Virtual fundamental classes via dg-manifolds

Details Virtual fundamental classes via dg-manifolds Virtual fundamental classes via dg-manifolds

2007-03-08

arxiv.org/abs/math/0703214v2

Mathematics

Algebraic Geometry

24 pages

Scientific paper

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-Fantechi [BF] or Li-Tian [LT]. 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 [BF] 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".

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