Chow-Witt rings and topology of flag varieties
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2023
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866916480807862272 |
|---|---|
| author | Hudson, Thomas Matszangosz, Ákos K. Wendt, Matthias |
| author_facet | Hudson, Thomas Matszangosz, Ákos K. Wendt, Matthias |
| contents | The paper computes the Witt-sheaf cohomology rings of partial flag varieties in type A in terms of the Pontryagin classes of the subquotient bundles. The proof is based on a Leray-Hirsch-type theorem for Witt-sheaf cohomology for the maximal rank cases, and a detailed study of cohomology ring presentations and annihilators of characteristic classes for the general case. The computations have consequences for the topology of real flag manifolds: we show that all torsion in the integral cohomology is 2-torsion, which was not known in full generality previously. This allows for example to compute the Poincaré polynomials of complete flag varieties for cohomology with twisted integer coefficients. The computations also allow to describe the Chow-Witt rings of flag varieties, and we sketch an enumerative application to counting flags satisfying multiple incidence conditions to given hypersurfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_11003 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Chow-Witt rings and topology of flag varieties Hudson, Thomas Matszangosz, Ákos K. Wendt, Matthias Algebraic Geometry Algebraic Topology 14M15, 14F25, 19G12, 57T15 The paper computes the Witt-sheaf cohomology rings of partial flag varieties in type A in terms of the Pontryagin classes of the subquotient bundles. The proof is based on a Leray-Hirsch-type theorem for Witt-sheaf cohomology for the maximal rank cases, and a detailed study of cohomology ring presentations and annihilators of characteristic classes for the general case. The computations have consequences for the topology of real flag manifolds: we show that all torsion in the integral cohomology is 2-torsion, which was not known in full generality previously. This allows for example to compute the Poincaré polynomials of complete flag varieties for cohomology with twisted integer coefficients. The computations also allow to describe the Chow-Witt rings of flag varieties, and we sketch an enumerative application to counting flags satisfying multiple incidence conditions to given hypersurfaces. |
| title | Chow-Witt rings and topology of flag varieties |
| topic | Algebraic Geometry Algebraic Topology 14M15, 14F25, 19G12, 57T15 |
| url | https://arxiv.org/abs/2302.11003 |