Chern Classes of Toric Variety Bundles

TL;DR

This paper proves a formula for the total Chern class of toric variety bundles, generalizing known results and applying to various classes of algebraic varieties, with implications for understanding their topological and geometric properties.

Contribution

It provides a new formula for the total Chern class of toric variety bundles, extending previous results and applying to toroidal horospherical varieties and quasitoric manifolds.

Findings

Derived a formula for the total Chern class of tangent bundles in toric bundles.

Generalized the description of Chern classes to a broader class of varieties.

Provided a new proof of Masuda's formula for equivariant Chern classes.

Abstract

In this paper, we resolve a conjecture of Khovanskii--Monin on the Chern classes of toric variety bundles. The main result is a formula for the total Chern class of the tangent bundle of a toric variety bundle in terms of the total Chern class of the base and the total Chern class of the toric fibre. The result serves as a simultaneous generalization of the description of the total Chern class of a projectivized vector bundle and of the formula for the total Chern class of a toric variety in terms of the Chern classes of the toric divisors. We also establish a topological version of this statement for stably complex quasitoric manifolds. As an immediate application, we obtain a formula for the total Chern class of a toroidal horospherical variety in terms of the Chern classes of the generalized flag variety and the total Chern class of the toric fibre, as well as a new proof of Masuda's formula for equivariant Chern classes. This paper is written with a view towards finding polytopal models for various numeric invariants of spherical varieties.