On motivic Joyce-Song formula for the Behrend function identities

Abstract

We prove the version of Joyce-Song formula for the Behrend function identities in the motivic setting. The main method we use is the proof of Kontsevich-Soibelman conjecture about the motivic Milnor fibers by Q. T. Le, who uses the method of motivic integration for formal schemes and Cluckers-Loeser's motivic constructible functions. In the Appendix the motivic formula can be used to provide a different proof that there is an algebra homomorphism of Kontsevich-Soibelman from the motivic Hall algebra of the abelian category of coherent sheaves on a Calabi-Yau threefold $Y$ to the motivic quantum torus of $Y$.

34 pages, major revision of the paper

References in corpus (9)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Donaldson-Thomas invariants via microlocal geometry
• Desingularization of quasi-excellent schemes in characteristic zero
• A trace formula for rigid varieties, and motivic Weil generating series for formal schemes
• Generalized Donaldson-Thomas theory over fields K $\ne$ C
• The motivic Thom-Sebastiani theorem for regular and formal functions
• Pro-Chern-Schwartz-MacPherson class for DM stacks
• Formal and rigid geometry: an intuitive introduction, and some applications
• Note on the motivic DT/PT correspondence and the motivic Flop formula

Cited by in corpus (4)

• Hall algebras in the derived category and higher rank DT invariants
• The moduli space of stable coherent sheaves via non-archimedean geometry
• Motivic virtual signed Euler characteristics and applications to Vafa-Witten invariants
• Note on the motivic DT/PT correspondence and the motivic Flop formula