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arXiv · 2506.20848

Chern Classes of Toric Variety Bundles

Gregory Taroyan
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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.

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Gregory Taroyan. 2025-06-25. Chern Classes of Toric Variety Bundles. https://arxiv.org/abs/2506.20848

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