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Chern Classes of Toric Variety Bundles

Gregory Taroyan

TL;DR

This work resolves the Khovanskii--Monin conjecture by proving a universal formula for the total Chern class of the tangent bundle of toric variety bundles, $c(TE)=\pi^*c(TB)\cdot \prod_{\rho\in \Sigma(1)}(1+[\widetilde{D_\rho}])$, and then extends the result to unitary generalized quasitoric manifold bundles. The authors derive vertical-subbundle descriptions, recover the known Chern-class formulas for complete toric varieties, and illustrate how the fibered-toric framework reduces complex equivariant problems to non-equivariant ones. They present several important consequences, including a new proof of Masuda's equivariant Chern-class formula and a method to compute Chern classes for toroidal horospherical varieties by combining base and fibre data. The approach links toric geometry with generalized quasitoric topology and paves the way for polytopal models of numeric invariants in spherical varieties.

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.

Chern Classes of Toric Variety Bundles

TL;DR

This work resolves the Khovanskii--Monin conjecture by proving a universal formula for the total Chern class of the tangent bundle of toric variety bundles, , and then extends the result to unitary generalized quasitoric manifold bundles. The authors derive vertical-subbundle descriptions, recover the known Chern-class formulas for complete toric varieties, and illustrate how the fibered-toric framework reduces complex equivariant problems to non-equivariant ones. They present several important consequences, including a new proof of Masuda's equivariant Chern-class formula and a method to compute Chern classes for toroidal horospherical varieties by combining base and fibre data. The approach links toric geometry with generalized quasitoric topology and paves the way for polytopal models of numeric invariants in spherical varieties.

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.

Paper Structure

This paper contains 17 sections, 28 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Toric Bundles
  3. Generalities on principal bundles
  4. Toric variety bundles
  5. Fibered toric varieties
  6. Chern Classes of Toric Variety Bundles
  7. Vertical subbundles
  8. Chern classes of complete toric varieties
  9. Khovanskii--Monin conjecture
  10. Chern Classes of Unitary Generalized Quasitoric Manifold Bundles
  11. Generalized quasitoric manifolds
  12. Fibered generalized quasitoric manifolds
  13. Formula for the total Chern class
  14. Concluding Remarks
  15. Equivariant Chern classes of toric varieties and quasitoric manifolds
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\pi:E_\Sigma\to B$ be a toric variety bundle. Then, the total Chern class of the tangent bundle of $E$ is given by the following formula. Here $[\widetilde{D_\rho}]$ is the cohomology class of the toric divisor associated to the ray $\rho$ in the fan $\Sigma$ corresponding to the toric variety bundle $E_\Sigma\to X_\Xi$. Here the class $\pi^*c(TB)$ denotes the pullback of the total Chern cla

Theorems & Definitions (57)

  • Theorem : \ref{['thm:chern_class_formula']}
  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Example 2.1.4
  • Proposition 2.2.2
  • proof
  • Proposition 2.2.3
  • proof
  • Definition 2.3.1: Khovanskii_Monin
  • ...and 47 more