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 [Invent. Math 128 (1997) 45-88] or Li and Tian
[J. Amer. Math. Soc. 11 (1998) 119-174]. Our class is initially defined in
–theory
as the class of the structure sheaf of the dg–manifold. We compare our construction
with that of Behrend and Fantechi 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".