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| Main Author: | |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2204.09824 |
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| _version_ | 1866917744924950528 |
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| author | Chen, Yuhang |
| author_facet | Chen, Yuhang |
| contents | We study the orbifold Hirzebruch-Riemann-Roch (HRR) theorem for quotient Deligne-Mumford stacks, explore its relation with the representation theory of finite groups, and derive a new orbifold HRR formula via an orbifold Mukai pairing. As a first application, we use this formula to compute the dimensions of $ G $-equivariant moduli spaces of stable sheaves on a $ K3 $ surface $ X $ under the action of a finite subgroup $ G $ of its symplectic automorphism group. We then apply the orbifold HRR formula to reproduce the number of fixed points on $ X $ when $ G $ is cyclic without using the Lefschetz fixed point formula. We prove that under some mild conditions, equivariant moduli spaces of stable sheaves on $ X $ are irreducible symplectic manifolds deformation equivalent to Hilbert schemes of points on $ X $ via a connection between Gieseker and Bridgeland moduli spaces, as well as the derived McKay correspondence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_09824 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Orbifold Hirzebruch-Riemann-Roch for quotient Deligne-Mumford stacks and equivariant moduli theory on $ K3 $ surfaces Chen, Yuhang Algebraic Geometry We study the orbifold Hirzebruch-Riemann-Roch (HRR) theorem for quotient Deligne-Mumford stacks, explore its relation with the representation theory of finite groups, and derive a new orbifold HRR formula via an orbifold Mukai pairing. As a first application, we use this formula to compute the dimensions of $ G $-equivariant moduli spaces of stable sheaves on a $ K3 $ surface $ X $ under the action of a finite subgroup $ G $ of its symplectic automorphism group. We then apply the orbifold HRR formula to reproduce the number of fixed points on $ X $ when $ G $ is cyclic without using the Lefschetz fixed point formula. We prove that under some mild conditions, equivariant moduli spaces of stable sheaves on $ X $ are irreducible symplectic manifolds deformation equivalent to Hilbert schemes of points on $ X $ via a connection between Gieseker and Bridgeland moduli spaces, as well as the derived McKay correspondence. |
| title | Orbifold Hirzebruch-Riemann-Roch for quotient Deligne-Mumford stacks and equivariant moduli theory on $ K3 $ surfaces |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2204.09824 |