2017/10/24 by Yunfeng Jiang, Jiang, Yunfeng · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.1710.08987
openalex publication_date 2017/10/24 · openalex created_date 2020/05/01 · openalex updated_date 2026/07/28
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 χy-genus for Vafa-Witten invariants of K3 surfaces, which is the same (at instanton branch) as the K-theoretical Vafa-Witten invariants of Thomas.