A decomposition theorem for the affine Springer fibers
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916202698244096 |
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| author | Chen, Zongbin |
| author_facet | Chen, Zongbin |
| contents | According to Laumon, an affine Springer fiber is homeomorphic to the universal abelian covering of the compactified Jacobian of a spectral curve. We construct equivariant deformations $f_{n}:\overline{\mathcal{P}}_{n}\to \mathcal{B}_{n}$ of the finite abelian coverings of this compactified Jacobian, and decompose the complex $Rf_{n,*}\mathbf{Q}_{\ell}$ as direct sum of intersection complexes. Pass to the limit, we obtain a similar expression for the homology of the affine Springer fibers. A quite surprising consequence is that we can reduce the homology to its $Λ^{0}$-invariant subspace. As an application, we get a sheaf-theoretic reformulation of the purity hypothesis of Goresky, Kottwitz and MacPherson. In an attempt to solve it, we propose a conjecture about the punctural weight of the intermediate extension of a smooth $\ell$-adic sheaf of pure weight. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08225 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A decomposition theorem for the affine Springer fibers Chen, Zongbin Algebraic Geometry According to Laumon, an affine Springer fiber is homeomorphic to the universal abelian covering of the compactified Jacobian of a spectral curve. We construct equivariant deformations $f_{n}:\overline{\mathcal{P}}_{n}\to \mathcal{B}_{n}$ of the finite abelian coverings of this compactified Jacobian, and decompose the complex $Rf_{n,*}\mathbf{Q}_{\ell}$ as direct sum of intersection complexes. Pass to the limit, we obtain a similar expression for the homology of the affine Springer fibers. A quite surprising consequence is that we can reduce the homology to its $Λ^{0}$-invariant subspace. As an application, we get a sheaf-theoretic reformulation of the purity hypothesis of Goresky, Kottwitz and MacPherson. In an attempt to solve it, we propose a conjecture about the punctural weight of the intermediate extension of a smooth $\ell$-adic sheaf of pure weight. |
| title | A decomposition theorem for the affine Springer fibers |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2404.08225 |