On the cohomological representations of finite automorphism groups of singular curves and compact complex spaces
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912810440589312 |
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| author | Liu, Qing Liu, Wenfei |
| author_facet | Liu, Qing Liu, Wenfei |
| contents | Let G be a finite group acting tamely on a proper reduced curve C over an algebraically closed field. We study the G-module structure on the cohomology groups of a G-equivariant locally free sheaf F on C, and give formulas of Chevalley--Weil type, with values in the Grothendieck ring R_k(G)_Q of finitely generated G-modules. We also give a similar formula for the singular cohomology of compact complex spaces.
The focus is on the case where C is nodal. Using the Chevalley--Weil formula, we compute the G-invariant part of the global sections of the pluricanonical bundle ω_C^{\otimes m}. In turn, we use the formula for m=2 to compute the equivariant deformation space of a stable G-curve C. We also obtain numerical criteria for the presence of any given irreducible representation in space of the global sections of ω_C\otimes F, where F is an ample locally free G-sheaf on C. Some new phenomena, pathological compared to the smooth curve case, are discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12290 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the cohomological representations of finite automorphism groups of singular curves and compact complex spaces Liu, Qing Liu, Wenfei Algebraic Geometry 14H37, 14J50, 32M18 Let G be a finite group acting tamely on a proper reduced curve C over an algebraically closed field. We study the G-module structure on the cohomology groups of a G-equivariant locally free sheaf F on C, and give formulas of Chevalley--Weil type, with values in the Grothendieck ring R_k(G)_Q of finitely generated G-modules. We also give a similar formula for the singular cohomology of compact complex spaces. The focus is on the case where C is nodal. Using the Chevalley--Weil formula, we compute the G-invariant part of the global sections of the pluricanonical bundle ω_C^{\otimes m}. In turn, we use the formula for m=2 to compute the equivariant deformation space of a stable G-curve C. We also obtain numerical criteria for the presence of any given irreducible representation in space of the global sections of ω_C\otimes F, where F is an ample locally free G-sheaf on C. Some new phenomena, pathological compared to the smooth curve case, are discussed. |
| title | On the cohomological representations of finite automorphism groups of singular curves and compact complex spaces |
| topic | Algebraic Geometry 14H37, 14J50, 32M18 |
| url | https://arxiv.org/abs/2507.12290 |