Cell decomposition and dual boundary complexes of character varieties
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917763007643648 |
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| author | Su, Tao |
| author_facet | Su, Tao |
| contents | The weak geometric P=W conjecture of L. Katzarkov, A. Noll, P. Pandit, and C. Simpson asserts that for any smooth Betti moduli space $\mathcal{M}_B$ of complex dimension $d$ over a punctured Riemann surface, the dual boundary complex $\mathbb{D}\partial\mathcal{M}_B$ is homotopy equivalent to a $(d-1)$-dimensional sphere. Here, we consider $\mathcal{M}_B$ as a generic $GL_n(\mathbb{C})$-character variety defined on a Riemann surface of genus $g$, with local monodromies specified by generic semisimple conjugacy classes at $k$ punctures.
In this article, we establish the weak geometric P=W conjecture for all \emph{very generic} $\mathcal{M}_B$ in the sense that at least one conjugacy class is regular semisimple. A crucial step is to establish a stronger form of A. Mellit's cell decomposition theorem, i.e. we decompose $\mathcal{M}_B$ (without passing to a vector bundle) into locally closed subvarieties of the form $(\mathbb{C}^{\times})^{d-2b}\times\mathcal{A}$, where $\mathcal{A}$ is stably isomorphic to $\mathbb{C}^b$. A second ingredient involves a motivic characterization of the integral cohomology of dual boundary complexes developed in a subsequent article [Su24]. Following C. Simpson's strategy, the proof is now an inductive computation of the dual boundary complexes from such a cell decomposition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_16657 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Cell decomposition and dual boundary complexes of character varieties Su, Tao Algebraic Geometry Algebraic Topology Representation Theory 14M35 (Primary) 14C30, 14F45 (Secondary) The weak geometric P=W conjecture of L. Katzarkov, A. Noll, P. Pandit, and C. Simpson asserts that for any smooth Betti moduli space $\mathcal{M}_B$ of complex dimension $d$ over a punctured Riemann surface, the dual boundary complex $\mathbb{D}\partial\mathcal{M}_B$ is homotopy equivalent to a $(d-1)$-dimensional sphere. Here, we consider $\mathcal{M}_B$ as a generic $GL_n(\mathbb{C})$-character variety defined on a Riemann surface of genus $g$, with local monodromies specified by generic semisimple conjugacy classes at $k$ punctures. In this article, we establish the weak geometric P=W conjecture for all \emph{very generic} $\mathcal{M}_B$ in the sense that at least one conjugacy class is regular semisimple. A crucial step is to establish a stronger form of A. Mellit's cell decomposition theorem, i.e. we decompose $\mathcal{M}_B$ (without passing to a vector bundle) into locally closed subvarieties of the form $(\mathbb{C}^{\times})^{d-2b}\times\mathcal{A}$, where $\mathcal{A}$ is stably isomorphic to $\mathbb{C}^b$. A second ingredient involves a motivic characterization of the integral cohomology of dual boundary complexes developed in a subsequent article [Su24]. Following C. Simpson's strategy, the proof is now an inductive computation of the dual boundary complexes from such a cell decomposition. |
| title | Cell decomposition and dual boundary complexes of character varieties |
| topic | Algebraic Geometry Algebraic Topology Representation Theory 14M35 (Primary) 14C30, 14F45 (Secondary) |
| url | https://arxiv.org/abs/2307.16657 |