The Blowup Formula for the Instanton Part of Vafa-Witten Invariants on Projective Surfaces
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866915318577758208 |
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| author | Kuhn, Nikolas Leigh, Oliver Tanaka, Yuuji |
| author_facet | Kuhn, Nikolas Leigh, Oliver Tanaka, Yuuji |
| contents | We prove a blow-up formula for the generating series of virtual $χ_y$-genera for moduli spaces of sheaves on projective surfaces, which is related to a conjectured formula for topological $χ_y$-genera of Göttsche. Our formula is a refinement of one by Vafa-Witten relating to S-duality. We prove the formula simultaneously in the setting of Gieseker stable sheaves on polarised surfaces and also in the setting of framed sheaves on $\mathbb{P}^2$. The proof is based on the blow-up algorithm of Nakajima-Yoshioka for framed sheaves on $\mathbb{P}^2$, which has recently been extend to the setting of Gieseker $H$-stable sheaves on $H$-polarised surfaces by Kuhn-Tanaka. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_12953 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Blowup Formula for the Instanton Part of Vafa-Witten Invariants on Projective Surfaces Kuhn, Nikolas Leigh, Oliver Tanaka, Yuuji Algebraic Geometry 14D20, 14D21, 14J60, 14J80, 14J81 We prove a blow-up formula for the generating series of virtual $χ_y$-genera for moduli spaces of sheaves on projective surfaces, which is related to a conjectured formula for topological $χ_y$-genera of Göttsche. Our formula is a refinement of one by Vafa-Witten relating to S-duality. We prove the formula simultaneously in the setting of Gieseker stable sheaves on polarised surfaces and also in the setting of framed sheaves on $\mathbb{P}^2$. The proof is based on the blow-up algorithm of Nakajima-Yoshioka for framed sheaves on $\mathbb{P}^2$, which has recently been extend to the setting of Gieseker $H$-stable sheaves on $H$-polarised surfaces by Kuhn-Tanaka. |
| title | The Blowup Formula for the Instanton Part of Vafa-Witten Invariants on Projective Surfaces |
| topic | Algebraic Geometry 14D20, 14D21, 14J60, 14J80, 14J81 |
| url | https://arxiv.org/abs/2205.12953 |