Motivic virtual signed Euler characteristics and applications to Vafa-Witten invariants

Abstract

For any scheme $M$ with a perfect obstruction theory, Jiang and Thomas associate a scheme $N$ with symmetric perfect obstruction theory. The scheme $N$ is a cone over $M$ given by the dual of the obstruction sheaf of $M$, and contains $M$ as its zero section. Locally $N$ is the critical locus of a regular function. In this note we prove that $N$ is a $d$-critical scheme in the sense of Joyce. By assuming an orientation on $N$ there exists a global motive for $N$ locally given by the motive of vanishing cycles of the local regular function. We prove a motivic localization formula under the good and circle compact $\C^*$-action for $N$. When taking Euler characteristic the weighted Euler characteristic of $N$ weighted by the Behrend function is the signed Euler characteristic of $M$ by motivic method. As applications we calculate the motivic generating series of the motivic Vafa-Witten invariants for K3 surfaces. This motivic series gives the result of the ${\chi }_y$-genus for Vafa-Witten invariants of K3 surfaces, which is the same (at instanton branch) as the K-theoretical Vafa-Witten invariants of Thomas.

26 pages, newly revised version, added an application to motivic Vafa-Witten invariants, and calculated the motivic series for a case of K3 surfaces proving Goschett-Kool conjecture. Comments are very welcome

References in corpus (7)

• Donaldson-Thomas invariants via microlocal geometry
• Vafa-Witten invariants for projective surfaces I: stable case
• A trace formula for rigid varieties, and motivic Weil generating series for formal schemes
• Vafa-Witten invariants for projective surfaces II: semistable case
• Sheaf counting on local K3 surfaces
• Semi-Perfect Obstruction theory and DT Invariants of Derived Objects
• The moduli space of stable coherent sheaves via non-archimedean geometry

Cited by in corpus (1)

• Note on the motivic DT/PT correspondence and the motivic Flop formula