Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Virtual signed Euler characteristics

Yunfeng Jiang, Richard P Thomas

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 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.

Virtual signed Euler characteristics

Abstract

Roughly speaking, to any space with perfect obstruction theory we associate a space with symmetric perfect obstruction theory. It is a cone over given by the dual of the obstruction sheaf of , and contains 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 we associate its -shifted cotangent bundle . By localising from to its -fixed locus this gives five notions of virtual signed Euler characteristic of : (1) The Ciocan-Fontanine-Kapranov/Fantechi-Göttsche signed virtual Euler characteristic of defined using its own obstruction theory, (2) Graber-Pandharipande's virtual Atiyah-Bott localisation of the virtual cycle of to , (3) Behrend's Kai-weighted Euler characteristic localisation of the virtual cycle of to , (4) Kiem-Li's cosection localisation of the virtual cycle of to , (5) times by the topological Euler characteristic of . Our main result is that (1)=(2) and (3)=(4)=(5). The first two are deformation invariant while the last three are not.

Paper Structure

This paper contains 9 sections, 6 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. The construction of $N$
  3. The four virtual signed Euler characteristics
  4. The signed virtual Euler characteristic of Ciocan-Fontanine-Kapranov and Fantechi-Göttsche
  5. Graber-Pandharipande localisation
  6. Kai localisation
  7. Kiem-Li's cosection localisation
  8. Equality of Kai and Kiem-Li localisations
  9. Ordinary signed Euler characteristic

Key Result

Theorem 1.2

Suppose that $M$ is a projective scheme with perfect obstruction theory $E^{\bullet}$ arising as $\pi_0$ of a quasi-smooth derived scheme. Then $(1)=(2)$ and $(3)=(4)=(-1)^{\operatorname{vd}\space}e(M)$. The first two are deformation invariant while the last two are not.

Theorems & Definitions (13)

  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.8
  • proof
  • Proposition 3.3
  • proof
  • Example 3.1
  • Example 3.2
  • Lemma 4.4
  • ...and 3 more