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Refined sheaf counting on local K3 surfaces

Richard P. Thomas

Abstract

We compute all refined sheaf counting invariants -- Vafa-Witten, reduced DT, stable pairs and Gopakumar-Vafa -- for all classes on local $K3$ surfaces. Along the way we develop rank 0 Vafa-Witten theory on $K3$ surfaces. An important feature of the calculation is that the ``instanton contribution" -- of sheaves supported scheme theoretically on $S$ -- to any of the invariants depends only on the square of the class, not its divisibility.

Refined sheaf counting on local K3 surfaces

Abstract

We compute all refined sheaf counting invariants -- Vafa-Witten, reduced DT, stable pairs and Gopakumar-Vafa -- for all classes on local surfaces. Along the way we develop rank 0 Vafa-Witten theory on surfaces. An important feature of the calculation is that the ``instanton contribution" -- of sheaves supported scheme theoretically on -- to any of the invariants depends only on the square of the class, not its divisibility.
Paper Structure (8 sections, 5 theorems, 64 equations)

This paper contains 8 sections, 5 theorems, 64 equations.

Table of Contents

  1. Vafa-Witten invariants in all ranks
  2. Stable case
  3. Semistable case
  4. Vanishing and multiple cover formula
  5. The instanton contribution
  6. Wall crossing
  7. Deformations, autoequivalences and wall crossing
  8. Computation on $S$

Key Result

Theorem 1

KVW is true: for a Mukai vector $v\in H^*(S,\mathbb Z)$,

Theorems & Definitions (11)

  • Theorem 1
  • Remark 2
  • Remark 2.2
  • Theorem 2.5
  • proof
  • Proposition 3.9
  • Proposition 3.12
  • proof
  • Lemma 4.5
  • proof
  • ...and 1 more