Geometry & Topology

Virtual fundamental classes via dg–manifolds

Ionuţ Ciocan-Fontanine and Mikhail Kapranov

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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".

Article information

Geom. Topol., Volume 13, Number 3 (2009), 1779-1804.

Received: 28 March 2008
Accepted: 19 February 2009
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20]
Secondary: 14A20: Generalizations (algebraic spaces, stacks)

virtual class dg-manifold cobordism


Ciocan-Fontanine, Ionuţ; Kapranov, Mikhail. Virtual fundamental classes via dg–manifolds. Geom. Topol. 13 (2009), no. 3, 1779--1804. doi:10.2140/gt.2009.13.1779.

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