Virtual signed Euler characteristics
  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 virtual signed Euler characteristic of $M$:   (1) The Ciocan-Fontanine-Kapranov/Fantechi-G\"ottsche 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. 
Yunfeng Jiang, Richard P Thomas
math.AG