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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
Published electronically: October 21, 2016
MathSciNet review: 3607000
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Abstract | References | Additional Information


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
MR Author ID: 714489

Richard P. Thomas
Affiliation: Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
MR Author ID: 636321

Received by editor(s): September 2, 2014
Published electronically: October 21, 2016
Article copyright: © Copyright 2016 University Press, Inc.