Cohomology rings of quasitoric bundles

TL;DR

This paper generalizes the Bernstein-Kushnirenko-Khovanskii (BKK) theorem to quasitoric bundles, providing a new algebraic description of their cohomology rings using differential operators and volume polynomials.

Contribution

It extends the BKK theorem and cohomology ring descriptions from toric varieties to quasitoric bundles, broadening the algebraic tools available for these geometric objects.

Findings

Generalized BKK theorem for quasitoric bundles

Explicit presentation of cohomology rings using differential operators

Description of cohomology rings via volume polynomials

Abstract

The classical BKK theorem computes the intersection number of divisors on toric variety in terms of volumes of corresponding polytopes. 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 Poincar\'e duality to give a description of cohomology rings of quasitoric bundles.