=2,\dots,m$. We can
construct a line bundle over $B_{m}$ from the integers
$(\ell_{1},\dots,\ell_{m})$ by

$\mathbf{L}=((\mathbb{C}^{n_{1}+1})^{\times}\times\cdots\times(\mathbb{C}^{n_{m}+1})^{\times})\times_{G}\mathbb{C},$

where $G=(\mathbb{C}^{\times})^{m}$ acts by

$((\mathbf{z}_{1},\dots,\mathbf{z}_{m}),v)\cdot\mathbf{a}=(\mathbf{z}_{1}^{\prime},\mathbf{z}_{2}^{\prime},\dots,\mathbf{z}_{m}^{\prime},a_{1}^{\ell_{1}}\cdots
a_{m}^{\ell_{m}}v).$ (2.1)

###  2.2 Torus action on generalized Bott towers

Let $N=\sum_{j=1}^{m}n_{j}$ and let $T^{N}=S^{1}\times\cdots\times S^{1}$. We
consider the action of $T^{N}$ on $B_{m}$ as follows:

$(\mathbf{t}_{1},\dots,\mathbf{t}_{m})\cdot[\mathbf{z}_{1},\dots,\mathbf{z}_{m}]=[\mathbf{t}_{1}\cdot\mathbf{z}_{1},\dots,\mathbf{t}_{m}\cdot\mathbf{z}_{m}],$

where $\mathbf{t}_{i}=(t_{i,1},\dots,t_{i,n_{i}})$ and
$\mathbf{t}_{i}\cdot\mathbf{z}_{i}=(z_{i,0},t_{i,1}z_{i,1},\dots,t_{i,n_{i}}z_{i,n_{i}})$
for $i=1,\dots,m$. Also we consider the action of $T=T^{N}\times S^{1}$ on
$\mathbf{L}$ as follows:

$(\mathbf{t}_{1},\dots,\mathbf{t}_{m},t_{m+1})\cdot[\mathbf{z}_{1},\dots,\mathbf{z}_{m},v]=[\mathbf{t}_{1}\cdot\mathbf{z}_{1},\dots,\mathbf{t}_{m}\cdot\mathbf{z}_{m},t_{m+1}v].$
(2.2)

###  2.3 Generalized twisted cubes

###### Definition 2.2

A generalized twisted cube $C$ is defined to be the set of
$x=(x_{1,1},\dots,x_{m,n_{m}})$
$\in\mathbb{R}^{N}$ which satisfies

$\displaystyle A_{i}(x)\leq\sum_{k=1}^{n_{i}}x_{i,k}\leq 0,\,\,x_{i,k}\leq
0\,\,\,\,(1\leq k\leq n_{i})$ $\displaystyle{\rm
or}\,\,0<\sum_{k=1}^{n_{i}}x_{i,k}<A_{i}(x),\,\,x_{i,k}>0\,\,\,\,(1\leq k\leq
n_{i}),$ (2.3)

for all $1\leq i\leq m$, where

$A_{i}(x)=\begin{cases}-\ell_{m}&(i=m)\\\
-(\ell_{i}+\sum_{j=i+1}^{m}\sum_{k=1}^{n_{j}}c_{i,j}^{(k)}x_{j,k})&(1\leq
i\leq m-1).\end{cases}$

###### Remark 2.3

(i) The generalized twisted cube is a special case of multi-polytope defined
in [7]. In particular, it is a special case of twisted polytope defined in
[9].

(ii) When $n_{i}=1$ for all $1\leq i\leq m$, the generalized twisted cube is
the twisted cube given in [6, (2.21)].

###### Definition 2.4

We define ${\rm sgn}(x_{i,k})=1$ for $x_{i,k}>0$ and ${\rm sgn}(x_{i,k})=-1$
for $x_{i,k}\leq 0$. The density function of the generalized twisted cube is
then defined to be $\rho(x)=(-1)^{N}\prod_{i,k}{\rm sgn}(x_{i,k})$ when $x\in
C$ and $0$ elsewhere.

###### Example 2.5

Suppose that $m=2,n_{1}=1,n_{2}=2,\ell_{1}=1$, and $\ell_{2}=2$. We set
$c_{1,2}^{(1)}=2$ and $c_{1,2}^{(2)}=-1$. Then the generalized twisted cube is
the set of $x=(x_{1,1},x_{2,1},x_{2,2})$ which satisfies

  * •

$-2\leq x_{2,1}+x_{2,2}\leq 0,\,\,x_{2,1},x_{2,2}\leq 0,$

  * •

$-1-2x_{2,1}+x_{2,2}\leq x_{1,1}\leq
0\,\,\textup{or}\,\,0<x_{1,1}<-1-2x_{2,1}+x_{2,2}.$

In Figure 1, the black dots represent the lattice points of the sign $+1$ and
the white dots represent the sign $-1$.

O$(0,0,-2)$$(-3,0,-2)$$(x_{1,1},x_{2,1},x_{2,2})=(0,-2,0)$$(3,-2,0)$

Figure 1

###### Example 2.6

Suppose that $m=2,n_{1}=2,n_{2}=1,\ell_{1}=2$, and $\ell_{2}=-6$. We set
$c_{1,2}^{(1)}=-1$. Then the generalized twisted cube is the set of
$x=(x_{1,1},x_{1,2},x_{2,1})$ which satisfies

  * •

$0<x_{2,1}<6,$

  * •

$-2+x_{2,1}\leq x_{1,1}+x_{1,2}\leq 0,\,\,x_{1,1},x_{1,2}\leq
0\,\,\textup{or}\,\,0<x_{1,1}+x_{1,2}<-2+x_{2,1},\,\,x_{1,1},x_{1,2}>0.$

In Figure 2, the white dots represent the sign $-1$.

O$(0,-2,0)$$(0,0,2)$$(x_{1,1},x_{1,2},x_{2,1})=(0,0,6)$$(0,4,6)$$(4,0,6)$$(1,1,5)$

Figure 2

###### Example 2.7

Suppose that $m=2,n_{1}=n_{2}=2,\ell_{1}=1$, and $\ell_{2}=2$. We set
$c_{1,2}^{(1)}=2$ and $c_{1,2}^{(2)}=-1$. Then the generalized twisted cube is
the set of $x=(x_{1,1},x_{1,2},x_{2,1},x_{2,2})$ which satisfies

  * •

$-2\leq x_{2,1}+x_{2,2}\leq 0,\,\,x_{2,1},x_{2,2}\leq 0,$

  * •

$-1-2x_{2,1}+x_{2,2}\leq x_{1,1}+x_{1,2}\leq 0,\,\,x_{1,1},x_{1,2}\leq 0\\\
\textup{or}\,\,0<x_{1,1}+x_{1,2}<-1-2x_{2,1}+x_{2,2},\,\,x_{1,1},x_{1,2}>0.$

The lattice points in the generalized twisted cube represent the sign $-1$.

###  2.4 Equivariant index

Let $\mathbf{L}$ be a holomorphic line bundle over a generalized Bott manifold
$B_{m}$ with the action of the torus $T$ as in (2.2). Let
$\mathcal{O}_{\mathbf{L}}$ be the sheaf of holomorphic sections. The
equivariant index of a generalized Bott manifold is the formal sum of
representation of $T$:

${\rm
index}(B_{m},\mathcal{O}_{\mathbf{L}})=\sum(-1)^{i}H^{i}(B_{m},\mathcal{O}_{\mathbf{L}})$

The character of the equivariant index is the function $\chi:T\to\mathbb{C}$
which is given by $\chi=\sum(-1)^{i}\chi^{i}$ where
$\chi^{i}(a)=\text{trace}\\{a:H^{i}(B_{m},\mathcal{O}_{\mathbf{L}})\to
H^{i}(B_{m},\mathcal{O}_{\mathbf{L}})\\}$ for $a\in T$. Let $\mathfrak{t}$ be
the Lie algebra of $T$ and let $\mathfrak{t}^{\ast}$ be its dual space. Every
$\mu$ in the integral weight lattice $\ell^{\ast}\subset i\mathfrak{t}^{\ast}$
defines a homomorphism $\lambda^{\mu}:T\to S^{1}$. We can write
$\chi=\sum_{\mu\in\ell^{\ast}}m_{\mu}\lambda^{\mu}$. The coefficients are
given by a function $\text{mult}\,:\ell^{\ast}\to\mathbb{Z}$, sending
$\mu\mapsto m_{\mu}$, called the multiplicity function for the equivariant
index.

##  3 Main theorems

###  3.1 Multiplicity function of the equivariant index

We will show that the multiplicity function of the equivariant index of a
generalized Bott manifold is given by the density function of a generalized
twisted cube $C$. In particular, all the weights occur with a multiplicity
$-1,0$, or $1$.

###### Theorem 3.1

Fix integers $\\{c_{i,j}^{(k)}\\}$ and $\\{\ell_{j}\\}$. Let $\mathbf{L}\to
B_{m}$ be the corresponding line bundle over a generalized Bott manifold. Let
$\rho\,:\,\mathbb{R}^{N}\to\\{-1,0,1\\}$ be the density function of the
generalized twisted cube $C$ which is determined by these integers as in
(2.2). Consider the torus action of $T=T^{N}\times S^{1}$ as in (2.2). Then
the multiplicity function for $\ell^{\ast}\cong\mathbb{Z}^{N}\times\mathbb{Z}$
is given by

${\rm mult}(x,k)=\begin{cases}\rho(x)&(k=1)\\\ 0&(k\neq 1).\end{cases}$

Proof ; We compute $H^{\ast}(B_{m},\mathcal{O}_{\mathbf{L}})$. Take the
covering $\tilde{\mathcal{U}}=\\{U_{r_{1}}\times\cdots\times U_{r_{m}}\\}$ of
$(\mathbb{C}^{n_{1}+1})^{\times}\times\cdots\times(\mathbb{C}^{n_{m}+1})^{\times}$
for $r_{1},\dots,r_{m}\in\\{0,1,\dots,n_{\ell}\\}\,\,(\ell=1,\dots,m)$, where
$U_{r_{j}}=\underbrace{\mathbb{C}\times\cdots\times\mathbb{C}}_{r_{j}}\times\mathbb{C}^{\times}\times\underbrace{\mathbb{C}\times\cdots\times\mathbb{C}}_{n_{\ell}-r_{j}}$.
This descends to the covering $\mathcal{U}$ of $B_{m}$; every intersection of
sets in $\mathcal{U}$ is isomorphic to a product of $\mathbb{C}$’s and
$\mathbb{C}^{\times}$’s. The coverings $\tilde{\mathcal{U}}$ and $\mathcal{U}$
are the Leray coverings ([5]).

Let $\mathcal{O}$ be the sheaf of holomorphic functions, and let
$G=(\mathbb{C}^{\times})^{m}$. Since holomorphic sections of
$\mathcal{O}_{\mathbf{L}}$ are given by holomorphic sections of $\mathcal{O}$
which are $G$-invariant with respect to the action (2.1) ([9]),
$H^{\ast}(\mathcal{U},\mathcal{O}_{\mathbf{L}})$ is isomorphic to the
$G$-invariant part of $H^{\ast}(\tilde{\mathcal{U}},\mathcal{O})$. By the
Leray theorem, $H^{\ast}(B_{m},\mathcal{O}_{\mathbf{L}})$ is isomorphic to the
$G$-invariant part of
$H^{\ast}((\mathbb{C}^{n_{1}+1})^{\times}\times\cdots\times(\mathbb{C}^{n_{m}+1})^{\times},\mathcal{O})$.

In order to compute
$H^{\ast}((\mathbb{C}^{n_{1}+1})^{\times}\times\cdots\times(\mathbb{C}^{n_{m}+1})^{\times},\mathcal{O})$,
we compute $H^{\ast}((\mathbb{C}^{n+1})^{\times},\mathcal{O})$. Let
$\mathcal{U}^{\prime}=\\{U_{0},U_{1},\dots,U_{n}\\}$ be the covering of
$(\mathbb{C}^{n+1})^{\times}$, let
$j_{0},j_{1},\dots,j_{k}\in\\{0,1,\dots,n\\}$ for $k=0,1,\dots,n$ and let
$U_{j_{0}j_{1}\cdots j_{k}}=U_{j_{0}}\cap U_{j_{1}}\cap\cdots\cap U_{j_{k}}$.
Let $I=(i_{0},i_{1},\dots,i_{n})\in\mathbb{Z}^{n+1}$. The holomorphic
functions on $U_{j_{0}j_{1}\cdots j_{k}}$ are given by

$\Gamma_{\rm hol}(U_{j_{0}j_{1}\cdots
j_{k}})=\left\\{\sum_{I\in\mathbb{Z}^{n+1},i_{\ell}\geq 0(\ell\neq
j_{0},j_{1},\dots,j_{k})}a_{I}z_{0}^{i_{0}}z_{1}^{i_{1}}\cdots
z_{n}^{i_{n}}\right\\}.$

Consider the Čech cochain complex

$0\to\check{C}^{0}(\mathcal{U}^{\prime},\mathcal{O})\stackrel{{\scriptstyle\delta^{0}}}{{\to}}\check{C}^{1}(\mathcal{U}^{\prime},\mathcal{O})\stackrel{{\scriptstyle\delta^{1}}}{{\to}}\cdots\stackrel{{\scriptstyle\delta^{n-1}}}{{\to}}\check{C}^{n}(\mathcal{U}^{\prime},\mathcal{O})\stackrel{{\scriptstyle\delta^{n}}}{{\to}}0,$

where $\check{C}^{i}(\mathcal{U}^{\prime},\mathcal{O})=\oplus\Gamma_{\rm
hol}(U_{j_{0}j_{1}\cdots j_{i}})\,\,(i=0,\dots,n)$. The map
$\delta^{p}:\check{C}^{p}(\mathcal{U}^{\prime},\mathcal{O})\to\check{C}^{p+1}(\mathcal{U}^{\prime},\mathcal{O})$
is given by $\\{f_{j_{0}j_{1}\cdots j_{p}}\\}\mapsto\\{g_{j_{0}j_{1}\cdots
j_{p+1}}\\},g_{j_{0}j_{1}\cdots
j_{p+1}}=\sum(-1)^{k}f_{j_{0}j_{1}\cdots\hat{j_{k}}\cdots j_{p+1}}$. Recall
that
$H^{0}((\mathbb{C}^{n+1})^{\times},\mathcal{O})={\rm Ker}\,\delta^{0}$, and
$H^{n}((\mathbb{C}^{n+1})^{\times},\mathcal{O})={\rm Coker}\,\delta^{n-1}$.
The torus $T^{n+1}=(S^{1})^{n+1}$ acts on the holomorphic functions by
$((t_{0},\dots,t_{n})\cdot
f)(z_{0},\dots,z_{n})=f(t_{0}^{-1}z_{0},\dots,t_{n}^{-1}z_{n})$. This action
descends to the cohomology. The corresponding weight spaces for the weight
$I\in\mathbb{Z}^{n+1}$ are

$\displaystyle H^{0}((\mathbb{C}^{n+1})^{\times},\mathcal{O})_{I}$
$\displaystyle=\begin{cases}{\rm span}(z_{0}^{-i_{0}}\cdots
z_{n}^{-i_{n}})&(I\in\mathbb{Z}_{\leq 0}^{n+1})\\\
0&\text{otherwise}\end{cases}$ $\displaystyle
H^{n}((\mathbb{C}^{n+1})^{\times},\mathcal{O})_{I}$
$\displaystyle=\begin{cases}{\rm span}(z_{0}^{-i_{0}}\cdots
z_{n}^{-i_{n}})&(I\in\mathbb{Z}_{>0}^{n+1})\\\ 0&\text{otherwise}.\end{cases}$

We now prove $H^{q}((\mathbb{C}^{n+1})^{\times},\mathcal{O})=0$ for $1\leq
q\leq n-1$. Let $\Delta$ be the fan of $(\mathbb{C}^{n+1})^{\times}$, and let
$|\Delta|=\cup_{\sigma\in\Delta}\sigma$. Let

$Z(I):=\\{v\in|\Delta|\,;\,\langle I,v\rangle\leq\varphi(v)\\},$

where $\varphi$ is the support function. From [4],

$H^{q}((\mathbb{C}^{n+1})^{\times},\mathcal{O})_{I}=H^{q}(|\Delta|,|\Delta|\setminus
Z(I)\,;\,\mathbb{C}).$

Since $\mathcal{O}$ is the sheaf of holomorphic function, $\varphi(v)=0$ for
all $v\in|\Delta|$. In the case that $i_{j}\leq 0$ for all $j$, since
$|\Delta|$ is contractible,

$H^{q}((\mathbb{C}^{n+1})^{\times},\mathcal{O})_{I}=0\,\,(q\geq 1).$

In the case that $i_{j}>0$ for all $j$, $Z(I)=\\{0\\}$. Since
$|\Delta|\setminus\\{0\\}$ is homotopic to $S^{n-1}$,

$H^{q}((\mathbb{C}^{n+1})^{\times},\mathcal{O})_{I}=0\,\,(q\neq n).$

In other case, since $|\Delta|\setminus Z(I)$ is path-connected and
contractible,

$H^{q}((\mathbb{C}^{n+1})^{\times},\mathcal{O})_{I}=0$

for all $q$.

We now compute
$H^{\ast}((\mathbb{C}^{n_{1}+1})^{\times}\times\cdots\times(\mathbb{C}^{n_{m}+1})^{\times},\mathcal{O})$.
Consider the natural action of $T^{N+m}=(S^{1})^{N+m}$ on the holomorphic
function. The weights are multi-indices $I^{\prime}\in\mathbb{Z}^{N+m}$; we
write $I^{\prime}=(\mathbf{i}^{\prime}_{1},\dots,\mathbf{i}^{\prime}_{m})$,
where $\mathbf{i}^{\prime}_{j}=(i_{j,0},i_{j,1},\dots,i_{j,n_{j}})$ for
$j=1,\dots,m$. From the cohomology of $(\mathbb{C}^{n+1})^{\times}$ that we
have computed and from the Künneth formula ([1]), it follows that

$H^{q}((\mathbb{C}^{n_{1}+1})^{\times}\times\cdots\times(\mathbb{C}^{n_{m}+1})^{\times},\mathcal{O})_{I^{\prime}}=\begin{cases}\text{span}(z_{1,0}^{-i_{1,0}}z_{1,1}^{-i_{1,1}}\cdots
z_{m,n_{m}}^{-i_{m,n_{m}}})\\\ 0.\end{cases}$

The former occurs if for all $\ell$ we have ${\rm sgn}(i_{\ell,0})={\rm
sgn}(i_{\ell,1})=\cdots={\rm sgn}(i_{\ell,n_{\ell}})=:\varepsilon_{\ell}$,
here $q=\sum_{\\{\ell\,|\,\varepsilon_{\ell}=1,1\leq\ell\leq m\\}}n_{\ell}$,
and $q=0$ when $\varepsilon_{\ell}=-1$ for all $\ell$. In particular,
$(-1)^{q}=(-1)^{N}\prod_{1\leq\ell\leq m,1\leq p\leq n_{\ell}}{\rm
sgn}(i_{\ell,p})$.

The action (2.1) induces an action on functions given by

$\displaystyle(a_{k}f)(z_{1,0},\dots,z_{m,n_{m}})=a_{k}^{\ell_{k}}f(z_{1,0},\dots,z_{k-1,n_{k-1}},$
$\displaystyle
z_{k,0}a_{k}^{-1},z_{k,1}a_{k}^{-1},\dots,z_{k,n_{k}}a_{k}^{-1},$
$\displaystyle\dots,z_{\ell,0},a_{k}^{-c_{k,\ell}^{(1)}}z_{\ell,1},\dots,a_{k}^{-c_{k,\ell}^{(n_{\ell})}}z_{\ell,n_{\ell}},\dots).$

The monomial $z_{1,0}^{-i_{1,0}}z_{1,1}^{-i_{1,1}}\cdots
z_{m,n_{m}}^{-i_{m,n_{m}}}$ is then a weight vector with a weight whose $k$-th
coordinate is
$\ell_{k}+i_{k,0}+\cdots+i_{k,n_{k}}+\sum_{\ell=k+1}^{m}\sum_{p=1}^{n_{\ell}}c_{k,\ell}^{(p)}i_{\ell,p}$.
Thus the $G$-invariant part of
$H^{\ast}((\mathbb{C}^{n_{1}+1})^{\times}\times\cdots\times(\mathbb{C}^{n_{m}+1})^{\times},\mathcal{O})$
consists of those monomials $z_{1,0}^{-i_{1,0}}z_{1,1}^{-i_{1,1}}\cdots
z_{m,n_{m}}^{-i_{m,n_{m}}}$ for which

$\displaystyle\ell_{1}+i_{1,0}+\cdots+i_{1,n_{1}}+\sum_{\ell=2}^{m}\sum_{p=1}^{n_{\ell}}c_{1,\ell}^{(p)}i_{\ell,p}$
$\displaystyle=0$
$\displaystyle\ell_{2}+i_{2,0}+\cdots+i_{2,n_{2}}+\sum_{\ell=3}^{m}\sum_{p=1}^{n_{\ell}}c_{2,\ell}^{(p)}i_{\ell,p}$
$\displaystyle=0$ (3.1) $\displaystyle\vdots$
$\displaystyle\ell_{m}+i_{m,0}+\cdots+i_{m,n_{m}}$ $\displaystyle=0.$

The action (2.2) induces a $T$ action on the functions given by

$\displaystyle((t_{1,1},\dots,t_{m,n_{m}},t_{m+1})\cdot
f)(z_{1,0},\dots,z_{m,n_{m}})$
$\displaystyle=t_{m+1}f(z_{1,0},t_{1,1}^{-1}z_{1,1},\dots,z_{m,0},t_{m,1}^{-1}z_{m,1},\dots,t_{m,n_{m}}^{-1}z_{m,n_{m}}).$

The weight of the monomial $z_{1,0}^{-i_{1,0}}z_{1,1}^{-i_{1,1}}\cdots
z_{m,n_{m}}^{-i_{m,n_{m}}}$ with respect to this $T$ action is
$(\mathbf{i}_{1},\mathbf{i}_{2},\dots,\mathbf{i}_{m},1)$, where
$\mathbf{i}_{j}=(i_{j,1},\dots,i_{j,n_{j}})$ for $j=1,\dots,m$. Thus the index
of $(B_{m},\mathcal{O}_{\mathbf{L}})$ is given by the set of
$x=(x_{1,1},\dots,x_{m,n_{m}},1)=(i_{1,1},\dots,i_{m,n_{m}},1)$ for which
there exist $(i_{1,0},\dots,i_{m,0})$ such that (3.1) is satisfied and such
that ${\rm sgn}(i_{\ell,0})={\rm sgn}(i_{\ell,1})=\cdots={\rm
sgn}(i_{\ell,n_{\ell}})$ for all $\ell$. This is exactly the set (2.2).
Therefore the multiplicity of the equivariant index is
$(-1)^{N}\prod_{1\leq\ell\leq m,1\leq p\leq n_{\ell}}{\rm
sgn}(i_{\ell,p})=(-1)^{N}\prod_{1\leq\ell\leq m,1\leq p\leq n_{\ell}}{\rm
sgn}(x_{\ell,p})=\rho(x)$. $\square$

###  3.2 Character formula for the equivariant index

In the following the theorem we give a formula for the character
$\chi\,:\,T\to\mathbb{C}$ of the equivariant index of a generalized Bott
manifold. For every integral weight $\mu\in\ell^{\ast}$ we have a homomorphism
$\lambda^{\mu}\,:\,T\to S^{1}$. We denote the integral combinations of these
$\lambda^{\mu}$’s by $\mathbb{Z}[T]$. Then $\chi\in\mathbb{Z}[T]$ is given by
$\chi=\sum_{\mu\in\ell^{\ast}}m_{\mu}\lambda^{\mu}$ where $m_{\mu}={\rm
mult}(\mu)$.

###### Definition 3.2

Let $\\{e_{1,1},\dots,e_{m,n_{m}},e_{m+1}\\}$ be the standard basis in
$\mathbb{R}^{N+1}$, $x_{i}=\left(x_{i,1},\dots,\right.$
$\left.x_{i,n_{i}}\right)$ and $e_{i}=(e_{i,1},\dots,e_{i,n_{i}})$. Let
$\Delta_{n,r}^{-}=\left\\{z=(z_{1},\dots,z_{n})\in\mathbb{Z}_{\leq
0}^{n}\,\middle|\,z_{1}+\cdots+z_{n}=-r\right\\}$, and let
$\Delta_{n,r}^{+}=\left\\{z=(z_{1},\dots,z_{n})\in\mathbb{Z}_{>0}^{n}\,\middle|\,z_{1}+\cdots+z_{n}=r-1\right\\}$.
Then the operators $D_{i}:\mathbb{Z}[T]\to\mathbb{Z}[T]$ are defined using
$c_{i,j}^{(k)}$ and $\ell_{j}$ in the following way:

$D_{i}(\lambda^{\mu})=\begin{cases}\displaystyle\sum_{0\leq r\leq
k_{i}}\sum_{x_{i}\in\Delta_{n_{i},r}^{-}}\lambda^{\mu+\langle
x_{i},e_{i}\rangle}&\text{if}\,\,k_{i}\geq 0\\\ 0&\text{if}\,\,-n_{i}\leq
k_{i}\leq-1\\\ \displaystyle\sum_{n_{i}+1\leq r\leq-
k_{i}}\sum_{x_{i}\in\Delta_{n_{i},r}^{+}}(-1)^{n_{i}}\lambda^{\mu+\langle
x_{i},e_{i}\rangle}&\text{if}\,\,k_{i}\leq-n_{i}-1,\end{cases}$

where the functions $k_{i}$ are defined as follows: if
$\mu=e_{m+1}+\sum_{j=i+1}^{m}\sum_{k=1}^{n_{j}}x_{j,k}e_{j,k}$, then
$k_{i}(\mu)=\ell_{i}+\sum_{j=i+1}^{m}\sum_{k=1}^{n_{j}}c_{i,j}^{(k)}x_{j,k}$.

From Theorem 3.1, we immediately obtain the following theorem.

###### Theorem 3.3

Consider the action of the torus $T$ on $\mathbf{L}\to B_{m}$ as in (2.2).
Denote the $(N+1)$-th component of the standard basis in $\mathbb{R}^{N+1}$ by
$e_{m+1}$. Then the character is given by the following element of
$\mathbb{Z}[T]$:

$\chi=D_{1}\cdots D_{m}(\lambda^{e_{m+1}}).$

###### Remark 3.4

When $n_{i}=1$ for all $i$, the operator $D_{i}$ is given by

$D_{i}(\lambda^{\mu})=\begin{cases}\lambda^{\mu}+\lambda^{\mu-
e_{i,1}}+\cdots+\lambda^{\mu-k_{i}e_{i,1}}&{\it if}\,\,k_{i}\geq 0\\\ 0&{\it
if}\,\,k_{i}=-1\\\
-\lambda^{\mu+e_{i,1}}-\lambda^{\mu+2e_{i,1}}-\cdots-\lambda^{\mu-(k_{i}+1)e_{i,1}}&{\it
if}\,\,k_{i}\leq-2.\end{cases}$

We can check that this operator agrees with the one in [6, Proposition 2.32].

###### Example 3.5

Suppose that $m=2,n_{1}=1$, and $n_{2}=2$. We set
$\ell_{1}=1,\ell_{2}=2,c_{1,2}^{(1)}=2$, and $c_{1,2}^{(2)}=-1$ as in Example
2.5. Then the corresponding character $\chi$ is given by

$\displaystyle\chi$ $\displaystyle=D_{1}D_{2}(\lambda^{e_{3}})$
$\displaystyle=D_{1}(\lambda^{e_{3}}+\lambda^{e_{3}-e_{2,1}}+\lambda^{e_{3}-e_{2,2}}+\lambda^{e_{3}-2e_{2,1}}+\lambda^{e_{3}-e_{2,1}-e_{2,2}}+\lambda^{e_{3}-2e_{2,2}})$
$\displaystyle=\lambda^{e_{3}}+\lambda^{e_{3}-e_{1,1}}+\lambda^{e_{3}-e_{2,2}}+\lambda^{e_{3}-e_{2,2}-e_{1,1}}+\lambda^{e_{3}-e_{2,2}-2e_{1,1}}-\lambda^{e_{3}-2e_{2,1}+e_{1,1}}-\lambda^{e_{3}-2e_{2,1}+2e_{1,1}}$
$\displaystyle\,\,\,\,+\lambda^{e_{3}-e_{2,1}-e_{2,2}}+\lambda^{e_{3}-2e_{2,2}}+\lambda^{e_{3}-2e_{2,2}-e_{1,1}}+\lambda^{e_{3}-2e_{2,2}-2e_{1,1}}+\lambda^{e_{3}-2e_{2,2}-3e_{1,1}}.$

###### Example 3.6

Suppose that $m=2,n_{1}=2$, and $n_{2}=1$. We set $\ell_{1}=2,\ell_{2}=-6$,
and $c_{1,2}^{(1)}=-1$ as in Example 2.6. Then the corresponding character
$\chi$ is given by

$\displaystyle\chi$ $\displaystyle=D_{1}D_{2}(\lambda^{e_{3}})$
$\displaystyle=D_{1}(-\lambda^{e_{3}+e_{2,1}}-\lambda^{e_{3}+2e_{2,1}}-\lambda^{e_{3}+3e_{2,1}}-\lambda^{e_{3}+4e_{2,1}}-\lambda^{e_{3}+5e_{2,1}})$
$\displaystyle=-\lambda^{e_{3}+e_{2,1}}-\lambda^{e_{3}+e_{2,1}-e_{1,1}}-\lambda^{e_{3}+e_{2,1}-e_{1,2}}-\lambda^{e_{3}+2e_{2,1}}-\lambda^{e_{3}+5e_{2,1}+e_{1,1}+e_{1,2}}.$

###### Example 3.7

Suppose that $m=2,n_{1}=2$, and $n_{2}=2$. We set
$\ell_{1}=1,\ell_{2}=2,c_{1,2}^{(1)}=2$, and $c_{1,2}^{(2)}=-1$ as in Example
2.7. Then the corresponding character $\chi$ is given by

$\displaystyle\chi$ $\displaystyle=D_{1}D_{2}(\lambda^{e_{3}})$
$\displaystyle=D_{1}(\lambda^{e_{3}}+\lambda^{e_{3}-e_{2,1}}+\lambda^{e_{3}-e_{2,2}}+\lambda^{e_{3}-2e_{2,1}}+\lambda^{e_{3}-e_{2,1}-e_{2,2}}+\lambda^{e_{3}-2e_{2,2}})$
$\displaystyle=\lambda^{e_{3}}+\lambda^{e_{3}-e_{1,1}}+\lambda^{e_{3}-e_{1,2}}+\lambda^{e_{3}-e_{2,2}}+\lambda^{e_{3}-e_{2,2}-e_{1,1}}+\lambda^{e_{3}-e_{2,2}-e_{1,2}}+\lambda^{e_{3}-e_{2,2}-2e_{1,1}}$
$\displaystyle\,\,\,\,+\lambda^{e_{3}-e_{2,2}-e_{1,1}-e_{1,2}}+\lambda^{e_{3}-e_{2,2}-2e_{1,2}}+\lambda^{e_{3}-2e_{2,1}+e_{1,1}+e_{1,2}}+\lambda^{e_{3}-e_{2,1}-e_{2,2}}+\lambda^{e_{3}-2e_{2,2}}$
$\displaystyle\,\,\,\,+\lambda^{e_{3}-2e_{2,2}-e_{1,1}}+\lambda^{e_{3}-2e_{2,2}-e_{1,2}}+\lambda^{e_{3}-2e_{2,2}-2e_{1,1}}+\lambda^{e_{3}-2e_{2,2}-e_{1,1}-e_{1,2}}+\lambda^{e_{3}-2e_{2,2}-2e_{1,2}}$
$\displaystyle\,\,\,\,+\lambda^{e_{3}-2e_{2,2}-3e_{1,1}}+\lambda^{e_{3}-2e_{2,2}-2e_{1,1}-e_{1,2}}+\lambda^{e_{3}-2e_{2,2}-e_{1,1}-2e_{1,2}}+\lambda^{e_{3}-2e_{2,2}-3e_{1,2}}.$

###### Remark 3.8

We gave the formula for the character using the Demazure-type operators. On
the other hand, the character is also given by the localization formula ([7,
Corollary 7.4]). For example, when we set the parameters as in Example 3.5,
the character is computed using the localization formula as follows:

$\displaystyle\chi=\lambda^{e_{3}}$
$\displaystyle\left(\frac{1}{(1-\lambda^{-e_{1,1}})(1-\lambda^{-e_{2,1}})(1-\lambda^{-e_{2,2}})}+\frac{\lambda^{-2e_{2,2}}}{(1-\lambda^{-e_{1,1}})(1-\lambda^{-e_{2,1}+e_{2,2}})(1-\lambda^{e_{2,2}})}\right.$
$\displaystyle\left.+\frac{\lambda^{-2e_{2,1}}}{(1-\lambda^{-e_{1,1}})(1-\lambda^{e_{2,1}-e_{2,2}})(1-\lambda^{e_{2,1}})}+\frac{\lambda^{-e_{1,1}}}{(1-\lambda^{e_{1,1}})(1-\lambda^{2e_{1,1}-e_{2,1}})(1-\lambda^{-e_{1,1}-e_{2,2}})}\right.$
$\displaystyle\left.+\frac{\lambda^{-3e_{1,1}-2e_{2,2}}}{(1-\lambda^{e_{1,1}})(1-\lambda^{3e_{1,1}-e_{2,1}+e_{2,2}})(1-\lambda^{e_{1,1}+e_{2,2}})}\right.$
$\displaystyle\left.+\frac{\lambda^{3e_{1,1}-2e_{2,1}}}{(1-\lambda^{e_{1,1}})(1-\lambda^{-3e_{1,1}+e_{2,1}-e_{2,2}})(1-\lambda^{-2e_{1,1}+e_{2,1}})}\right).$

We can check that this result agrees with the result in Example 3.5.

## References

  * [1] A. Andreotti, and F. Norguet, Cycles of algebraic manifolds and $\partial\bar{\partial}$-cohomology, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze $3^{e}$ série, 25(1) (1971), 59–114.
  * [2] S. Choi, M. Masuda, and D. Y. Suh, Quasitoric manifolds over a product of simplices, Osaka J. Math., 47(1) (2010), 109–129.
  * [3] S. Choi, M. Masuda, and D. Y. Suh, Topological classification of generalized Bott towers, Trans. Amer. Math. Soc., 362(2) (2010), 1097–1112.
  * [4] W. Fulton, Introduction to Toric Varieties, Ann. of Math. Studies, Princeton University Press, 1993.
  * [5] P. Griffiths, and J. Harris, Principles of Algebraic Geometry, Wiley, New York, 1978.
  * [6] M. Grossberg, and Y. Karshon, Bott towers, complete integrability, and the extended character of representations, Duke math. J., 76(1) (1994), 23–58.
  * [7] A. Hattori, and M. Masuda, Theory of multi-fans, Osaka J. Math., 40 (2003), 1–68.
  * [8] T. Hwang, E. Lee, and D. Y. Suh, The Gromov width of generalized Bott manifolds, International Math. Research Notices, 2021(9) (2019), 7096–7131.
  * [9] Y. Karshon, and S. Tolman, The moment map and line bundles over presymplectic toric manifolds, J. Differential geometry, 38 (1993), 465–484.

Yuki Sugiyama
Department of Mathematics, Graduate School of Science and Engineering, Chuo
University,
Kasuga, Bunkyo-Ku, Tokyo, 112–8551 Japan.
e-mail<EMAIL_ADDRESS>