# Cohomology rings of quasitoric bundles

Askold Khovanskii  Department of Mathematics, University of Toronto, Toronto,
Canada; Moscow Independent University, Moscow, Russia.
<EMAIL_ADDRESS>,  Ivan Limonchenko  Department of Mathematics,
University of Toronto, Toronto, Canada; Faculty of Computer Science, HSE
University, Moscow, Russia<EMAIL_ADDRESS>and  Leonid Monin  Max Planck
Institute for Mathematics in the Sciences, Leipzig, Germany
<EMAIL_ADDRESS>
###### Abstract.

The classical BKK theorem computes the intersection number of divisors on
toric variety in terms of volumes of corresponding polytopes. In [PK92b], it
was observed by Pukhlikov and the first author that the BKK theorem leads to a
presentation of the cohomology ring of toric variety as a quotient of the ring
of differential operators with constant coefficients by the annihilator of an
explicit polynomial.

In this paper we generalize this construction to the case of quasitoric
bundles. These are fiber bundles with generalized quasitoric manifolds as
fibers. First we obtain a generalization of the BKK theorem to this case. Then
we use recently obtained descriptions of the graded-commutative algebras which
satisfy Poincaré duality to give a description of cohomology rings of
quasitoric bundles.

###### Key words and phrases:

Quasitoric manifolds, quasitoric bundles, moment-angle-complexes, Stanley-
Reisner rings

###### 2020 Mathematics Subject Classification:

57S12, 13F55, 55N45

##  1\. Introduction

In this paper we study cohomology rings of quasitoric bundles, a class of
fiber bundles over a closed orientable smooth manifold with a generalized
quasitoric manifold as a fiber. A generalized quasitoric manifold is a
generalization of a toric variety to smooth category, hence the first examples
of a quasitoric bundle are a nonsingular complete toric variety and a
quasitoric manifold, both viewed as fiber bundles over a point.

Description of cohomology rings of a toric variety. Let us start with a brief
recollection of the classical toric case. The cohomology ring of a smooth
toric variety has several descriptions. In this paper we mostly focus on the
following three descriptions: Stanley-Reisner, Brion’s and Pukhlikov-
Khovanskii.

Let $T\simeq({\mathbb{C}}^{*})^{n}$ be an algebraic torus with character
lattice $M$ and lattice of one-parameter subgroups $N$. Further, let
$X_{\Sigma}$ be a smooth projective toric variety given by a fan
$\Sigma\subseteq N_{\mathbb{R}}\coloneqq N\otimes_{\mathbb{Z}}{\mathbb{R}}$.
Denote the rays of $\Sigma$ by $\Sigma(1)=\\{\rho_{1},\ldots,\rho_{s}\\}$ and
their primitive generators in $N$ by $e_{1},\ldots,e_{s}$. Recall the
following well-known description of the cohomology ring of $X_{\Sigma}$ (see,
for instance, [CLS11, Theorem 12.4.1]):

$H^{*}(X_{\Sigma},{\mathbb{R}})\simeq{\mathbb{R}}[x_{1},\ldots,x_{s}]/(I+J)\eqqcolon
R_{\Sigma},$

where $I$ is generated by monomials $x_{i_{1}}\cdots x_{i_{t}}$ such that
$\rho_{i_{1}},\ldots,\rho_{i_{t}}\in\Sigma(1)$ are distinct and do not form a
cone in $\Sigma$ and
$J=\mathopen{}\mathclose{{}\left\langle\sum_{i=1}^{s}\chi(e_{i})x_{i}\colon\chi\in
M}\right\rangle$. Note that $I$ depicts the Stanley-Reisner ideal of the fan
$\Sigma$ and therefore we refer to this description as the _Stanley-Reisner
description_ of $H^{*}(X_{\Sigma},{\mathbb{R}})$.

On the other hand, Brion used explicit localization theorem in equivariant
cohomology to compute the cohomology ring of a smooth toric variety. Let $M$
be the character lattice of the torus $T$ and set
$M_{\mathbb{R}}=M\otimes{\mathbb{R}}$ and
$N_{\mathbb{R}}={\mathrm{Hom}}_{\mathbb{Z}}(M,{\mathbb{R}})$. Let $\Sigma$ be
a smooth projective fan in $N_{\mathbb{R}}$. A map $f\colon
N_{\mathbb{R}}\to{\mathbb{R}}$ is _piecewise polynomial_ if for any
$\sigma\in\Sigma$, the map $f|_{\sigma}\colon\sigma\to{\mathbb{R}}$ extends to
a polynomial function on the linear space
${\operatorname{span}}_{\mathbb{R}}\\{\sigma\\}$, i.e. a piecewise polynomial
function $f$ on $\Sigma$ is a collection of compatible polynomial functions
$f_{\sigma}\colon\sigma\to{\mathbb{R}}$.

Let us denote by $\mathrm{PP}_{\Sigma}$ the ring of piecewise polynomial
functions on $\Sigma$ with respect to pointwise addition and multiplication.
Note that the character lattice $M$ naturally injects in
$\mathrm{PP}_{\Sigma}$ as global linear functions on $N_{\mathbb{R}}$. Brion
shows that one has the following description of the cohomology ring of toric
variety:

$H^{*}(X_{\Sigma},{\mathbb{R}})\simeq\mathrm{PP}_{\Sigma}/\langle M\rangle,$

where $\langle M\rangle$ is the ideal generated by
$M\subset\mathrm{PP}_{\Sigma}$.

Given line bundles $L_{1},\ldots,L_{t}$ on $X_{\Sigma}$, one can directly
compute a top degree cup product $c_{1}(L_{1})^{k_{1}}\cdots
c_{1}(L_{t})^{k_{t}}$ in $H^{*}(X_{\Sigma},{\mathbb{R}})$ by using the BKK
theorem [Kou76] (see also [Ber75, BKH76]). Here, $c_{1}(L_{i})\in
H^{2}(X_{\Sigma},{\mathbb{R}})$ denotes the first Chern class of $L_{i}$. More
precisely, as any line bundle on $X_{\Sigma}$ is the difference of two ample
line bundles, it suffices to evaluate products $c_{1}(L_{1})^{k_{1}}\cdots
c_{1}(L_{t})^{k_{t}}$ with all $L_{i}$ ample. As is well-known in toric
geometry, the ample line bundles $L_{i}$ correspond to polytopes $P_{i}$ whose
normal fans coarsen the fan $\Sigma$. Using the BKK Theorem, we obtain

$c_{1}(L_{1})^{k_{1}}\cdots c_{1}(L_{t})^{k_{t}}=n!\cdot
V(\underbrace{P_{1},\ldots,P_{1}}_{k_{1}\;\text{times}},\ldots,\underbrace{P_{t},\ldots,P_{t}}_{k_{t}\;\text{times}})$

where $V(P_{1},\ldots,P_{1},\ldots,P_{t},\ldots,P_{t})$ denotes the mixed
volume of the $n$-tuple $(P_{1},\ldots,P_{1},\ldots,P_{t},\ldots,P_{t})$. In
[PK92b, Section 1.4], Pukhlikov and the first author observed that the
information on these cup products suffices to regain a description of the
cohomology ring $H^{*}(X_{\Sigma},{\mathbb{R}})$. More precisely, they showed
that

$H^{*}(X_{\Sigma},{\mathbb{R}})\simeq{\mathrm{Diff}}({\mathcal{P}}_{\Sigma})/{\mathrm{Ann}}({\mathrm{Vol}}),$

where ${\mathrm{Diff}}({\mathcal{P}}_{\Sigma})$ is the ring of differential
operators with constant coefficients on the space of virtual polytopes
associated with $\Sigma$ and ${\mathrm{Ann}}({\mathrm{Vol}})$ is the ideal of
differential operators which annihilate the volume polynomial. As virtual
polytopes play a crucial role in this description, we refer to it as the
_virtual polytope description_ of the cohomology ring
$H^{*}(X_{\Sigma},{\mathbb{R}})$.

Cohomology rings of quasitoric bundles. Now let $T\simeq(S^{1})^{n}$ be a
compact torus. A quasitoric manifold is a smooth $2n$-dimensional manifold
with a smooth effective locally standard $T$-action such that the orbit space
is diffeomorphic to a simple polytope $P$. This notion was introduced as a
topological counterpart of a nonsingular projective toric variety (or, toric
manifold) with the restricted action of
$(S^{1})^{n}\subset({\mathbb{C}}^{*})^{n}$ in the seminal paper [DJ91].
Quasitoric manifolds have been studied intensively since 1990s and have
already found various applications in homotopy theory [CMS08, HKS16, HK17],
unitary [BPR06, LW16] and special unitary bordism [LP16, LLP18], hyperbolic
geometry [BP16, BEM+17, BGLV20], and other areas of research.

Quasitoric manifolds acquire smooth structures as orbit spaces of moment-angle
manifolds by freely acting tori of maximal possible rank [BP02, BPR06]. In
Section 2, we define a generalized quasitoric manifold to be a partial
quotient obtained as an orbit space of a moment-angle-complex over a
starshaped sphere by a freely acting torus of the maximal possible rank. It is
a locally standard torus manifold [Mas99, HM03, MP06] with a poset of
characteristic submanifolds being isomorphic to a complete simplicial fan.

Similar to quasitoric manifolds, generalized quasitoric manifolds have a
combinatorial description: an omnioriented generalized quasitoric manifold is
uniquely determined by a pair $(\Sigma,\Lambda)$, where $\Sigma$ is a complete
simplicial fan and $\Lambda\colon\Sigma(1)\to N$ is a map such that
$\Lambda(\rho_{1}),\ldots,\Lambda(\rho_{l})$ can be extended to a basis of
$N$, whenever $\rho_{1},\ldots,\rho_{l}\in\Sigma(1)$ form a cone.

In particular, for a smooth complete fan $\Sigma\subset N$ and
$\Lambda(\rho)=e_{\rho}$, the primitive ray generator, the corresponding
generalized quasitoric manifold is equivariantly diffeomorphic to a
nonsingular complete toric variety $X_{\Sigma}$ (considered with the
restricted action of $(S^{1})^{n}\subset({\mathbb{C}}^{*})^{n}$).

For a principal $T$-bundle $p\colon E\to B$ over a closed orientable real
manifold $B$ and a generalized quasitoric manifold $X_{\Sigma,\Lambda}$, let
$E_{\Sigma,\Lambda}\coloneqq(E\times X_{\Sigma,\Lambda}/T)$ be the associated
quasitoric bundle (see Section 3 for details).

A particular example of quasitoric bundles is formed by toric bundles, where
we consider associated bundles $E_{\Sigma}\coloneqq(E\times X_{\Sigma}/T)$
coming from smooth complete toric varieties. A generalization of the Stanley-
Reisner type description for the cohomology ring
$H^{*}(E_{\Sigma},{\mathbb{R}})$ was obtained by Sankaran and Uma [SU03]. In
[Hof19], it was noticed that Brion’s description also generalizes to the case
of toric bundles. Finally, in [HKM20] a generalization of the virtual polytope
description was obtained for toric bundles with smooth projective fibers.
Descriptions of cohomology rings obtained in [Hof19] and [HKM20] showed to be
useful for computation of the ring of conditions of horospherical homogeneous
spaces. Such a computation involves taking the direct limit of cohomology
rings of toric bundles with the same base, which is descriptions of [Hof19,
HKM20] are more suitable.

On the other side, in [DJ91] the Stanley-Reisner description was extended to
the cohomology rings of quasitoric manifolds and in [DKU19] it was further
extended to the case of quasitoric bundles. Moreover, in [AM16] the cohomology
rings of quasitoric manifolds were described via volume polynomial on the
space of analogous multi-polytopes.

In this paper, we complete this picture and obtain a generalization of all the
three descriptions of the cohomology ring of a smooth toric variety to the
case of quasitoric bundles. In particular, we extend the results of [HKM20] to
the case of toric bundles with complete (but not necessarily projective)
fibers and give an independent proof of the results of [DKU19] in the
quasitoric bundle case.

Our main tool in the computation of cohomology rings is a topological version
of BKK theorem for quasitoric bundles which computes top-degree intersection
numbers of cohomology classes of $E_{\Sigma,\Lambda}$ (Theorem 3.5).

More precisely, we reduce the computation of intersection numbers on the
quasitoric bundle to intersection numbers on the base. This provides a “BKK
-type theorem” for any choice of a cohomology class in the base $\gamma\in
H^{*}(B,{\mathbb{R}})$.

We also provide an equivalent version of the generalized BKK Theorem in
Section 3. It can be made as follows. Suppose the torus $T$ has rank $n$ and
the real dimension of $B$ is $k$. Similar to the toric case, a multi-polytope
$\Delta$ defines a cohomology class $\rho(\Delta)\in
H^{2}(E_{\Sigma,\Lambda},{\mathbb{R}})$ on the quasitoric bundle $p\colon
E_{\Sigma,\Lambda}\to B$. For any given $j$, we define a map which associates
to a multi-polytope $\Delta$ a cohomology class $\gamma_{2j}(\Delta)\in
H^{2j}(B,{\mathbb{R}})$ such that

$\rho(\Delta)^{n+j}\cdot p^{*}(\gamma)=\gamma_{2j}(\Delta)\cdot\gamma\text{,}$

for any $\gamma\in H^{k-2j}(B,{\mathbb{R}})$. Here “$\cdot$” denotes the cup
product on the respective cohomology ring. We call the class
$\gamma_{2j}(\Delta)$ _the horizontal part_ of $\rho(\Delta)^{n+j}$. In
Theorem 3.6, we compute the horizontal part of $\rho(\Delta)^{n+j}$ for any
multi-polytope $\Delta$.

######  Remark 1.1.

In this paper we deal with a topological version of BKK theorem. In
particular, quasitoric bundles are locally trivial fiber bundles in smooth
category over a smooth compact base. Thus, Theorems 3.5 and 3.6 are natural
extensions of the topological BKK theorems for toric bundles obtained in
[HKM20].

On the other hand, if one works only with algebraic toric bundles, one can use
the theory of Newton-Okounkov bodies to relax assumptions on the base to be
smooth or compact. We will address this direction in the future work.

Our version of the BKK theorem allows us to give a Pukhlikov-Khovanskii type
description of the cohomology ring of quasitoric bundles. To do so we use the
description of graded commutative algebras which satisfy Poincaré duality
obtained in [HKM20]. We then use this description to further prove a version
of Stanley-Reisner and Brion descriptions of the cohomology ring of a
quasitoric bundle in Theorems 6.4 and 6.5 respectively.

In the case when the base $B$ is an even-dimensional manifold (e.g. a
quasitoric manifold), the subring

$A=\bigoplus H^{2i}(E_{\Sigma,\Lambda},{\mathbb{R}})$

of even-dimensional cohomology classes is a commutative Gorenstein ring. Using
our version of the BKK theorem stated in Section 3 and proved in Section 4 and
the exact computation of a Macaulay’s inverse system from [KM21] and recalled
in Section 5, we obtain an alternative description of the ring $A$ (see
Theorem 6.3) in Section 6.

### Notations

We will denote by $\cdot$ the cup product in cohomology. For a manifold $B$
and a cohomology class $\gamma\in H^{*}(B,{\mathbb{R}})$, we will denote by
$\langle\gamma,[B]\rangle$ the pairing of $\gamma$ with the fundamental class
of $B$.

### Acknowledgements

The authors thank the Fields Institute for Research in Mathematical Sciences
for the opportunity to work on this research project during the Thematic
Program on Toric Topology and Polyhedral Products. We are grateful to Anton
Ayzenberg, Victor Buchstaber, Johannes Hofscheier and Taras Panov for several
helpful and inspiring conversations.

The first author is partially supported by the Canadian Grant No. 156833-17.
The research of the second author has been funded within the framework of the
HSE University Basic Research Program and is also supported in part by Young
Russian Mathematics award.

##  2\. Generalized quasitoric manifolds

In this section we introduce generalized quasitoric manifolds and discuss
various approaches to describing their cohomology.

###  2.1. Definition

We are going to define a notion, which is slightly more general than that of a
quasitoric manifold; the methods we develop in this paper will work in this
generality.

Let $K$ be an abstract simplicial complex of dimension $n-1$ on the vertex set
$[s]=\\{1,2,\ldots,s\\}$. Recall that its moment-angle-complex
$\mathcal{Z}_{K}$ is defined to be the $(s+n)$-dimensional cellular subspace
in the unitary polydisc $(D^{2})^{s}\subset{\mathbb{C}}^{s}$ given by the
formula $\bigcup_{I\in K}\prod\limits_{i=1}^{s}{Y_{i}}$, where $Y_{i}=D^{2}$,
if $i\in I$ and $Y_{i}=S^{1}$, otherwise.

There is a natural (coordinatewise) action of the compact torus $T^{s}$ on
$\mathcal{Z}_{K}$ and the orbit space $\mathcal{Z}_{K}/T^{s}$ is homeomorphic
to the cone over the barycentric subdivision of $K$. When $K=\partial P^{*}$
is a polytopal sphere, the moment-angle-complex $\mathcal{Z}_{K}$ is
homeomorphic to the moment-angle manifold $\mathcal{Z}_{P}$ and therefore
acquires a smooth structure.

Let $\Sigma$ be a complete simplicial fan in ${\mathbb{R}}^{n}$ with the set
of rays (1-dimensional generators) $\Sigma(1)=\\{\rho_{1},\ldots,\rho_{s}\\}$.
The intersection $K_{\Sigma}$ of the fan $\Sigma$ with the unit sphere
$S^{n-1}\subset{\mathbb{R}}^{n}$ is, by definition, a starshaped triangulated
sphere. Note that already in dimension 3, there exist non-polytopal starshaped
spheres. Panov and Ustinovsky [PU12] proved that $\mathcal{Z}_{K}$ has a
smooth structure if and only if $K$ is a starshaped sphere.

Throughout the paper we denote by $M$ and $N$ the character and cocharacter
lattices of $T^{n}$, respectively. Suppose $K=K_{\Sigma}$ is a starshaped
sphere and $\Lambda:\Sigma(1)\to N$ is a _characteristic map_ , i.e. such a
map that the collection of vectors

$\Lambda(\rho_{1}),\ldots,\Lambda(\rho_{k})$

can be completed to a basis of the cocharacter lattice $N$, whenever
$\rho_{1},\ldots,\rho_{k}$ generate a cone in $\Sigma$.

Then the $(s-n)$-dimensional subtorus $H:=\ker\exp\Lambda\subset T^{s}$ acts
freely on $\mathcal{Z}_{K}$ and the smooth manifold
$X_{\Sigma,\Lambda}:=\mathcal{Z}_{K}/H$ will be called a _generalized
quasitoric manifold_.

######  Remark 2.1.

The maximal rank $s(K)$ of a freely acting subtorus in $T^{s}$ is called the
_Buchstaber number_ of $K$; it satisfies the double inequality: $1\leq
s(K)\leq s-n$. If $\Sigma$ is a normal fan of a convex simple $n$-dimensional
polytope $P$ with $s$ facets and $s(\partial P^{*})=s-n$, then
$X_{\Sigma,\Lambda}$ is a _quasitoric manifold_ of $P$. This justifies the
term ’generalized quasitoric manifold’ we suggested above. Naturally, a
quasitoric manifold serves as a key example and the most important particular
case of a generalized quasitoric manifold.

A closed connected codimension-$k$ submanifold of $X_{\Sigma,\Lambda}$ is
called _characteristic_ if it is fixed pointwise by a subtorus of dimension
$k$ in $T^{n}$. Any such can be uniquely represented as a transverse
intersection of $k$ characteristic submanifolds of codimension-2. It is easy
to see that the face poset of characteristic submanifolds in
$X_{\Sigma,\Lambda}$ is isomorphic to $K_{\Sigma}$.

In what follows we will always assume that our generalized quasitoric
manifolds are _omnioriented_ ; as in the case of a quasitoric manifold, we say
that $X_{\Sigma,\Lambda}$ is omnioriented if an orientation is specified for
$X_{\Sigma,\Lambda}$ and for each of the $s$ codimension-2 characteristic
submanifolds $D_{i}$. The choice of this extra data is convenient for two
reasons. First, it allows us to view the circle fixing $D_{i}$ as an element
in the lattice $N={\mathrm{Hom}}(S^{1},T^{n})\simeq{\mathbb{Z}}^{n}$. But even
more importantly, the choice of omniorientation defines the fundamental class
$[X_{\Sigma,\Lambda}]$ of $X_{\Sigma,\Lambda}$ as well as cohomology classes
$[D_{i}]$ dual to the characteristic submanifolds.

We further assume that $\Sigma\subset{\mathbb{R}}^{n}$ and $N_{\mathbb{R}}$
are endowed with orientation. This defines a sign for each collection of rays
$\rho_{i_{1}},\ldots,\rho_{i_{n}}$ forming a maximal cone of $\Sigma$ in the
following way. Let $I=\\{i_{1},\ldots,i_{n}\\}$ be a set of indices ordered
such that the collection of rays $\rho_{i_{1}},\ldots,\rho_{i_{n}}$ is
positively oriented in ${\mathbb{R}}^{n}$, then

${\mathrm{sign}}(I)=\det(\Lambda(\rho_{i_{1}}),\ldots,\Lambda(\rho_{i_{n}}))=\pm
1.$

It is easy to see, that for codimension 2 characteristic submanifolds
$D_{i_{1}},\ldots,D_{i_{n}}$ in a generalized quasitoric manifold one has

$[D_{i_{1}}]\cdots[D_{i_{n}}]=\begin{cases}{\mathrm{sign}}(I)\quad\text{ if
}\rho_{i_{1}}\ldots,\rho_{i_{n}}\text{ form a cone in }\Sigma\\\
0\qquad\qquad\text{otherwise}\end{cases}$

###  2.2. Cohomology ring: Stanley-Reisner-type description

In this subsection we give a description of cohomology rings of generalized
quasitoric manifolds based on the notion of a Stanley-Reisner ring of a
simplicial complex.

Let $K$ be a simplicial complex on the vertex set $[s]$. The quotient algebra

${\mathbb{Z}}[K]:={\mathbb{Z}}[x_{1},\ldots,x_{s}]/I,$

where $I$ is the monomial ideal generated by all square-free monomials
$x_{i_{1}}\ldots x_{i_{t}}$ with $\\{i_{1},\ldots,i_{t}\\}\notin K$, is called
the _Stanley-Reisner ring_ (or, _face ring_) of $K$ (with integer
coefficients). In what follows we assume that ${\mathbb{Z}}[K]$ is a graded
algebra with $\deg x_{i}=2,1\leq i\leq s$. Denote by
${\mathbb{Z}}[\Sigma]:={\mathbb{Z}}[K_{\Sigma}]$.

Recall that any generalized quasitoric manifold $X_{\Sigma,\Lambda}$ with
$\dim\Sigma=n$ is a $2n$-dimensional _locally standard torus manifold_ ; that
is, a smooth connected closed orientable $2n$-dimensional manifold with an
effective smooth locally standard action of the compact torus $T^{n}$ such
that the fixed point set $M^{T}$ is non-empty and finite.

Moreover, since the dual $Q^{\vee}$ to the manifold with corners
$Q:=X_{\Sigma,\Lambda}/T^{n}$ is isomorphic to the starshaped sphere
$K_{\Sigma}$, the orbit space $Q$ is a _homology polytope_ ; that is, a
manifold with corners in which all faces, including $Q$ itself are acyclic,
and all nonempty intersections of faces are connected.

The next result is a particular case of a theorem due to Masuda and Panov
[MP06], who computed (equivariant) cohomology rings of torus manifolds. It
describes the (equivariant) cohomology ring of $X_{\Sigma,\Lambda}$ in terms
of the Stanley-Reisner ring ${\mathbb{Z}}[K_{\Sigma}]$ of $K_{\Sigma}$.

######  Theorem 2.2.

  1. (1)

Equivariant cohomology ring $H^{*}_{T^{n}}(X_{\Sigma,\Lambda})$ is isomorphic
to ${\mathbb{Z}}[K_{\Sigma}]$;

  2. (2)

Ordinary cohomology ring $H^{*}(X_{\Sigma,\Lambda})$ is isomorphic to
${\mathbb{Z}}[K_{\Sigma}]/J$, where

$J=\mathopen{}\mathclose{{}\left\langle\sum_{i=1}^{s}\chi(v_{i})x_{i}\colon\chi\in
M}\right\rangle\text{ and }v_{i}:=\Lambda(\rho_{i}),1\leq i\leq s.$

When $\Sigma$ is a normal fan of a convex simple polytope $P$ with $s(\partial
P^{*})=s-n$, these results were obtained by Davis and Januszkiewicz [DJ91].
This description of the cohomology ring yields an algorithm to compute
products of top-degree cohomology classes in
$H^{*}(X_{\Sigma,\Lambda},{\mathbb{R}})$. It can be done as follows.

As $R_{\Sigma,\Lambda}:=H^{*}(X_{\Sigma,\Lambda},{\mathbb{R}})$ is generated
in degree 1, it suffices to consider monomials $x_{i_{1}}^{k_{1}}\ldots
x_{i_{t}}^{k_{t}}$. Recall that the graded piece of top degree of
$R_{\Sigma,\Lambda}$ is one-dimensional and is generated by $x_{i_{1}}\cdots
x_{i_{n}}$, for any collection $i_{1},\ldots,i_{n}$ of indices such that the
rays $\rho_{i_{1}},\ldots,\rho_{i_{n}}$ generate a full-dimensional cone in
$\Sigma$. Indeed, all such monomials $x_{i_{1}}\cdots x_{i_{n}}$ yield the
same element in $R_{\Sigma,\Lambda}$. Therefore, the evaluation of a monomial
$x_{i_{1}}^{k_{1}}\ldots x_{i_{t}}^{k_{t}}$ in the top degree amounts to
expressing it as a linear combination of square free monomials.

For a monomial $x_{i_{1}}^{k_{1}}\cdots x_{i_{t}}^{k_{t}}$, let
$m=\sum_{i=1}^{t}(k_{i}-1)$ be its _multiplicity_ , so that being square free
is equivalent to having multiplicity $0$. To simplify notation, consider the
monomial $x_{1}^{k_{1}}\cdots x_{t}^{k_{t}}$ (always possible by reordering
the variables). Suppose $x_{1}^{k_{1}}\cdots x_{t}^{k_{t}}$ is a monomial with
multiplicity $m>0$ and $k_{1}>1$. The goal is to express it as a linear
combination of monomials of smaller multiplicity. If
$\rho_{1},\ldots,\rho_{t}$ do not form a cone, then this monomial is equal to
zero in $R_{\Sigma,\Lambda}$ and we are done. Otherwise, the set
$\\{v_{1},\ldots,v_{t}\\}$ can be extended to a lattice basis of $N$, and
therefore there is $\chi\in M$ such that $\chi(e_{1})=1$ and $\chi(e_{j})=0$,
for $j=2,\ldots,t$. Note that $\chi$ induces a linear relation in $J$, so that
we obtain

$x_{1}^{k_{1}}\ldots x_{t}^{k_{t}}=x_{1}^{k_{1}-1}\ldots
x_{t}^{k_{t}}\cdot\mathopen{}\mathclose{{}\left(-\sum_{k=t+1}^{s}\chi(v_{k})x_{k}}\right)\text{.}$

This is a linear combination of monomials of multiplicity $m-1$. Applying this
procedure inductively, we end up with a linear combination of square free
monomials, and thus obtain an evaluation of $x_{1}^{k_{1}}\cdots
x_{t}^{k_{t}}$.

###  2.3. Cohomology ring: volume polynomial description

Another description of the cohomology ring of $X_{\Sigma,\Lambda}$ comes from
the volume polynomial on the space of analogous multi-polytopes. To a
simplicial fan $\Sigma$ with the characteristic map $\Lambda$ one can
associate a vector space of analogous multi-polytopes
${\mathcal{P}}_{\Sigma,\Lambda}$. Multi-fans and multi-polytopes were
introduced in [HM03].

Similar to a space of virtual polytopes, the space
${\mathcal{P}}_{\Sigma,\Lambda}$ can be identified with the space of
$\Lambda$-piecewise linear functions on $\Sigma$.

######  Definition 2.3.

A function $h:\Sigma\to{\mathbb{R}}$ is called _$\Lambda$ -piecewise linear_
if its restriction $f|_{\sigma}$ to any cone $\sigma$ of $\Sigma$ is a
pullback of a linear function on $N$:

$f|_{\sigma}=(\Lambda^{*}\chi)|_{\sigma},\text{ for some }\chi\in
M_{\mathbb{R}}.$

A $\Lambda$-piecewise linear function is called _integer_ , if for any cone
$\sigma$, the corresponding linear function $\chi_{\sigma}\in M$ is integer.

We define the _space of analogous multi-polytopes_ as the space of all
$\Lambda$-piecewise linear functions on $\Sigma$:

${\mathcal{P}}_{\Sigma,\Lambda}=\mathopen{}\mathclose{{}\left\\{h\colon\Sigma\to{\mathbb{R}}\colon\Lambda-\text{piecewise
linear function on}\;\Sigma}\right\\}.$

Any element of ${\mathcal{P}}_{\Sigma,\Lambda}$ is determined by a choice of a
point on each ray from $\Sigma(1)$.

The above description generalizes the description of virtual polytopes via
their support functions. In Section 4 we give a more detailed introduction to
multi-polytopes. In particular, we discuss their geometric description as
convex chains. By analogy with virtual polytopes, we will usually denote a
multi-polytope by $\Delta$ and the corresponding $\Lambda$-piecewise linear
function by $h_{\Delta}$.

Let $X_{\Sigma,\Lambda}$ be a generalized quasitoric manifold and let
$D_{1},\ldots,D_{s}$ be characteristic submanifolds of $X_{\Sigma,\Lambda}$ of
codimension $2$. Since $X_{\Sigma,\Lambda}$ is omnioriented, every multi-
polytope $\Delta\in{\mathcal{P}}_{\Sigma,\Lambda}$ defines a degree 2
cohomology class of $X_{\Sigma,\Lambda}$ via

$\Delta\mapsto\sum_{i=1}^{s}h_{\Delta}(e_{i})[D_{i}],$

where $[D_{i}]\in H^{2}(X_{\Sigma,\Lambda},{\mathbb{R}})$ is a class Poincaré
dual to the characteristic submanifold $D_{i}$. For quasitoric manifolds, it
was shown in [DJ91] that the cohomology ring
$H^{*}(X_{\Sigma,\Lambda},{\mathbb{R}})$ is generated by the classes
$[D_{1}],\ldots,[D_{s}]$ and, in particular, cohomology is generated by its
degree-two part.

Commutative graded rings with Poincaré duality generated in degree 1 admit a
convenient description via Macaulay’s inverse systems. The following theorem
is implicit in [PK92b] and forms a key idea to get a description of the
cohomology ring $H^{*}(X_{\Sigma},{\mathbb{R}})$. The following version is
taken from [Kav11, Theorem 1.1].

######  Theorem 2.4.

Suppose $A=\bigoplus_{i=0}^{n}A_{i}$ is a graded finite dimensional
commutative algebra over a field ${\mathbb{K}}$ of characteristic $0$ such
that $A$ is generated (as an algebra) by the elements $A_{1}$ of degree one,
$A_{0}\simeq A_{n}\simeq{\mathbb{K}}$, and the bilinear map $A_{i}\times
A_{n-i}\to A_{n}$ is non-degenerate for any $i=0,\ldots,n$ (Poincaré duality).
Then

$A\simeq{\mathbb{K}}[t_{1},\ldots,t_{r}]/\\{p(t_{1},\ldots,t_{r})\in{\mathbb{K}}[t_{1},\ldots,t_{r}]\colon
p(\tfrac{\partial}{\partial x_{1}},\ldots,\tfrac{\partial}{\partial
x_{r}})f(x_{1},\ldots,x_{r})=0\\}$

where we identify $A_{1}$ with ${\mathbb{K}}^{r}$ via a basis
$v_{1},\ldots,v_{r}$ and define $f\colon
A_{1}\simeq{\mathbb{K}}^{r}\to{\mathbb{K}}$ as the polynomial given by
$f(x_{1},\ldots,x_{r})=(x_{1}v_{1}+\ldots+x_{r}v_{r})^{n}\in A_{n}\simeq k$.

######  Remark 2.5.

Instead of identifying $A_{1}$ with ${\mathbb{K}}^{r}$, the previous theorem
accepts a basis-free formulation in terms of ${\mathrm{Diff}}(A_{1})$, the
ring of differential operators with constant coefficients:
$A\simeq{\mathrm{Diff}}(A_{1})/\\{p\in{\mathrm{Diff}}(A_{1})\colon p(f)=0\\}$.
In particular, we formulate Theorems 5.8 and 5.9 which generalize Theorem 2.4.

Theorem 2.4 together with Theorem 2.2 imply that to compute the cohomology
ring of $X_{\Sigma,\Lambda}$ it is enough to know the self-intersection index
of characteristic submanifolds $D_{1},\ldots,D_{s}$. This was done by
Ayzenberg and Masuda [AM16]. They defined the notion of a volume polynomial of
a multi-polytope and obtained the following generalization of the BKK theorem
to (generalized) quasitoric manifolds.

######  Theorem 2.6.

Let $X_{\Sigma,\Lambda}$ be a (generalized) quasitoric manifold and let
$\Delta\in{\mathcal{P}}_{\Sigma,\Lambda}$ be a multi-polytope. Then

$\left(\sum_{i=1}^{s}h_{\Delta}(e_{i})[D_{i}]\right)^{n}=n!\cdot{\mathrm{Vol}}(\Delta).$

######  Corollary 2.7.

Let $X_{\Sigma,\Lambda}$ be a generalized quasitoric manifold, then the
cohomology ring $H^{*}(X_{\Sigma,\Lambda},{\mathbb{R}})$ can be computed as

$H^{*}(X_{\Sigma,\Lambda},{\mathbb{R}})\simeq{\mathrm{Diff}}({\mathcal{P}}_{\Sigma,\Lambda})/{\mathrm{Ann}}({\mathrm{Vol}}),$

where ${\mathrm{Ann}}({\mathrm{Vol}})$ is the ideal of differential operators
annihilating the volume polynomial on ${\mathcal{P}}_{\Sigma,\Lambda}$.

###  2.4. Cohomology ring: Brion-type description

In this subsection we give the third description of the cohomology ring of a
quasitoric manifold via the ring of piecewise polynomial functions on the
multi-fan $(\Sigma,\Lambda)$.

For a multi-fan $(\Sigma,\Lambda)$, we call a continuous function
$f:\Sigma\to{\mathbb{R}}$ ($\Lambda$-)_piecewise polynomial_ if its
restriction $f|_{\sigma}$ to any cone $\sigma$ of $\Sigma$ is a pullback of a
polynomial on $N_{\mathbb{R}}$ via $\Lambda$:

$f|_{\sigma}=\Lambda^{*}g_{\sigma},\quad g_{\sigma}\in{\mathbb{R}}[M].$

The set of all piecewise polynomial functions on $(\Sigma,\Lambda)$ forms an
${\mathbb{R}}$-algebra with respect to pointwise addition and multiplication.
Let us denote the algebra of piecewise polynomial functions on
$(\Sigma,\Lambda)$ by $\mathrm{PP}_{\Sigma,\Lambda}$.

######  Theorem 2.8.

As before, let $X_{\Sigma,\Lambda}$ be a generalized quasitoric manifold given
by the multi-fan $(\Sigma,\Lambda)$. Then equivariant cohomology ring of
$X_{\Sigma,\Lambda}$ is given by the ring of piecewise polynomial functions on
$(\Sigma,\Lambda)$:

$H^{*}_{T}(X_{\Sigma,\Lambda},{\mathbb{R}})\simeq\mathrm{PP}_{\Sigma,\Lambda}.$

Furthermore, the cohomology ring $H^{*}(X_{\Sigma,\Lambda},{\mathbb{R}})$ is
given by

$H^{*}(X_{\Sigma,\Lambda},{\mathbb{R}})\simeq\mathrm{PP}_{\Sigma,\Lambda}/\langle
M\rangle.$

###### Proof.

A generalized quasitoric manifold $X_{\Sigma,\Lambda}$ has only finitely many
$T$-fixed points, and for each $p\in X_{\Sigma,\Lambda}^{T}$ the tangent space
$T_{p}X_{\Sigma,\Lambda}$ is decomposed as a direct sum of irreducible
representations of $T$ given by linearly independent characters. Moreover,
$H^{odd}(X_{\Sigma,\Lambda})=0$ hence $X_{\Sigma,\Lambda}$ is an
equivariantly-formal GKM manifold. The description of equivariant cohomology
of $X_{\Sigma,\Lambda}$ follows from computation of equivariant cohomology of
the corresponding GKM-graph ([GKM98, Kur09]).

Indeed, the fixed points of $T$-action on $X_{\Sigma,\Lambda}$ are in
bijection with maximal cones of $\Sigma$ and for the two maximal cones
$\sigma_{1},\sigma_{2}$ connected by a codimension 1 face, the label of the
corresponding edge in the GKM-graph is given by a lattice generator of
$(\sigma_{1}\cup\sigma_{2})^{\perp}$. Hence we have:

$H^{*}_{T}(X_{\Sigma,\Lambda}^{T},{\mathbb{R}})\simeq\left\\{(f_{p})_{p\in
X_{\Sigma,\Lambda}}\in{\mathrm{Sym}}(M)\times\ldots\times{\mathrm{Sym}}(M)\,\Big{|}\,f_{p_{\sigma_{1}}}|_{\sigma_{1}\cap\sigma_{2}}=f_{p_{\sigma_{2}}}|_{\sigma_{1}\cap\sigma_{2}}\right\\}\simeq\mathrm{PP}_{\Sigma,\Lambda}$

and

$H^{*}(X_{\Sigma,\Lambda},{\mathbb{R}})\simeq\mathrm{PP}_{\Sigma,\Lambda}/\langle
M\rangle.$

∎

##  3\. BKK theorems for quasitoric bundles

In this section, we first collect several facts on quasitoric bundles which
will be used below. Then we formulate the main results of this paper: Theorems
3.5 and 3.6. Details and proofs will be given in Section 4, and in Section 6
these theorems will be used to describe cohomology rings of toric bundles.

###  3.1. Preliminaries on quasitoric bundles

Let $G$ be a topological group and let $p\colon E\to B$ be a _principal
$G$-bundle_ over a topological space $B$. Recall that to any principal
$G$-bundle $p\colon E\to B$ and any topological space $X$ equipped with a
continuous action by $G$, one associates a fiber bundle by introducing a
(right) action on the product $E\times X$:

$(e,x)\cdot g\coloneqq(e\cdot g,g^{-1}\cdot x)\text{.}$

The _associated fiber bundle_ is given as the quotient
$E\times_{G}X\coloneqq(E\times X)/G$. It is a fiber bundle with fiber $X$. If
$G=T\simeq(S^{1})^{n}$ is an $n$-dimensional torus, then a $T$–principal
bundle is also called a _torus bundle_.

Crucial to the understanding of the cohomology of fiber bundles is the
following theorem (see, for instance, [Hat02, Theorem 4D.1]):

######  Theorem 3.1 (Leray–Hirsch).

Let $E$ be a fiber bundle with fiber $F$ over a compact manifold $B$. If there
are global cohomology classes $u_{1},\ldots,u_{r}$ on $E$ whose restrictions
$i^{*}(u_{i})$ form a basis for the cohomology of each fiber $F$ (where
$i\colon F\to E$ is the inclusion), then we have an isomorphism of vector
spaces:

$H^{*}(B,{\mathbb{R}})\otimes H^{*}(F,{\mathbb{R}})\to
H^{*}(E,{\mathbb{R}});\quad\sum_{i,j}b_{i}\otimes
i^{*}(u_{j})\mapsto\sum_{i,j}p^{*}(b_{i})\cdot u_{j}\text{.}$

######  Corollary 3.2.

If $T$ is a torus, $p\colon E\to B$ a torus bundle as in Theorem 3.1, and $X$
a generalized $T$–quasitoric manifold, then as a group the cohomology of
$E_{X}=E\times_{T}X$ is given by

$H^{*}(E_{X},{\mathbb{R}})\simeq H^{*}(B,{\mathbb{R}})\otimes
H^{*}(X,{\mathbb{R}})\text{.}$

###### Proof.

As a group, the cohomology of generalized quasitoric manifolds is generated by
classes Poincaré dual to characteristic submanifolds. For any characteristic
submanifold $X_{\sigma}$ of $X$, let $E_{\sigma}=E\times_{T}X_{\sigma}$ be the
associated bundle which is a submanifold of $E_{X}$. Let us choose a linear
basis $X_{\sigma_{1}},\ldots,X_{\sigma_{r}}$ of $H^{*}(X,{\mathbb{R}})$, then
the cohomology classes $u_{1},\ldots,u_{r}$ Poincaré dual to
$E_{\sigma_{1}},\ldots,E_{\sigma_{r}}$ satisfy the condition of Theorem 3.1.
The statement follows. ∎

Corollary 3.2 yields a description of the cohomology _group_ of $E_{X}$.
Crucial for this description is the map which associates to any characteristic
submanifold $X_{\sigma}$ of $X$ the Poincaré dual of the corresponding
$T$-invariant submanifold of $E_{X}$. By restricting this map to codimension
$2$ characteristic manifolds and extending using linearity, we obtain

$\rho\colon{\mathcal{P}}_{\Sigma,\Lambda}\to
H^{2}(E_{X},{\mathbb{R}})\qquad\text{(where $(\Sigma,\Lambda)$ is the multi-
fan of $X=X_{\Sigma,\Lambda}$)}$

which plays an important role in our description of the cohomology ring of
$E_{X}$. We provide more details for $\rho$. Recall that
$\rho_{1},\ldots,\rho_{r}$ denote the rays of $\Sigma$ with

$\Lambda(\rho_{1})=e_{1},\ldots,\Lambda(\rho_{r})=e_{r}.$

Let $D_{1},\ldots,D_{r}$ be the corresponding codimension 2 characteristic
submanifolds in $X$. We also write $D_{i}$ for the submanifold
$E\times_{T}D_{i}$ of $E_{X}$, note that it comes with orientation induced by
omniorientation of $X$. Then for $\Delta\in{\mathcal{P}}_{\Sigma,\Lambda}$, we
have:

$\rho(\Delta)=\sum_{i=1}^{r}h_{\Delta}(e_{i})[D_{i}]\in
H^{2}(E_{X},{\mathbb{R}}),$

where $[D_{i}]$ is the class Poincaré dual to $D_{i}\subseteq E_{X}$ and
$h_{\Delta}\colon N_{\mathbb{R}}\to{\mathbb{R}}$ is the support function of
multi-polytope $\Delta$.

As before, any character $\mu\in M$ of $T$ can be viewed as a multi-polytope,
since $\Lambda^{*}\mu$ is a $\Lambda$-piecewise linear function on $\Sigma$.
Therefore, the map $\rho$ defines a group homomorphism $\rho:M\to
H^{2}(E_{X},{\mathbb{Z}})$, which we denote by the same symbol. We extend this
map by linearity to a map of vector spaces $\rho:M_{\mathbb{R}}\to
H^{2}(E_{X},{\mathbb{R}})$

On the other hand, any character $\lambda\in M$ defines a one–dimensional
representation ${\mathbb{C}}_{\lambda}$ of $T$, namely $t\cdot z=\lambda(t)z$
for $t\in T$, and $z\in{\mathbb{C}}_{\lambda}$. If ${\mathcal{L}}_{\lambda}$
denotes the associated complex line bundle on $B$, i.e.
${\mathcal{L}}_{\lambda}\simeq E\times_{T}{\mathbb{C}}_{\lambda}$, then
${\mathcal{L}}_{\lambda+\mu}={\mathcal{L}}_{\lambda}\otimes{\mathcal{L}}_{\mu}$,
and thus we obtain a group homomorphism:

$c\colon M\to H^{2}(B,{\mathbb{Z}}),\quad\lambda\mapsto
c_{1}({\mathcal{L}}_{\lambda}),$

where $c_{1}({\mathcal{L}}_{\lambda})$ is the first Chern class. By linearity,
we extend the homomorphism to a map of vector spaces:

$c\colon M_{\mathbb{R}}\to H^{2}(B,{\mathbb{R}}).$

The following observation about connection between $\rho(\cdot)$ and
$c(\cdot)$ will be crucial for our approach.

######  Proposition 3.3.

Let $X$ be a generalized quasitoric bundle given by a characteristic pair
$(\Sigma,\Lambda)$ and let $p\colon E_{X}\to B$ be as before. Then for any
character $\lambda\in M$:

$p^{*}c(\lambda)=\rho(\lambda),$

where on the right hand side of the equality $\lambda$ is regarded as a
virtual polytope.

###### Proof.

A character $\lambda$ of $T$ defines an equivarient line bundle $L_{\lambda}$
on the generalized quasitoric manifold $X$. The associated fiber bundle
$E\times_{T}L_{\lambda}$ is a line bundle over $E_{X}$. Moreover, the Chern
class of $E\times_{T}L_{\lambda}$ given by
$c_{1}(E\times_{T}L_{\lambda})=\rho(\lambda)$. It is easy to see that
$E\times_{T}L_{\lambda}$ is a pullback of line bundle
${\mathcal{L}}_{\lambda}$ on $B$

$E\times_{T}L_{\lambda}=p^{*}{\mathcal{L}}_{\lambda}.$

Hence the proposition follows from naturality of Chern classes. ∎

We will finish this subsection with the following observation. The map of
lattices $c\colon M\to H^{2}(B,{\mathbb{Z}})$ is an invariant that uniquely
describes torus bundles in the topological category. Indeed, principle
$G$-bundles over a smooth manifold $B$ are classified by the homotopy classes
of maps $f:B\to BG$ to the classifying space. Since the classifying space of a
compact torus $T$ is homotopy equivalent to
${\mathbb{C}}{\mathbb{P}}^{\infty}\times\ldots\times{\mathbb{C}}{\mathbb{P}}^{\infty}$
and, in particular, is an Eilenberg–MacLane space $K({\mathbb{Z}}^{n},2)$,
homotopy classes of maps are uniquely determined by

$c:=f^{*}:H^{2}(BT)\simeq M(T)\to H^{2}(B)$

We arrive at the following result.

######  Proposition 3.4.

Let $B$ be a closed oriented manifold and $T$ be a compact torus with lattice
of algebraic characters $M(T)$. Then $T$-principal bundles over $B$ are in
bijection with homomorphisms $c\colon M(T)\to H^{2}(B,{\mathbb{Z}})$.

###  3.2. BKK theorems for quasitoric bundles

As before, let $p\colon E\to B$ be a principal torus bundle with respect to a
torus $T\simeq(S^{1})^{n}$ over a compact smooth orientable manifold $B$ of
real dimension $k$. Let $M$ be the character lattice of $T$ and
$(\Sigma,\Lambda)$ be a complete multi-fan which gives rise to a generalized
quasitoric manifold $X=X_{\Sigma,\Lambda}$. Let $E_{X}$ be the total space of
the associated quasitoric bundle. Note that $E_{X}$ is a compact smooth
orientable manifold of real dimension $k+2n$. To keep notation simple we
denote the projection map of the toric bundle by $p\colon E_{X}\to B$ as well.

Our main theorems show that a choice of a natural number $i\leq\tfrac{k}{2}$
and $\gamma\in H^{k-2i}(B,{\mathbb{R}})$ gives rise to a BKK -type theorem.
First, we define two functions $I_{\gamma}$ and $F_{\gamma}$ on
${\mathcal{P}}_{\Sigma,\Lambda}$ as follows.

Let $f_{\gamma}\colon M_{\mathbb{R}}\to{\mathbb{R}}$ be given by

$f_{\gamma}(x)=\langle c(x)^{i}\cdot\gamma,[B]\rangle\text{,}$

where “$\cdot$” denotes the cup product of the cohomology ring
$H^{*}(B,{\mathbb{R}})$. Since $c\colon M_{\mathbb{R}}\to
H^{2}(B,{\mathbb{R}})$ is a linear map, $f_{\gamma}$ is a homogeneous
polynomial of degree $i$ on $M_{\mathbb{R}}$. This leads to the definition of
$I_{\gamma}$:

$I_{\gamma}\colon{\mathcal{P}}_{\Sigma,\Lambda}\to{\mathbb{R}};\quad
I_{\gamma}(\Delta)\coloneqq\int_{\Delta}f_{\gamma}(x)\mathop{}\\!d\mu\quad\text{for}\;\Delta\in{\mathcal{P}}_{\Sigma,\Lambda},$

where $\mu$ denotes the Lebesgue measure on $M_{\mathbb{R}}$ normalized with
respect to the lattice $M$, i.e. a cube spanned by an affine lattice basis of
$M$ has volume $1$. By Theorem 4.1, $I_{\gamma}$ is well defined homogeneous
polynomial of degree $n+i$ on ${\mathcal{P}}_{\Sigma,\Lambda}$.

Recall the definition of $\rho\colon{\mathcal{P}}_{\Sigma,\Lambda}\to
H^{2}(E_{X},{\mathbb{R}})$ from Section 3.1. This leads to the definition of
the function $F_{\gamma}$:

$F_{\gamma}\colon{\mathcal{P}}_{\Sigma,\Lambda}\to{\mathbb{R}};\quad
F_{\gamma}(\Delta)\coloneqq\langle\rho(\Delta)^{n+i}\cdot
p^{*}(\gamma),[E_{X}]\rangle.$

Clearly, $F_{\gamma}$ is a homogeneous polynomial of degree $n+i$ on
${\mathcal{P}}_{\Sigma,\Lambda}$.

The main result of this section is the following analog of the BKK theorem for
quasitoric bundles. Indeed, it expresses certain intersection numbers of
cohomology classes as mixed integrals.

######  Theorem 3.5.

The polynomials $I_{\gamma}$ and $F_{\gamma}$ are proportional with
coefficient of proportionality given by:

$(n+i)!\cdot I_{\gamma}(\Delta)=i!\cdot F_{\gamma}(\Delta)\qquad\text{for any
$\Delta\in{\mathcal{P}}_{\Sigma,\Lambda}$.}$

In particular, the polarizations of $I_{\gamma}$ and $F_{\gamma}$ are
proportional multilinear forms, i.e. for any
$\Delta_{1},\ldots,\Delta_{n+i}\in{\mathcal{P}}_{\Sigma,\Lambda}$

$(n+i)!\cdot I_{\gamma}(\Delta_{1},\ldots,\Delta_{n+i})=i!\cdot
F_{\gamma}(\Delta_{1},\ldots,\Delta_{n+i}).$

For the reader’s convenience, we recall the concept of polarization (or
equivalently mixed integrals). Let $V$ be a vector space. Recall that for a
homogeneous polynomial $f\colon V\to{\mathbb{R}}$ of degree $m$, the
_polarization of $f$_ is the unique symmetric multilinear form $g\colon
V^{m}\to{\mathbb{R}}$ such that $g(v,\ldots,v)=f(v)$. It is a well-known fact
that for any vector space $V$ and any homogeneous polynomial $f$ of degree
$m$, the polarization exists and can be defined as follows:

(1) $g(v_{1},\ldots,v_{m})=\frac{1}{m!}L_{v_{1}}\ldots L_{v_{m}}f,$

where by $L_{v}$ we denote the Lie derivative in direction of $v$. For a
homogeneous polynomial $f\colon M_{\mathbb{R}}\to{\mathbb{R}}$, we will call
the polarization of $I_{f}$ on the space of analogous multi-polytopes
${\mathcal{P}}_{\Sigma,\Lambda}$ the _mixed integral of $f$_.

We conclude this section with an alternative interpretation of Theorem 3.5
which can be favourable for certain applications. By the Leray-Hirsch Theorem
(see Corollary 3.2), $H^{*}(E_{X},{\mathbb{R}})\simeq
H^{*}(B,{\mathbb{R}})\otimes H^{*}(X,{\mathbb{R}})$, and so for
$\Delta\in{\mathcal{P}}_{\Sigma,\Lambda}$, the cycle $\rho(\Delta)^{n+i}\in
H^{2n+2i}(E_{X},{\mathbb{R}})$ can be written as

$\rho(\Delta)^{n+i}=b_{2n+2i}\otimes x_{0}+b_{2n+2i-2}\otimes
x_{2}+\ldots+b_{2i+2}\otimes x_{2n-2}+b_{2i}\otimes x_{2n},$

with $b_{s}\in H^{s}(B,{\mathbb{R}})$ and $x_{r}\in H^{r}(X,{\mathbb{R}})$. As
$X$ is a quasitoric manifold, its cohomology groups in odd degrees vanish, and
therefore $x_{2k+1}=0$ for any $k$. If $x_{2n}$ is normalized such that it is
dual to a point, we call $b_{2i}$ the _horizontal part_ of
$\rho(\Delta)^{n+i}$. Equivalently, the horizontal part $b_{2i}$ of
$\rho(\Delta)^{n+i}$ is the unique class in $H^{2i}(B,{\mathbb{R}})$ such that

$\langle\rho(\Delta)^{n+i}\cdot p^{*}(\eta),[E_{X}]\rangle=\langle
b_{2i}\cdot\eta,[B]\rangle\text{,}$

for any $\eta\in H^{k-2i}(B,{\mathbb{R}})$. Then Theorem 3.5 accepts the
following reformulation.

######  Theorem 3.6.

For any $\Delta\in{\mathcal{P}}_{\Sigma,\Lambda}$, the horizontal part of
$\rho(\Delta)^{n+i}$ can be computed as

$b_{2i}=\frac{(n+i)!}{i!}\int_{\Delta}c(x)^{i}\mathop{}\\!dx\text{.}$

Note that $c(\cdot)^{i}\colon M_{\mathbb{R}}\to H^{2i}(B,{\mathbb{R}})$ is a
vector valued map whose components (after choosing suitable coordinates) are
given by homogeneous polynomials of degree $i$. Thus the integral in Theorem
3.6 exists. Furthermore, although we show that Theorem 3.5 implies Theorem
3.6, in fact they are equivalent.

###### Proof.

Since $H^{*}(B,{\mathbb{R}})$ satisfies Poincaré duality, it suffices to check
that for any $\gamma\in H^{k-2i}(B,{\mathbb{R}})$, we have

$\left\langle i!\cdot\gamma\cdot
b_{2i},[B]\right\rangle=(n+i)!\left\langle\gamma\cdot\int_{\Delta}c(x)^{i}\mathop{}\\!dx,[B]\right\rangle,$

Recall from Theorem 3.1, that there are $u_{1},\ldots,u_{r}\in
H^{*}(E_{X},{\mathbb{R}})$ such that the restrictions $i^{*}(u_{i})$ form a
basis for the cohomology of each fiber $X$ where $i\colon X\to E_{X}$ is the
inclusion. In particular, there are
$y_{i}\in{\mathbb{R}}u_{1}\oplus\ldots\oplus{\mathbb{R}}u_{r}$ such that
$i^{*}(y_{i})=x_{i}$. Since $x_{2n}$ is Poincaré dual to a point (say to a
torus fixed point $x\in X$), it follows that $y_{2n}=E\times_{T}\\{x\\}$, i.e.
$y_{2n}=[S]^{*}$ is the class Poincaré dual to a section of $p$.

Let $[{\mathrm{pt}}]^{*}\in H^{k}(B,{\mathbb{R}})$ be the class dual to a
point in $B$. For $\gamma\in H^{k-2i}(B,{\mathbb{R}})$, let $\gamma\cdot
b_{2i}=a\cdot[{\mathrm{pt}}]^{*}$ for some real number $a\in{\mathbb{R}}$.
Then,

$\displaystyle p^{*}(\gamma)\cdot\rho(\Delta)^{n+i}$
$\displaystyle=p^{*}(\gamma)\cdot(p^{*}(b_{2n+2i})\cdot
y_{0}+p^{*}(b_{2n+2i-2})\cdot y_{2}+\ldots+p^{*}(b_{2i+2})\cdot
y_{2n-2}+p^{*}(b_{2i})\cdot y_{2n})$ $\displaystyle=p^{*}(\gamma\cdot
b_{2i})\cdot y_{2n}=a\cdot p^{*}([{\mathrm{pt}}]^{*})\cdot
E\times_{T}\\{x\\}=a\cdot[X]^{*}\cdot[S]^{*}=a\in
H^{2n+k}(E_{X},{\mathbb{R}})\simeq{\mathbb{R}}\text{,}$

where $[X]^{*}$ is the class dual to a fiber of $p$. Hence, we get

$\langle\gamma\cdot b_{2i},[B]\rangle=\langle
p^{*}(\gamma)\cdot\rho(\Delta)^{n+i},[E_{X}]\rangle=F_{\gamma}(\Delta)\text{.}$

On the other hand,

$\left\langle\gamma\cdot\int_{\Delta}c(x)^{i}\mathop{}\\!dx,[B]\right\rangle=\int_{\Delta}\langle\gamma\cdot
c(x)^{i},[B]\rangle\mathop{}\\!dx=I_{\gamma}(\Delta).$

The statement follows by Theorem 3.5. ∎

Like Theorem 3.5, Theorem 3.6 admits a polarized version.

##  4\. Proof of the BKK theorems

This section is devoted to the proof of Theorem 3.5. Before we begin with the
proof, let us summarize needed results on the polynomial measures on the space
of analogous multi-polytopes. For the details and proofs please see [KLM21,
Kho21].

As before, let $(\Sigma,\Lambda)$ be a characteristic pair and let
$\Sigma(1)=\\{\rho_{1},\ldots,\rho_{s}\\}$ be the set of rays of $\Sigma$ as
before. To a pair $(\Sigma,\Lambda)$ we associate the space of _multi-
polytopes_ ${\mathcal{P}}_{\Sigma,\Lambda}$ which is identified with the space
of $\Lambda$-piecewise linear functions on $\Sigma$.

Note that since the fan $\Sigma$ is simplicial, a $\Lambda$-piecewise linear
function on $\Sigma$ is uniquely determined by its values on the unit vectors
in the directions of rays of $\Sigma$. Therefore the space
${\mathcal{P}}_{\Sigma,\Lambda}$ is naturally isomorphic to ${\mathbb{R}}^{s}$
with the natural coordinates $h_{1},\ldots,h_{s}$ given by evaluation of
$\Lambda$-piecewise linear functions on the unit vectors in the directions of
rays $\rho_{1},\ldots,\rho_{s}$.

There is a correspondence between the cones of $\Sigma$ and faces of the
multi-polytope $\Delta\in{\mathcal{P}}_{\Sigma,\Lambda}$. Indeed, for each
cone $\sigma\in\Sigma$ of dimension $i$ there exists an affine subspace
$M_{\sigma}\subset M_{\mathbb{R}}$ of codimension $i$ with a multi-polytope
$\Delta_{\sigma}$ in it. We will only use this correspondence in the case of
maximal dimensional cones. Let $\sigma$ be a maximal cone of $\Sigma$. Without
loss of generality we may assume that $\sigma$ is generated by the rays
$\rho_{1},\ldots,\rho_{n}$. Then the corresponding affine subspace
$M_{\sigma}$ is zero-dimensional given by the system of linear equations

$\langle\Lambda(\rho_{i}),x\rangle=h_{\Delta}(e_{i}),\quad i=1,\ldots,n,$

and we identify the corresponding vertex of $\Delta$ with the unique solution
of the system above.

The main result we will need is the following construction. Let
$f:M_{\mathbb{R}}\to{\mathbb{R}}$ be a homogeneous polynomial of degree $d$,
then there is a well-defined integration functional on the space of virtual
polytopes:

$I_{f}(\Delta)=\int_{\Delta}f\mathop{}\\!d\mu,\quad\text{for
}\Delta\in{\mathcal{P}}_{\Sigma,\Lambda}.$

The main result we will need is the following theorem.

######  Theorem 4.1.

Let $f\colon M_{\mathbb{R}}\to{\mathbb{R}}$ be a homogeneous polynomial of
degree $d$, then the function
$I_{f}\colon{\mathcal{P}}_{\Sigma,\Lambda}\to{\mathbb{R}}$ is a homogeneous
polynomial of degree $n+d$ on ${\mathcal{P}}_{\Sigma,\Lambda}$.

We will further need the following result on the functional $I_{f}$. Let
$\partial_{i}=\partial/\partial_{h_{i}}$ be the partial derivatives along the
coordinate vectors of ${\mathcal{P}}_{\Sigma,\Lambda}\simeq{\mathbb{R}}^{s}$.

######  Lemma 4.2.

Let $I=\\{i_{1},\ldots,i_{r}\\}\subseteq\\{1,\ldots,s\\}$ be a subset and
$k_{1},\ldots,k_{r}$ positive integers. Let
$\Delta\in{\mathcal{P}}_{\Sigma,\Lambda}$ be a multi-polytope and
$\rho_{i_{1}},\ldots,\rho_{i_{r}}$ do not span a cone in $\Sigma$, then we
have

$\partial_{i_{1}}^{k_{1}}\cdots\partial_{i_{r}}^{k_{r}}\left(I_{f}|_{{\mathcal{P}}_{\Sigma}}\right)(\Delta)=0\text{.}$

However, if $r=n$ and $\rho_{i_{1}},\ldots,\rho_{i_{n}}$ span a cone in
$\Sigma$ dual to the vertex $A\in M_{\mathbb{R}}$, we have

$\partial_{I}\left(I_{f}|_{{\mathcal{P}}_{\Sigma}}\right)(\Delta)={\mathrm{sign}}(I)f(A)\cdot|\det(e_{i_{1}},\ldots,e_{i_{n}})|\text{.}$

###  4.1. Proof of Theorem 3.5

For the reader’s convenience, we recall the used notation. Let $p\colon E\to
B$ be a principal torus bundle with respect to an $n$-dimensional torus $T$
over a smooth compact orientable manifold $B$ of real dimension $k$. The
character lattice of $T$ we denote by $M$. Let
$\Sigma\subseteq{\mathbb{R}}^{n}$ be a simplicial complete fan with rays
$\rho_{1},\ldots,\rho_{s}$ and let $\Lambda:\Sigma(1)\to M$ be a
characteristic map. Let $X=X_{\Sigma,\Lambda}$ be the generalized quasitoric
manifold corresponding to $(\Sigma,\Lambda)$ and denote the total space of the
associated quasitoric bundle by $E_{X}$. To keep notation simple we use the
same notation $p\colon E_{X}\to B$ for the projection map of the toric bundle.
Fix $i\leq\frac{k}{2}$ and a class $\gamma\in H^{k-2i}(B,{\mathbb{R}})$. The
rays $\rho_{i}$ correspond to codimension $2$ characteristic submanifolds in
$X$ which give rise to codimension $2$ submanifolds $D_{i}$ in $E_{X}$. For a
multi-polytope $\Delta\in{\mathcal{P}}_{\Sigma,\Lambda}$, we introduced

$\rho(\Delta)=\sum_{i=1}^{s}h_{\Delta}(e_{i})[D_{i}]\in
H^{2}(E_{X},{\mathbb{R}})\text{,}$

where $[D_{i}]$ is the class dual to $D_{i}\subseteq E_{X}$ (oriented using
omniorientation of $X_{\Sigma,\Lambda}$) and $h_{\Delta}$ is the support
function of $\Delta$. Further, we introduce

$F_{\gamma}\colon{\mathcal{P}}_{\Sigma,\Lambda}\to{\mathbb{R}};F_{\gamma}(\Delta)=\langle\rho(\Delta)^{n+i}\cdot
p^{*}(\gamma),[E_{X}]\rangle\text{.}$

We noticed that $F_{\gamma}$ is a homogeneous polynomial of degree $n+i$ on
${\mathcal{P}}_{\Sigma,\Lambda}$.

Recall that for any character $\lambda\in M$ we have an associated complex
line bundle $\mathcal{L}_{\lambda}$ on $B$. Taking Chern classes and extending
by linearity, we obtain a morphism of vector spaces $c\colon M_{\mathbb{R}}\to
H^{2}(B,{\mathbb{R}})$. Let $f_{\gamma}\colon M_{\mathbb{R}}\to{\mathbb{R}}$
be the function $f_{\gamma}(x)=\langle c(x)^{i}\cdot\gamma,[B]\rangle$. We
defined a map $I_{\gamma}\colon{\mathcal{P}}_{\Sigma,\Lambda}\to{\mathbb{R}}$
to be

$I_{\gamma}(\Delta)=\int_{\Delta}f_{\gamma}(x)\mathop{}\\!d\mu\text{.}$

By Theorem 4.1, $I_{\gamma}$ is a homogeneous polynomial of degree $n+i$ on
${\mathcal{P}}_{\Sigma,\Lambda}$.

We prove Theorem 3.5 by induction on the parameter $0\leq i\leq\frac{k}{2}$.

Let us start with the base case $i=0$. If $i=0$, then $\gamma\in
H^{k}(B,{\mathbb{R}})$ is a multiple of the class dual to a point. For
simplicity, let us assume that this multiple is $1$. So $p^{*}(\gamma)$ is the
class dual to a fiber, which is a generilized quasitoric manifold, and hence
$\rho(\Delta)^{n}\cdot p^{*}(\gamma)$ coincides with the self-intersection
index of the codimesion 2 submanifold in $X$ corresponding to $\Delta$. By the
Theorem 2.6, this can be computed by the normalized volume of $\Delta$ which
equals to $n!\cdot I_{\gamma}(\Delta)$. In other words, for $i=0$, Theorem 3.5
reduces to Theorem 2.6.

As induction hypothesis, suppose that we know Theorem 3.5 for some $i-1\geq
0$. The induction step consists of proving that Theorem 3.5 is also true for
$i$. Since both $F_{\gamma}$ and $I_{\gamma}$ are homogeneous polynomials of
the same degree $n+i$, in order to show equality between $(n+i)!\cdot
I_{\gamma}(\Delta)$ and $i!\cdot F_{\gamma}(\Delta)$, it suffices to show that
all their partial derivatives of order $n$ coincide. In other words, it
suffices to consider differential monomials
$\partial_{i_{1}}^{k_{1}}\dots\partial_{i_{r}}^{k_{r}}$ where
$\partial_{i}=\partial/\partial_{h_{i}}$ are the partial derivatives along the
coordinate vectors of ${\mathcal{P}}_{\Sigma,\Lambda}\simeq{\mathbb{R}}^{s}$
and $\sum_{i=1}^{r}k_{i}=n$. Let us call the number $\sum_{i=1}^{r}(k_{i}-1)$
the _multiplicity of the monomial_
$\partial_{i_{1}}^{k_{1}}\dots\partial_{i_{r}}^{k_{r}}$. In particular, a
monomial has multiplicity $0$ if and only if it is square free.

The proof of equality of the partial derivatives of order $n$ of $(n+i)!\cdot
I_{\gamma}(\Delta)$ and $i!\cdot F_{\gamma}(\Delta)$ is by induction on the
multiplicity $m$ of the applied differential monomial. We will refer to the
induction over $i$ as the “outer induction” and we are going to call the
induction over $m$ the “inner induction”.

The base case of the inner induction (i.e., the case of square free
differential monomials) is treated in the subsections. The result of these
calculations is summarized in Corollary 4.4.

###  4.2. Differentiation with respect to square free monomials

The results of differentiation of $I_{\gamma}$ with respect to square free
differential monomials are given in Lemma 4.2. Now we verify the base case of
square free differential monomials for $F_{\gamma}$:

######  Lemma 4.3.

Let $I=\\{i_{1},\ldots,i_{r}\\}\subseteq\\{1,\ldots,s\\}$ be a subset such
that $\rho_{i_{1}},\ldots,\rho_{i_{r}}$ do not span a cone in $\Sigma$ and
$k_{1},\ldots,k_{r}$ are positive integers. If
$\Delta\in{\mathcal{P}}_{\Sigma,\Lambda}$ is a multi-polytope, then we have

$\partial_{i_{1}}^{k_{1}}\cdots\partial_{i_{r}}^{k_{r}}\left(F_{\gamma}|_{{\mathcal{P}}_{\Sigma,\Lambda}}\right)(\Delta)=0\text{.}$

If $r=n=\dim(N_{\mathbb{R}})$ and $\rho_{i_{1}},\ldots,\rho_{i_{n}}$ span a
cone in $\Sigma$ dual to the vertex $A\in M_{\mathbb{R}}$, we have

$\partial_{I}\left(F_{\gamma}|_{{\mathcal{P}}_{\Sigma,\Lambda}}\right)(\Delta)={\mathrm{sign}}(I)\frac{(n+i)!}{i!}f_{\gamma}(A)\text{.}$

###### Proof.

Without loss of generality we may assume that $i_{j}=j$ for $j=1,\ldots,r$.
For a monomial $\partial_{1}^{k_{1}}\cdots\partial_{r}^{k_{r}}$, let
$\partial_{I}=\partial_{1}\cdots\partial_{r}$ be the corresponding square free
monomial. It is enough to show that $\partial_{I}F_{\gamma}(\Delta)=0$ in
order to prove that
$\partial_{1}^{k_{1}}\cdots\partial_{r}^{k_{r}}F_{\gamma}(\Delta)=0$.

We compute the expansion of the polynomial $F_{\gamma}$ at $\Delta$. This
amounts to expressing $F_{\gamma}(\Delta+\sum_{i=1}^{s}\lambda_{i}h_{i})$ in
terms of the monomials
$\lambda_{1}^{\alpha_{1}}\cdots\lambda_{s}^{\alpha_{s}}$ for non-negative
integers $\alpha_{1},\ldots,\alpha_{s}$. A straightforward computation yields:

$\displaystyle F_{\gamma}\left(\Delta+\sum_{i=1}^{s}\lambda_{i}h_{i}\right)$
$\displaystyle=\left(\rho(\Delta)+\sum_{i=1}^{s}\lambda_{i}[D_{i}]\right)^{n+i}\cdot
p^{*}(\gamma)$
$\displaystyle=\sum_{\alpha_{0}+\alpha_{1}+\ldots+\alpha_{s}=n+i}\binom{n+i}{\alpha_{0},\alpha_{1},\ldots,\alpha_{s}}\cdot\rho(\Delta)^{\alpha_{0}}\cdot[D_{1}]^{\alpha_{1}}\cdots[D_{s}]^{\alpha_{s}}\cdot
p^{*}(\gamma)\cdot\lambda_{1}^{\alpha_{i}}\cdots\lambda_{s}^{\alpha_{s}}$
$\displaystyle=\frac{(n+i)!}{(n+i-r)!}\cdot\rho(\Delta)^{n+i-r}\cdot[D_{1}]\cdots[D_{r}]\cdot
p^{*}(\gamma)\cdot\lambda_{1}\cdots\lambda_{r}+\text{(other terms)}$

where
$\binom{n+i}{\alpha_{0},\alpha_{1},\ldots,\alpha_{s}}=\frac{(n+i)!}{\alpha_{0}!\cdot\alpha_{1}!\cdots\alpha_{s}!}$
denotes the usual multinomial coefficient. The derivative
$\partial_{I}F_{\gamma}(\Delta)$ is equal to the coefficient in front of the
monomial $\lambda_{1}\cdots\lambda_{r}$ in the expression
$F_{\gamma}(\Delta+\sum_{i=1}^{s}\lambda_{i}h_{i})$:

$\partial_{I}F_{\gamma}(\Delta)=\frac{(n+i)!}{(n+i-r)!}\left\langle\rho(\Delta)^{n+i-r}\cdot[D_{1}]\cdots[D_{r}]\cdot
p^{*}(\gamma),[E_{X}]\right\rangle\text{.}$

Characteristic submanifolds $D_{1},\ldots,D_{r}$ intersect transversely in
$E_{X}$, so the product $[D_{1}]\cdots[D_{r}]$ is the class Poincaré dual to
their intersection. In the case that $e_{1},\ldots,e_{r}$ do not generate a
cone in $\Sigma$ the such intersection is empty, and so
$\partial_{I}F_{\gamma}(\Delta)=0$.

For the proof of the second part, let $r=n=\dim(N_{\mathbb{R}})$. We have

$\partial_{I}F_{\gamma}(\Delta)=\frac{(n+i)!}{i!}\left\langle\rho(\Delta)^{i}\cdot[D_{1}]\cdots[D_{n}]\cdot
p^{*}(\gamma),[E_{X}]\right\rangle\text{.}$

If $e_{1},\ldots,e_{n}$ generate a cone in $\Sigma$ dual to the vertex $A$ of
$\Delta$, then $[D_{1}]\cdots[D_{n}]=[E_{A}]$, where $E_{A}=E\times_{T}A$ is
the torus invariant submanifold in $E_{X}$ corresponding to $A$. Note that the
omniorientation of $X$ defines an orientation of $E_{A}$, hence the class
$[E_{A}]$ is well-defined. In particular, the restriction of the projection
map $p\colon E_{A}\to B$ is a diffeomorphism which is orientation preserving
if ${\mathrm{sign}}(I)=1$ and orientaion reversing if ${\mathrm{sign}}(I)=-1$.

Now, let $\widetilde{\Delta}=\Delta-A$ be a multi-polytope which is the
translation of the multi-polytope $\Delta$ for which the vertex $A$ is at the
origin. Since the vertex of $\widetilde{\Delta}$ corresponding to $A$ is at
the origin, we get

$h_{\widetilde{\Delta}}(e_{1})=\ldots=h_{\widetilde{\Delta}}(e_{n})=0,\qquad\text{and
so}\qquad\rho(\widetilde{\Delta})=\sum_{j>n}h_{\widetilde{\Delta}}(e_{j})\cdot[D_{j}].$

Hence, by the first part, we get
$\rho(\widetilde{\Delta})\cdot[D_{1}]\cdots[D_{n}]=0$ as there is no cone in
$\Sigma$ with more than $n$ rays. Therefore:

$\rho(\Delta)^{i}\cdot[D_{1}]\cdots[D_{n}]\cdot
p^{*}(\gamma)=\rho(\widetilde{\Delta}+A)^{i}\cdot[D_{1}]\cdots[D_{n}]\cdot
p^{*}(\gamma)=\rho(A)^{i}\cdot[E_{A}]\cdot p^{*}(\gamma)\text{.}$

By Proposition 3.3, $\rho(A)=p^{*}c(A)$. Since $p\colon E_{A}\to B$ is a
diffeomorphism, we get:

$\left\langle\rho(A)^{i}\cdot
p^{*}(\gamma)\cdot[E_{A}],[E_{X}]\right\rangle=\left\langle(p^{*}c(A))^{i}\cdot
p^{*}(\gamma)\cdot[E_{A}],[E_{X}]\right\rangle=\left\langle
c(A)^{i}\cdot\gamma,{\mathrm{sign}}(I)[B]\right\rangle={\mathrm{sign}}(I)f_{\gamma}(A),$

and therefore
$\partial_{I}F_{\gamma}(\Delta)={\mathrm{sign}}(I)\frac{(n+i)!}{i!}f_{\gamma}(A)$.
∎

The base case of the inner induction is an immediate corollary of Lemmas 4.2
and 4.3:

######  Corollary 4.4 (Base case of the inner induction).

For any $i\leq\frac{k}{2}$, any $\gamma\in H^{k-2i}(B,{\mathbb{R}})$ and any
square free differential monomial
$\partial_{I}=\partial_{i_{1}}\dots\partial_{i_{n}}$ of order $n$ (where
$I=\\{i_{1},\ldots,i_{n}\\}\subseteq\\{1,\ldots,s\\}$), we have:

$\partial_{I}\left((n+i)!\cdot
I_{\gamma}(\Delta)\right)=\partial_{I}\left(i!\cdot
F_{\gamma}(\Delta)\right)\text{.}$

###  4.3.  The inner induction step

In this subsection, the index set $I\subseteq\\{1,\ldots,s\\}$ is considered
as being a multiset. As the induction hypothesis, suppose that
$\partial_{I}\left((n+i)!\cdot
I_{\gamma}(\Delta)\right)=\partial_{I}\left(i!\cdot
F_{\gamma}(\Delta)\right)$, for each differential monomial $\partial_{I}$ of
multiplicity $m-1\geq 0$. It remains to show that the equality is true for
differential monomials of multiplicity $m$. As before, to keep notation simple
assume $i_{j}=j$ for $j=1,\ldots,r$, so that
$\partial_{I}=\partial_{1}^{k_{1}}\ldots\partial_{r}^{k_{r}}$ for positive
integers $k_{1},\ldots,k_{r}$. By reordering the coordinates of
${\mathcal{P}}_{\Sigma,\Lambda}$, we may assume $k_{1}>1$. By Lemmas 4.2 and
4.3, it suffices to consider the case where the vectors $e_{1},\ldots,e_{r}$
form a cone in $\Sigma$ (as otherwise
$\partial_{I}I_{\gamma}(\Delta)=0=\partial_{I}F_{\gamma}(\Delta)$).

We are going to express $\partial_{1}$ as a combination of Lie derivative
$L_{v}$ (for some $v\in M_{\mathbb{R}}$) and other partial derivatives. Then
the (inner) induction step will follow by an explicit computation of
$L_{v}I_{\gamma}(\Delta)$ and $L_{v}F_{\gamma}(\Delta)$.

As $e_{1},\ldots,e_{r}$ form a cone in the smooth fan $\Sigma$, they can be
completed to a basis of $N_{\mathbb{R}}$. We take $v$ to be the first vector
of the dual basis of $M_{\mathbb{R}}$. Then $\langle v,e_{1}\rangle=1$, and
$\langle v,e_{j}\rangle=0$ for $j=2,\ldots,r$. Since $v\in M_{\mathbb{R}}$
(considered as an element of ${\mathcal{P}}_{\Sigma,\Lambda}$) is given by
$v=\sum_{i=1}^{s}\langle v,e_{i}\rangle h_{i}$, we obtain that
$L_{v}=\sum_{i=1}^{s}\langle v,e_{i}\rangle\partial_{i}$, and thus
$\partial_{1}=L_{v}-\sum_{j>r}\langle v,e_{j}\rangle\partial_{j}$. We get:

$\displaystyle\partial_{I}=\partial_{1}^{k_{1}}\ldots\partial_{r}^{k_{r}}$
$\displaystyle=\left(L_{v}-\sum_{j>r}\langle
v,e_{j}\rangle\partial_{j}\right)\partial_{1}^{k_{1}-1}\cdot\partial_{2}^{k_{2}}\cdots\partial_{r}^{k_{r}}$
$\displaystyle=L_{v}\cdot\partial_{1}^{k_{1}-1}\cdot\partial_{2}^{k_{2}}\cdots\partial_{r}^{k_{r}}-\sum_{j>r}\langle
v,e_{j}\rangle\cdot\partial_{1}^{k_{1}-1}\cdot\partial_{2}^{k_{2}}\cdots\partial_{r}^{k_{r}}\cdot\partial_{j}\text{.}$

Since $k_{1}>1$ and $j>r$, each monomial in the sum $\sum_{j>r}\langle
v,e_{j}\rangle\partial_{1}^{k_{1}-1}\cdot\partial_{2}^{k_{2}}\cdots\partial_{r}^{k_{r}}\cdot\partial_{j}$
has multiplicity $m-1$, so by the inner induction hypothesis, we get

$(n+i)!\cdot\left(\sum_{j>r}\langle
v,e_{j}\rangle\partial_{1}^{k_{1}-1}\cdot\partial_{2}^{k_{2}}\cdots\partial_{r}^{k_{r}}\cdot\partial_{j}\right)\cdot
I_{\gamma}(\Delta)=i!\cdot\left(\sum_{j>r}\langle
v,e_{j}\rangle\partial_{1}^{k_{1}-1}\cdot\partial_{2}^{k_{2}}\cdots\partial_{r}^{k_{r}}\cdot\partial_{j}\right)\cdot
F_{\gamma}(\Delta)\text{.}$

It remains to consider the first summand:

######  Lemma 4.5.

In the situation above, we have $(n+i)!\cdot L_{v}I_{\gamma}(\Delta)=i!\cdot
L_{v}F_{\gamma}(\Delta)$.

###### Proof.

A direct calculation shows:

$\displaystyle L_{v}I_{\gamma}(\Delta)$
$\displaystyle=\partial_{t}\Bigg{|}_{t=0}\left(\int_{\Delta+tv}c(x)^{i}\cdot\gamma\mathop{}\\!d\mu\right)=\partial_{t}\Bigg{|}_{t=0}\left(\int_{\Delta}c(x+tv)^{i}\cdot\gamma\mathop{}\\!d\mu\right)=$
$\displaystyle=\partial_{t}\Bigg{|}_{t=0}\left(\int_{\Delta}\left(\sum_{a=0}^{i}\binom{i}{a}t^{a}\cdot
c(v)^{a}c(x)^{i-a}\right)\cdot\gamma\mathop{}\\!d\mu\right)=\int_{\Delta}i\cdot
c(v)c(x)^{i-1}\cdot\gamma\mathop{}\\!d\mu$
$\displaystyle=i\cdot\int_{\Delta}c(x)^{i-1}\cdot(c(v)\cdot\gamma)\mathop{}\\!d\mu=i\cdot
I_{c(v)\cdot\gamma}(\Delta)\text{.}$

Similarly, by a direct calculation and Proposition 3.3, we get

$\displaystyle L_{v}F_{\gamma}(\Delta)$
$\displaystyle=\partial_{t}\Big{|}_{t=0}\Big{(}\rho(\Delta+tv)^{n+i}\cdot
p^{*}(\gamma)\Big{)}=\partial_{t}\Big{|}_{t=0}\Big{(}(\rho(\Delta)+t\rho(v))^{n+i}\cdot
p^{*}(\gamma)\Big{)}=$
$\displaystyle=\partial_{t}\Bigg{|}_{t=0}\left(\left(\sum_{a=0}^{n+i}\binom{n+i}{a}t^{a}\cdot\rho(v)^{a}\cdot\rho(\Delta)^{n+i-a}\right)\cdot
p^{*}(\gamma)\right)=(n+i)\cdot\rho(v)\cdot\rho(\Delta)^{n+i-1}\cdot
p^{*}(\gamma)$ $\displaystyle=(n+i)\cdot\rho(\Delta)^{n+i-1}\cdot
p^{*}(c(v)\cdot\gamma)=(n+i)\cdot F_{c(v)\gamma}(\Delta).$

Since $c(v)\cdot\gamma\in H^{k-2(i-1)}(B,{\mathbb{R}})$, by the induction
hypothesis of the outer induction, we have

$(n+i-1)!\cdot I_{c(v)\cdot\gamma}(\Delta)=(i-1)!\cdot
F_{c(v)\cdot\gamma}(\Delta).$

Therefore,

$(n+i)!\cdot L_{v}I_{\gamma}(\Delta)=(n+i)!\cdot i\cdot
I_{c(v)\cdot\gamma}(\Delta)=i!\cdot(n+i)\cdot
F_{c(v)\cdot\gamma}(\Delta)=i!\cdot L_{v}F_{\gamma}(\Delta)\text{.}\qed$

##  5\. Graded-commutative $n$-self-dual algebras

In this section we briefly summarize some algebraic results obtained in the
recent papers [HKM20] and [KM21] that we are going to rely on when describing
cohomology rings of quasitoric bundles in a way similar to the classical
approach by Khovanskii and Pukhlikov [PK92a].

###  5.1. The general case

Here, we introduce all the necessary algebraic preliminaries that will be used
to describe the entire cohomology rings of quasitoric bundles. Our
presentation follows that in [HKM20].

Recall that a graded ${\mathbb{K}}$-algebra $A=A^{0}\oplus
A^{1}\oplus\cdots\oplus A^{k}\oplus\cdots$ over a field ${\mathbb{K}}$ is
called _graded-commutative_ if for any pair of homogeneous elements $x$, $y$,
the following relation holds:

$xy=(-1)^{\deg(x)\deg(y)}yx$

where $\deg(x)$ (resp. $\deg(y)$) denotes the degree of $x$ (resp. of $y$).

######  Definition 5.1.

Let $n$ be a natural number. A graded-commutative ${\mathbb{K}}$-algebra $A$
is said to be _Poincaré $n$-self-dual_ (or just, _$n$ -self-dual_) if the
following conditions hold:

  1. (1)

The algebra $A$ has a multiplicative unit element $e$ which is homogeneous of
degree zero, i.e. $e\in A^{0}\subseteq A$.

  2. (2)

The homogeneous components $A^{k}$ for $k>n$ vanish, i.e. $A^{k}=0$ for $k>n$,
and $\dim_{{\mathbb{K}}}(A^{n})=1$.

  3. (3)

The pairing $A^{k}\times A^{n-k}\rightarrow A^{n}$ induced by multiplication
in the algebra $A$ is non-degenerate for $0\leq k\leq n$.

Here is a key example we are most interested in throughout this paper.

######  Example 5.2.

Let $M$ be a connected compact oriented $n$-dimensional manifold. By Poincaré
duality, the cohomology ring $A=H^{*}(M,{\mathbb{K}})$ is an $n$-self-dual
graded-commutative ${\mathbb{K}}$-algebra. Moreover, $A$ is equipped with two
extra structures:

  1. (1)

A ${\mathbb{K}}$-linear function $\ell_{*}\colon A\to{\mathbb{K}}$ given as
follows. For $\alpha\in A^{n}$, we let $\ell_{*}(\alpha)$ be equal to the
value of the cohomology class $\alpha$ on the fundamental class of the
oriented manifold $M$. For $\alpha\in A^{k},k\neq n$, we set
$\ell_{*}(\alpha)=0$. We extend $\ell_{*}$ to the whole $A$ by linearity.

  2. (2)

A ${\mathbb{K}}$-bilinear intersection form $F_{\ell_{*}}$ on $A$ defined by
the identity $F_{\ell_{*}}(\alpha,\beta)=\ell_{*}(\alpha\cdot\beta)$ (recall
that we denote the cup product in the cohomology ring $A$ by “$\cdot$”).

######  Remark 5.3.

Note that $F_{\ell_{*}}$ is non-degenerate; in fact, the non-degeneracy of
$F_{\ell_{*}}$ is equivalent to the Poincaré duality on $A$.

Let $B=B^{0}\oplus B^{1}\oplus\cdots\oplus B^{k}\oplus\cdots$ be a graded-
commutative ${\mathbb{K}}$-algebra with multiplicative unit element $e\in
B^{0}$ and $\dim_{{\mathbb{K}}}(B^{0})=1$. Our goal is to describe all
$n$-self dual factor-algebras of $B$. To that extent we introduce the
following notion.

######  Definition 5.4.

An ideal $I\subseteq B$ is called _$n$ -self-dual_ (or just, _$n$ -sd ideal_)
if $I$ is a two-sided homogeneous ideal in $B$ and the factor-algebra $A=B/I$
is $n$-self-dual.

It’s easy to see that if $I\subseteq B$ is an $n$-sd ideal, then $L=I\cap
B^{n}$ is a hyperplane in $B^{n}$. As hyperplanes are in correspondence with
linear functions, we introduce the next definition.

######  Definition 5.5.

A linear function $\ell\colon B\to{\mathbb{K}}$ on a graded-commutative
${\mathbb{K}}$-algebra $B$ is _$n$ -homogeneous_ if it is not identically
equal to zero on $B$, but for any $m\neq n$ its restriction to the homogeneous
component $B^{m}$ vanishes.

Denote by $L_{\ell}\subseteq B^{n}$ the hyperplane in $B^{n}$ defined by the
identity

$L_{\ell}:=\\{\ell=0\\}\cap B^{n}.$

Observe that any hyperplane $L\subseteq B^{n}$ is equal to a hyperplane of the
type $L_{\ell}$ for a unique (up to a non-zero scalar multiple)
$n$-homogeneous linear function $\ell$.

In the rest of this subsection, we explain how to obtain an $n$-self-dual
factor-algebra from an $n$-homogeneous linear function $\ell\colon
B\to{\mathbb{K}}$.

######  Definition 5.6.

For an $n$-homogeneous linear function $\ell\colon B\to{\mathbb{K}}$ let
$I_{1}(L_{\ell})$ and $I_{2}(L_{\ell})$ be the subsets of $B$ defined by the
following conditions:

  1. (1)

an element $a\in B$ belongs to $I_{1}(L_{\ell})$ if and only if $\ell(ab)=0$
for all $b\in B$;

  2. (2)

an element $b\in B$ belongs to $I_{2}(L_{\ell})$ if and only if $\ell(ab)=0$
for all $a\in B$.

For any hyperplane $L_{\ell}\subseteq B^{n}$ one has:

$I_{1}(L_{\ell})=I_{2}(L_{\ell})=:I(L_{\ell})\text{ is an }n-\text{sd ideal in
}B,$

where $\ell$ is an $n$-homogeneous linear function. Note that the ideals
$I_{1}(L_{\ell})$ and $I_{2}(L_{\ell})$ depend only on the hyperplane
$L:=L_{\ell}$, and so the notation $I(L):=I(L_{\ell})$ makes sense.

######  Definition 5.7.

Let $\ell\colon B\to{\mathbb{K}}$ be an $n$-homogeneous linear function on a
graded-commutative algebra $B$. The _Frobenius bilinear form_ $F_{\ell}\colon
B\times B\to{\mathbb{K}}$ associated with $\ell$ is the form defined by the
identity $F_{\ell}(a,b)=\ell(a\cdot b)$.

It is easy to see that the $n$-sd ideal $I(L)$ coincides with the left and
also the right kernel of $F_{\ell}$ for $L=L_{\ell}$.

It turns out that the set of $n$-sd factor-algebras $A=B/I$ equipped with non-
degenerate intersection forms is in one-to-one correspondence with the set of
$n$-homogeneous linear functions on $B$. More precisely, the following
fundamental result takes place.

######  Theorem 5.8.

Let $B$ be a graded-commutative algebra over a field ${\mathbb{K}}$, and let
$A$ be an $n$-sd factor-algebra of $B$ with a chosen isomorphism $\phi\colon
A^{n}\to{\mathbb{K}}$. Let $\rho\colon B\to A$ be the natural epimorphism of
algebras. Then

$A\simeq B/I(L_{\ell}),$

where $\ell$ is the $n$-homogeneous linear function on $B$ defined by

$\ell\colon B^{n}\to{\mathbb{K}},\quad\ell(b)=\phi(\rho(b)),$

and extended by 0 to $B^{i}$ with $i\neq n$.

Moreover, for any $n$-homogeneous linear function $\ell$, the algebra
$A=B/I(L_{\ell})$ is $n$-self dual and the Frobenius form $F_{\ell}$ induces a
non-degenerate pairing on $A$.

###  5.2. The even-dimensional case

In this subsection, we are going to introduce the algebraic results we need in
order to obtain a more convenient way to describe the subring $A^{*}(E_{X})$
of all even-dimensional classes in the cohomology ring $H^{*}(E_{X})$ of a
quasitoric bundle $E_{X}$ with an even-dimensional base $B$. Our presentation
follows that in [KM21].

Let ${\mathbb{K}}$ be a field of zero characteristic, in this paper we assume
${\mathbb{K}}={\mathbb{Q}},{\mathbb{R}}$, or ${\mathbb{C}}$. Throughout this
subsection $A$ will be a $2n$-self-dual graded-commutative algebra over
${\mathbb{K}}$ with vanishing odd-degree part (therefore, $A$ is a commutative
algebra in the usual sense).

Let $W\subset A$ be a vector subspace, which multiplicatively generates $A$.
Then $W$ has a natural graded vector space structure:
$W=\oplus_{k=0}^{n}W_{2k}$, where $W_{2k}:=W\cap A^{2k}$.

Let $\pi\colon V\to\oplus_{k=1}^{n}W_{2k}$ be a surjective linear mapping for
a finite-dimensional vector space $V$. This makes $V$ into a graded vector
space via $V=\oplus_{k=1}^{n}V_{2k}$, where $V_{2k}:=\pi^{-1}(W_{2k})$.

Observe that $\pi$ extends to an epimorphism of algebras

$\pi\colon{\mathrm{Sym}}(V)\to A.$

Furthermore, the grading on $V$ gives rise to a grading on
${\mathrm{Sym}}(V)$:

$\deg(v_{1}^{i_{1}}\ldots v_{n}^{i_{n}})=2\sum\limits_{k=1}^{n}ki_{k},\text{
for any }v_{k}\in V_{2k},$

making $\pi$ into a graded morphism.

Firstly, we are going to describe the ideal $I_{\ell}\subset{\mathrm{Sym}}(V)$
such that

$A\simeq{\mathrm{Sym}}(V)/I_{\ell}.$

Here, $\ell$ is defined on ${\mathrm{Sym}}(V)$ as a composition of maps:

$\ell=\alpha\circ\pi\colon{\mathrm{Sym}}(V)\to{\mathbb{K}},$

where $\alpha\colon A\to{\mathbb{K}}$ is a $2n$-homogeneous linear function
such that $\alpha\colon A^{2n}\simeq{\mathbb{K}}$. Note that $\ell$ is a
$2n$-quasi-homogeneous linear function.

Let us denote by $I_{\ell}$ the following ideal in ${\mathrm{Sym}}(V)$:

$I_{\ell}=\\{P\in{\mathrm{Sym}}(V)\,|\,\ell(P\cdot Q)=0\text{ for any
}Q\in{\mathrm{Sym}}(V)\\}.$

By Theorem 5.8 (see also [KKE21] for similar results), the kernel of the
graded epimorphism of algebras $\pi\colon{\mathrm{Sym}}(V)\to A$ is given by
the ideal just defined: $\mathrm{ker}(\pi)=I_{\ell}$, and hence

$A\simeq{\mathrm{Sym}}(V)/I_{\ell}.$

Below we describe the ideal $I_{\ell}$ in an alternative way, which is more
useful in some applications.

Observe that the symmetric algebra ${\mathrm{Sym}}(V)$ can be identified with
the space ${\mathrm{Diff}}(V)$ of differential operators with constant
coefficients on $V$ as follows. The zero degree component
${\mathrm{Sym}}_{0}(V)\simeq{\mathbb{K}}$ corresponds to operators of
multiplication by a number. For positive degree elements one can define a map

$\mathcal{D}\colon v_{1}\ldots v_{k}\mapsto L_{v_{1}}\ldots L_{v_{k}},$

where $L_{v}$ is the Gateaux derivative in the direction $v\in V$. For an
element $P\in{\mathrm{Sym}}(V)$ we denote by
$\mathcal{D}_{P}\in{\mathrm{Diff}}(V)$ the corresponding differential
operator.

Define a map

${\mathrm{Exp}}^{*}\colon{\mathrm{Sym}}(V)^{\vee}\to{\mathbb{K}}[[V]],\quad{\mathrm{Exp}}^{*}\colon\alpha\mapsto\sum\limits_{i=0}^{\infty}\alpha\left(\frac{x^{i}}{i!}\right).$

It turns out that ${\mathrm{Exp}}^{*}(\ell)$ is a quasi-homogeneous polynomial
of degree $2n$ for the $2n$-quasi-homogeneous linear function $\ell$.

Now we are ready to give an alternative description for the ideal $I_{\ell}$
in ${\mathrm{Sym}}(V)$.

######  Theorem 5.9.

Let $A$ be a $2n$-self-dual graded-commutative algebra over ${\mathbb{K}}$
with vanishing odd-degree part. Let $V$ be a finite-dimensional graded vector
space over ${\mathbb{K}}$ and $\pi\colon{\mathrm{Sym}}(V)\to A$ be a graded
epimorphism of algebras.

Suppose $\ell\colon{\mathrm{Sym}}(V)\to{\mathbb{K}}$ is a $2n$-quasi-
homogeneous linear function. Then we have

$A\simeq{\mathrm{Sym}}(V)/{\mathrm{Ann}}({\mathrm{Exp}}^{*}(\ell)),$

where

${\mathrm{Ann}}({\mathrm{Exp}}^{*}(\ell)):=\\{P\in{\mathrm{Sym}}(V)\,|\,\mathcal{D}_{P}({\mathrm{Exp}}^{*}(\ell))=0\\}.$

We call the quasi-homogeneous polynomial ${\mathrm{Exp}}^{*}(\ell)$ the
_potential_ of $A$ and denote it by $P_{A}$.

##  6\. Applications

###  6.1. Pukhlikov-Khovanskii description of cohomology ring of a quasitoric
bundle

In this section, we apply the results from Sections 4 and 5 to give a
Pukhlikov-Khovanskii-type description of the cohomology ring of a quasitoric
bundle that generalizes the descriptions obtained in [AM16] for quasitoric
manifolds and in [HKM20] for toric bundles.

Let, as before, $p\colon E\to B$ be a principal $T$-bundle, $\Sigma$ a
complete simplicial fan with $\Sigma(1)=\\{\rho_{1},\ldots,\rho_{r}\\}$,
$X=X_{\Sigma,\Lambda}$ a generalized quasitoric manifold, and $E_{X}$ the
corresponding quasitoric bundle.

By the Leray-Hirsch theorem (see Theorem 3.1) the cohomology ring
$H^{*}(E_{X},{\mathbb{R}})$ is a quotient of the polynomial algebra
$R[x_{1},\ldots,x_{r}]$, where $R=H^{*}(B,{\mathbb{R}})$ is the cohomology
ring of the base.

######  Theorem 6.1.

In the notation from above, the cohomology ring of $E_{X}$ is given by

$H^{*}(E_{X},{\mathbb{R}})\simeq R[x_{1},\ldots,x_{r}]/I(L_{\ell})$

where $\ell\colon R[x_{1},\ldots,x_{r}]\to{\mathbb{R}}$ is a
$(k+2n)$-homogeneous linear function defined by:

$\ell(\gamma\cdot x_{i_{1}}\cdots
x_{i_{s}})=I_{\gamma}(\rho_{i_{1}},\ldots,\rho_{i_{s}})$

for any monomial $\gamma\cdot x_{i_{1}}\cdots x_{i_{s}}$ with
$\deg(\gamma)+2s=k+2n$.

###### Proof.

By Theorem 5.8, as $H^{*}(E_{X},{\mathbb{R}})$ is a graded commutative
$(k+2n)$-self dual factor algebra of $R[x_{1},\ldots,x_{r}]$, it is given by
$R[x_{1},\ldots,x_{r}]/I(L_{\ell})$ for a $(k+2n)$-homogeneous linear function
$\ell$ which is obtained by pairing cohomology classes with the fundamental
class of $E_{X}$. Hence the statement follows by Theorem 3.5. ∎

For the next two theorems, let us assume that $\dim(B)=2k$ is even. In this
case, $\dim E_{X}=2(k+n)$ is also even. Then let us denote by
$A^{*}(B),A^{*}(E_{X})$ the corresponding rings of even cohomology classes:

$A^{*}(B)=\bigoplus_{i=0}^{2k}H^{2i}(B,{\mathbb{R}}),\quad
A^{*}(E_{X})=\bigoplus_{i=0}^{2(k+n)}H^{2i}(E_{X},{\mathbb{R}}).$

Rings $A^{*}(B),A^{*}(E_{X})$ are commutative graded rings satisfying Poincaré
duality. Hence we can use Theorem 5.9 to obtain their presentations.

Denote by $P_{B}$ the potential of $A^{*}(B)$ with respect to the generating
vector space $A^{*}(B)$ and by $P_{E_{X}}$ the potential of $A^{*}(E_{X})$
with respect to $A^{*}(B)\oplus{\mathcal{P}}_{\Sigma,\Lambda}$. The following
theorem provides a relation between potentials $P_{E_{X}}$ and $P_{B}$.

######  Theorem 6.2.

For any $\Delta\in{\mathcal{P}}_{\Sigma,\Lambda}$ and $\gamma\in A^{*}(B)$ we
have

$P_{E_{X}}(\gamma,\Delta)=\int_{\Delta}P_{B}(c(\lambda)+\gamma)\mathop{}\\!d\lambda.$

###### Proof.

The proof relies on Theorem 3.6 and will be obtain by a direct calculation.
Recall that by Theorem 5.9, the potentials $P_{E_{X}},P_{B}$ are given by

$P_{E_{X}}={\mathrm{Exp}}^{*}\langle[E_{X}],\cdot\rangle,\quad
P_{B}={\mathrm{Exp}}^{*}\langle[B],\cdot\rangle$

In other words, the evaluations of potentials $P_{E_{X}}$ and $P_{B}$ on
elements of $A^{*}(B)\oplus{\mathcal{P}}_{\Sigma,\Lambda}$ and $A^{*}(B)$
respectively are given by the following formulas

$P_{E_{X}}(\gamma,\Delta)=\left\langle[E_{X}],\sum_{j=0}^{\infty}\frac{(p^{*}(\gamma)+\rho(\Delta))^{j}}{j!}\right\rangle,\quad
P_{B}(\gamma)=\left\langle[B],\sum_{j=0}^{\infty}\frac{\gamma^{j}}{j!}\right\rangle$

By definition of horizontal parts $h_{i}(\Delta)$ of $\Delta$ we have

$P_{E_{X}}(\gamma,\Delta)=\left\langle[E_{X}],p^{*}(\exp(\gamma))\cdot\sum_{j=0}^{\infty}\frac{\rho(\Delta)^{j}}{j!}\right\rangle=\left\langle[B],\exp(\gamma)\cdot\sum_{j=0}^{\infty}\frac{h_{j}(\Delta)}{j!}\right\rangle.$

Now, by Theorem 3.6 we get $h_{j}(\Delta)=0$ for $j<n$ and

$h_{n+i}(\Delta)=\frac{(n+i)!}{i!}\int_{\Delta}c(\lambda)^{i}\mathop{}\\!d\lambda\text{,}$

otherwise. Hence we obtain

$\begin{split}P_{E_{X}}(\gamma,\Delta)=\left\langle[B],\exp(\gamma)\cdot\sum_{j=0}^{\infty}\frac{h_{j}(\Delta)}{j!}\right\rangle&=\left\langle[B],\exp(\gamma)\cdot\sum_{i=0}^{\infty}\frac{h_{n+i}(\Delta)}{(n+i)!}\right\rangle=\\\
\left\langle[B],\exp(\gamma)\cdot\sum_{i=0}^{\infty}\frac{1}{i!}\int_{\Delta}c(\lambda)^{i}\mathop{}\\!d\lambda\right\rangle&=\left\langle[B],\int_{\Delta}\exp(\gamma+c(\lambda))\mathop{}\\!d\lambda\right\rangle=\\\
\int_{\Delta}\langle[B],\exp(c(\lambda)+\gamma)\rangle\mathop{}\\!d\lambda&=\int_{\Delta}P_{B}(c(\lambda)+\gamma)\mathop{}\\!d\lambda.\end{split}$

This finishes the proof. ∎

The next result is a direct generalization of [KM21, Theorem 5.8] to
quasitoric bundles.

######  Theorem 6.3.

In the above notation, suppose $\dim B$ is even. Then the ring of even-degree
cohomology classes of a quasitoric bundle is given by

$A^{*}(E_{X})={\mathrm{Sym}}(A^{*}(B)\oplus{\mathcal{P}}_{\Sigma,\Lambda})/{\mathrm{Ann}}(P_{E_{X}}),$

where the quasi-homogeneous polynomial $P_{E_{X}}$ is given by

$P_{E_{X}}(\gamma,\Delta)=\int_{\Delta}P_{B}(c(\lambda)+\gamma)\mathop{}\\!d\lambda.$

###### Proof.

The proof goes by a direct application of Theorem 5.9 and Theorem 6.2. ∎

###  6.2. Stanley-Reisner description of cohomology ring of a quasitoric
bundle

In this subsection we provide a generalization of the Stanley-Reisner
description for the cohomology ring to quasitoric bundle case. A version of
Theorem 6.4 which works for more general torus manifold bundles was obtained
in [DKU19].

######  Theorem 6.4.

Let $E_{X}$ be a quasitoric bundle associated with a generalized quasitoric
manifold $X=X_{\Sigma,\Lambda}$ as before. Then $H^{*}(E_{X},{\mathbb{R}})$ is
isomorphic (as an $H^{*}(B,{\mathbb{R}})$-algebra) to the quotient of
$H^{*}(B,{\mathbb{R}})[x_{1},\ldots,x_{r}]$ by

$\mathopen{}\mathclose{{}\left(\left\langle x_{j_{1}}\cdots
x_{j_{k}}\colon\rho_{j_{1}},\ldots,\rho_{j_{k}}\;\text{do not span a cone
of}\;\Sigma\right\rangle+\left\langle
c\left(\lambda\right)-\sum_{i=1}^{n}\langle e_{i},\lambda\rangle
x_{i}\colon\lambda\in M\right\rangle}\right)\text{.}$

###### Proof.

The proof is identical to the proof of [HKM20, Theorem 3.5]. ∎

Note the similarities with the Stanley-Reisner description of the cohomology
ring of toric varieties and quasitoric manifolds. Indeed, the first ideal in
Theorem 6.4 corresponds to the Stanley-Reisner ideal of the corresponding
toric variety.

###  6.3. Brion description of cohomology ring of a quasitoric bundle

In this subsection we give a proof of a generalization of the Brion
description for a cohomology ring to the quasitoric bundle case.

######  Theorem 6.5.

Let $E_{X}$ be a quasitoric bundle associated with a generalized quasitoric
manifold $X=X_{\Sigma,\Lambda}$ as before. Then $H^{*}(E_{X},{\mathbb{R}})$ is
isomorphic (as an $H^{*}(B,{\mathbb{R}})$-algebra) to the quotient of
$H^{*}(B,{\mathbb{R}})\otimes\mathrm{PP}_{\Sigma,\Lambda}$ by the ideal

$\langle c\left(\lambda\right)\otimes
1-1\otimes\langle\cdot,\lambda\rangle\colon\lambda\in M\rangle,$

where $\langle\cdot,\lambda\rangle\in\mathrm{PP}^{1}_{\Sigma,\Lambda}$ is a
($\Lambda$–piecewise) linear function on $\Sigma$.

###### Proof.

For the generalized quasitoric manifold $X$, Theorem 2.2 (1) and Theorem 2.8
imply that

${\mathbb{R}}[\Sigma]\simeq
H^{*}_{T}(X;{\mathbb{R}})\simeq\mathrm{PP}_{\Sigma,\Lambda}.$

It is easy to see that the composition isomorphism of ${\mathbb{R}}$-algebras
is given by the mapping $v_{i}\mapsto\phi_{i}$, for all $1\leq i\leq s$. Hence
the set of piecewise linear functions $\\{\phi_{i}\colon 1\leq i\leq s\\}$
forms a basis of $\mathrm{PP}^{1}_{\Sigma,\Lambda}$.

Then any $\Lambda$-piecewise linear function $\langle\cdot,\lambda\rangle$ on
$\Sigma$ can be uniquely expressed as a linear combination of elements of this
basis, namely

$\langle\cdot,\lambda\rangle=\sum_{i=1}^{n}\langle
e_{i},\lambda\rangle\phi_{i}.$

It remains to apply the Stanley-Reisner description given by Theorem 6.4
proved in the previous subsection. ∎

## References

  * [AM16] Anton Ayzenberg and Mikiya Masuda.  Volume polynomials and duality algebras of multi-fans.  Arnold Mathematical Journal, 2(3):329–381, 2016.
  * [BEM+17] Victor M. Buchstaber, Nikolay Yu. Erokhovets, Mikiya Masuda, Taras E. Panov, and Seonjeong Park.  Cohomological rigidity of manifolds defined by 3-dimensional polytopes.  Russian Mathematical Surveys, 72:199–256, 2017.
  * [Ber75] D. N. Bernstein.  The number of roots of a system of equations.  Funkcional. Anal. i Priložen., 9(3):1–4, 1975.
  * [BGLV20] Djordje Baralić, Jelena Grbić, Ivan Yu. Limonchenko, and Aleksandar Vucić.  Toric objects associated with the dodecahedron.  Filomat, 34:2329–2356, 2020.
  * [BKH76] D. N. Bernstein, A. G. Kušnirenko, and A. G. Hovanskiĭ.  Newton polyhedra.  Uspehi Mat. Nauk, 31(3(189)):201–202, 1976.
  * [BP02] Victor M. Buchstaber and Taras E. Panov.  Torus actions and their applications in topology and combinatorics.  Number 24. American Mathematical Soc., 2002.
  * [BP16] Victor M. Buchstaber and Taras E. Panov.  On manifolds defined by 4-colourings of simple 3-polytopes.  Russian Mathematical Surveys, 71:1137–1139, 2016.
  * [BPR06] Victor M. Buchstaber, Taras E. Panov, and Nigel Ray.  Spaces of polytopes and cobordism of quasitoric manifolds.  Moscow Mathematical Journal, 7:219–242, 2006.
  * [CLS11] D. A. Cox, J. B. Little, and H. K. Schenck.  Toric varieties, volume 124 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, 2011.
  * [CMS08] Suyoung Choi, Mikiya Masuda, and Dong Youp Suh.  Quasitoric manifolds over a product of simplices.  arXiv: Algebraic Topology, 2008.
  * [DJ91] Michael W. Davis and Tadeusz Januszkiewicz.  Convex polytopes, coxeter orbifolds and torus actions.  Duke Mathematical Journal, 62(2):417–451, 1991.
  * [DKU19] Jyoti Dasgupta, Bivas Khan, and Vikraman Uma.  Cohomology of torus manifold bundles.  Mathematica Slovaca, 69(3):685–698, 2019.
  * [GKM98] Mark Goresky, Robert Kottwitz, and Robert MacPherson.  Equivariant cohomology, koszul duality, and the localization theorem.  Inventiones mathematicae, 131(1):25–84, 1998.
  * [Hat02] A. Hatcher.  Algebraic topology.  Cambridge University Press, Cambridge, 2002.
  * [HK17] Sho Hasui and Daisuke Kishimoto.  p-local stable cohomological rigidity of quasitoric manifolds.  Osaka Journal of Mathematics, 54(2):343–350, 2017.
  * [HKM20] Johannes Hofscheier, Askold Khovanskii, and Leonid Monin.  Cohomology rings of toric bundles and the ring of conditions.  arXiv preprint arXiv:2006.12043, 2020.
  * [HKS16] Sho Hasui, Daisuke Kishimoto, and Takashi Sato.  p-local stable splitting of quasitoric manifolds.  Osaka Journal of Mathematics, 53(3):843–854, 2016.
  * [HM03] Akio Hattori and Mikiya Masuda.  Theory of multi-fans.  Osaka Journal of Mathematics, 40(1):1–68, 2003.
  * [Hof19] J. Hofscheier.  The ring of conditions for horospherical homogeneous spaces.  In Proceedings of the conference Interactions with Lattice Polytopes Magdeburg, 2019.
  * [Kav11] K. Kaveh.  Note on cohomology rings of spherical varieties and volume polynomial.  J. Lie Theory, 21(2):263–283, 2011.
  * [Kho21] Askold Khovanskii.  Generalized virtual polyhedra and cohomology of torus manifolds.  Talk at International School ’Toric Topology and Combinatorics’, 2021\.
  * [KKE21] Boris Kazarnovskii, Askold Khovanskii, and Alexander Esterov.  Newton polytopes and tropical geometry.  Russian Mathematical Surveys, 76(1):91, 2021.
  * [KLM21] Askold Khovanskii, Ivan Limonchenko, and Leonid Monin.  Quasitoric manifolds and multi-polytopes.  In preparation, 2021.
  * [KM21] Askold Khovanskii and Leonid Monin.  Gorenstein algebras and toric bundles.  arXiv preprint arXiv:2106.15562, 2021.
  * [Kou76] A. G. Kouchnirenko.  Polyèdres de Newton et nombres de Milnor.  Invent. Math., 32(1):1–31, 1976.
  * [Kur09] Shintarô Kuroki.  Introduction to gkm theory.  Science and Technology, 2009(0063179):0063179, 2009.
  * [LLP18] Ivan Yu. Limonchenko, Zhi Lü, and Taras E. Panov.  Calabi yau hypersurfaces and su-bordism.  Proceedings of the Steklov Institute of Mathematics, 302:270–278, 2018.
  * [LP16] Zhi Lü and Taras Panov.  On toric generators in the unitary and special unitary bordism rings.  Algebraic & Geometric Topology, 16(5):2865–2893, 2016.
  * [LW16] Zhi Lü and Wei Wang.  Examples of quasitoric manifolds as special unitary manifolds.  Mathematical Research Letters, 23:1453–1468, 2016.
  * [Mas99] Mikiya Masuda.  Unitary toric manifolds, multi-fans and equivariant index.  Tohoku Mathematical Journal, 51:237–265, 1999.
  * [MP06] Mikiya Masuda and Taras Panov.  On the cohomology of torus manifolds.  Osaka Journal of Mathematics, 43(3):711–746, 2006.
  * [PK92a] A. V. Pukhlikov and A. G. Khovanskii.  Finitely additive measures of virtual polyhedra.  Algebra i Analiz, 4(2):161–185, 1992.
  * [PK92b] A. V. Pukhlikov and A. G. Khovanskiĭ.  The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes.  Algebra i Analiz, 4(4):188–216, 1992.
  * [PU12] Taras Panov and Yuri Ustinovsky.  Complex-analytic structures on moment-angle manifolds.  Mosc. Math. J., 12(1):149–172, 216, 2012.
  * [SU03] P. Sankaran and V. Uma.  Cohomology of toric bundles.  Comment. Math. Helv., 78(3):540–554, 2003.