Virtual fundamental classes via dg-manifolds
By Ionut Ciocan-Fontanine, M. Kapranov. 2007
111 citations recorded by the source corpus.
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
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/b91b817e4c3e85cdf47042af405e3b7def5576ef
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In Geometry and Topology
- Davesh Maulik
- Bumsig Kim
- Ionut Ciocan-Fontanine
- A. Okounkov
- R. Pandharipande
- Bertrand Toën
- D. Huybrechts
- Tobias Dyckerhoff
In Mathematical Physics
- Bumsig Kim
- Ionut Ciocan-Fontanine
- A. Okounkov
- Davesh Maulik
- R. Pandharipande
- Bertrand Toën
- Tobias Dyckerhoff
- D. Huybrechts
Citations in the corpus
- Lectures on K-theoretic computations in enumerative geometry (2015, background): “The virtual localization theorem of [19], see also [38] for a cohomological version, states that (3.”
- Stable quasimaps to GIT quotients (2011, background): “Specifically, (i) when X is a Grassmannian, the Quot schemes on C carry a perfect obstruction theory, see [34] and [9]; (ii) when X is a toric variety, the “toric compactifications” (or linear sigma models) of Givental and Morrison-Plesser also carry perfect obstruction theories; (iii) when X is the”
- Derived algebraic geometry (2014, background): “We refer to [29, 87] for more on the subject.”
- Derived Algebraic Geometry (2014, background): “We refer to [29, 87] for more on the subject.”