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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2109.11578 |
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| _version_ | 1866917866508386304 |
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| author | Wei, Chuanhao Yang, Ruijie |
| author_facet | Wei, Chuanhao Yang, Ruijie |
| contents | In this paper, we study the cohomology of semisimple local systems in the spirit of classical Hodge theory. On the one hand, we establish a generalization of Hodge-Riemann bilinear relations. For a semisimple local system on a smooth projective variety, we define a canonical isomorphism from the complex conjugate of its cohomology to the cohomology of the dual local system, which is a generalization of the classical Weil operator for pure Hodge structures. This isomorphism establishes a relation between the twisted Poincaré pairing, a purely topological object, and a positive definite Hermitian pairing. On the other hand, we prove a global invariant cycle theorem for semisimple local systems.
As an application, we give a new and geometric proof of Sabbah's Decomposition Theorem for the direct images of semisimple local systems under proper algebraic maps, without using the category of polarizable twistor D-modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_11578 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Cohomology of semisimple local systems and the Decomposition theorem Wei, Chuanhao Yang, Ruijie Algebraic Geometry Differential Geometry Representation Theory Primary: 14C30, Secondary: 32S60 In this paper, we study the cohomology of semisimple local systems in the spirit of classical Hodge theory. On the one hand, we establish a generalization of Hodge-Riemann bilinear relations. For a semisimple local system on a smooth projective variety, we define a canonical isomorphism from the complex conjugate of its cohomology to the cohomology of the dual local system, which is a generalization of the classical Weil operator for pure Hodge structures. This isomorphism establishes a relation between the twisted Poincaré pairing, a purely topological object, and a positive definite Hermitian pairing. On the other hand, we prove a global invariant cycle theorem for semisimple local systems. As an application, we give a new and geometric proof of Sabbah's Decomposition Theorem for the direct images of semisimple local systems under proper algebraic maps, without using the category of polarizable twistor D-modules. |
| title | Cohomology of semisimple local systems and the Decomposition theorem |
| topic | Algebraic Geometry Differential Geometry Representation Theory Primary: 14C30, Secondary: 32S60 |
| url | https://arxiv.org/abs/2109.11578 |