On multiple cover formula for local K3 gerbes

Abstract

We generalize the multiple cover formula of Y. Toda (proved by Maulik-Thomas) for counting invariants for semistable coherent sheaves on local K3 surfaces to semistable twisted sheaves over twisted local K3 surfaces. The formula has an application to prove any rank S-duality conjecture for K3 surfaces. CONTENTS 1. Introduction 1 1.1. Outline 3 1.2. Convention 3 Acknowledgments 4 2. Étale gerbes and stacks, notations 4 3. Stable pair theory on some threefold DM stacks 4 3.1. Stable pair theory 4 3.2. Stable pairs on Y = S ˆE. 5 3.3. The elliptic curve E-action 6 3.4. The reduced perfect obstruction theory 7 3.5. Symmetric obstruction theory of the quotient 8 4. Behrend equals to Reduced invariants of Oberdieck 13 5. Wall crossing for Calabi-Yau gerbes and Toda's multiple cover formula 15 5.1. Wall-crossing in D0-D2-D6 bound states for µ r -gerbes 15 5.2. Decomposition formula for étale gerbes 22 5.3. Multiple cover formula. 24 6. Multiple cover formula for twisted K3 surfaces 26 6.1. KKV formula for K3 surfaces and the theorem of Maulik-Thomas 26 6.2. Category of twisted sheaves on X = S ˆP1 26 6.3. The invariants N(v G ) 29 6.4. Twisted Hodge isometry 32 Appendix A. The invariants for twisted sheaves on twisted K3 surfaces 38 A.1. Review of stability conditions on twisted K3 surfaces 38 A.2. The moduli stack counting semistable objects 41 A.3. Counting twisted semistable objects and twisted semistable sheaves 42 References 46