Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Virtual signed Euler characteristics


Authors: Yunfeng Jiang and Richard P. Thomas
Journal: J. Algebraic Geom. 26 (2017), 379-397
DOI: https://doi.org/10.1090/jag/690
Published electronically: October 21, 2016
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Abstract | References | Additional Information

Abstract: Roughly speaking, to any space $ M$ with perfect obstruction theory we associate a space $ N$ with symmetric perfect obstruction theory. It is a cone over $ M$ given by the dual of the obstruction sheaf of $ M$ and contains $ M$ as its zero section. It is locally the critical locus of a function.

More precisely, in the language of derived algebraic geometry, to any quasi-smooth space $ M$ we associate its $ (\!-\!1)$-shifted cotangent bundle $ N$.

By localising from $ N$ to its $ \mathbb{C}^*$-fixed locus $ M$ this gives five notions of a virtual signed Euler characteristic of $ M$:

  1. The Ciocan-Fontanine-Kapranov/Fantechi-Göttsche signed virtual Euler characteristic of $ M$ defined using its own obstruction theory,
  2. Graber-Pandharipande's virtual Atiyah-Bott localisation of the virtual cycle of $ N$ to $ M$,
  3. Behrend's Kai-weighted Euler characteristic localisation of the virtual cycle of $ N$ to $ M$,
  4. Kiem-Li's cosection localisation of the virtual cycle of $ N$ to $ M$,
  5. $ (-1)^{\textrm {vd}}$ times by the topological Euler characteristic of $ M$.
Our main result is that (1)=(2) and (3)=(4)=(5). The first two are deformation invariant while the last three are not.

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Additional Information

Yunfeng Jiang
Affiliation: Department of Mathematics, University of Kansas, 405 Jayhawk Boulevard, Lawrence, Kansas 66045
Email: y.jiang@ku.edu

Richard P. Thomas
Affiliation: Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
Email: richard.thomas@imperial.ac.uk

DOI: https://doi.org/10.1090/jag/690
Received by editor(s): September 2, 2014
Published electronically: October 21, 2016
Article copyright: © Copyright 2016 University Press, Inc.

American Mathematical Society