Virtual signed Euler characteristics

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 ${\mathbb{C}}^{\ast }$-fixed locus $M$ this gives five notions of 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) $(-1{)}^{vd}$ 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.

Referees' corrections and improvements

References in corpus (2)

• Donaldson-Thomas invariants via microlocal geometry
• Deriving Deligne-Mumford Stacks with Perfect Obstruction Theories

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