Volume 13, issue 3 (2009)

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Virtual fundamental classes via dg–manifolds

Ionuţ Ciocan-Fontanine and Mikhail Kapranov

Geometry & Topology 13 (2009) 1779–1804
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 [Invent. Math 128 (1997) 45-88] or Li and Tian [J. Amer. Math. Soc. 11 (1998) 119-174]. 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 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".

Keywords
virtual class, dg-manifold, cobordism
Primary: 14F05
Secondary: 14A20
Publication
Accepted: 19 February 2009
Published: 16 March 2009
Proposed: Jim Bryan
Seconded: Peter Teichner, Haynes Miller
Authors
 Ionuţ Ciocan-Fontanine Department of Mathematics University of Minnesota 127 Vincent Hall 206 Church St SE Minneapolis, MN 55455 USA Mikhail Kapranov Department of Mathematics Yale University 10 Hillhouse Avenue New Haven, CT 06520 USA