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.

Key words: motivic Euler characteristics, dual obstruction cone, motivic Vafa-Witten invariants, K3 surfaces