Acta mathematica scientia,Series B ›› 2022, Vol. 42 ›› Issue (6): 2279-2300.

• Articles •

### MOTIVIC VIRTUAL SIGNED EULER CHARACTERISTICS AND THEIR APPLICATIONS TO VAFA-WITTEN INVARIANTS

Yunfeng JIANG

1. College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China;Department of Mathematics, University of Kansas, KS 66045, USA
• Received:2022-06-30 Online:2022-12-25 Published:2022-12-16
• Contact: Yunfeng JIANG, E-mail: y.jiang@ku.edu E-mail:y.jiang@ku.edu

Abstract: For any scheme M with a perfect obstruction theory, Jiang and Thomas associated a scheme N with a 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. There exists a global motive for N locally given by the motive of the vanishing cycle of the local regular function. We prove a motivic localization formula under the good and circle compact C*-action for N. When taking the 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, using the main theorem we study the motivic generating series of the motivic Vafa-Witten invariants for K3 surfaces.

CLC Number:

• 14N35
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