Algebraic geometry· 2026Q1
The blow-up formula for the instanton part of Vafa–Witten invariants on projective surfaces
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- 2026year
Short summary
A blow-up formula for the generating series of virtual χy-genera for moduli spaces of sheaves on projective surfaces is proven, refining a Vafa-Witten conjecture related to S-duality.
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Key points
- A blow-up formula for virtual χy-genera of sheaf moduli spaces on projective surfaces is proven.
- The formula refines the Vafa-Witten conjecture related to S-duality.
- The proof utilizes the blow-up algorithm of Nakajima-Yoshioka and its extension by Kuhn-Tanaka.
- The formula is demonstrated for both Gieseker stable sheaves and framed sheaves on P^2.
AI-generated from the title and abstract; the full text is not read.
Abstract
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 of Göttsche for topological χ y -genera.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 P 2 .The proof is based on the blow-up algorithm of Nakajima-Yoshioka for framed sheaves on P 2 , which has recently been extended to the setting of Gieseker H-stable sheaves on H-polarised surfaces by Kuhn-Tanaka.
The authors' abstract, as published at the source. Algebraic geometry, 2026 · DOI ↗
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Field: Geometry and Topology
Geometry and TopologyMathematics