Chern Classes of Toric Variety Bundles
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914115809705984 |
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| author | Taroyan, Gregory |
| author_facet | Taroyan, Gregory |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20848 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chern Classes of Toric Variety Bundles Taroyan, Gregory Algebraic Geometry Algebraic Topology Combinatorics 57S12, 14C17, 57R20 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. |
| title | Chern Classes of Toric Variety Bundles |
| topic | Algebraic Geometry Algebraic Topology Combinatorics 57S12, 14C17, 57R20 |
| url | https://arxiv.org/abs/2506.20848 |