Virtual fundamental classes via dg–manifolds

Abstract

We construct virtual fundamental classes for dg‐manifolds whose tangent sheaves have cohomology only in degrees 0 and 1. This condition is analogous to the existence of a perfect obstruction theory in the approach of Behrend and Fantechi [3] or Li and Tian [11]. Our class is initially defined in K ‐theory as the class of the structure sheaf of the dg‐manifold. We compare our construction with that of [3] as well as with the original proposal of Kontsevich. We prove a Riemann‐Roch type result for dg‐ manifolds which involves integration over the virtual class. We prove a localization theorem for our virtual classes. We also associate to any dg‐manifold of our type a cobordism class of almost complex (smooth) manifolds. This supports the intuition that working with dg‐manifolds is the correct algebro-geometric replacement of the analytic technique of “deforming to transversal intersection”. 14F05; 14A20