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 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.