[Paper Review] Virtual signed Euler characteristics

Yunfeng Jiang, Richard Thomas|arXiv (Cornell University)|Aug 11, 2014

Algebraic Geometry and Number Theory22 references8 citations

TL;DR

This paper constructs a cone $N$ over a space $M$ with perfect obstruction theory, equipped with a symmetric obstruction theory, such that $N$ is locally the critical locus of a function. By localizing the virtual cycle of $N$ to its $\mathbb{C}^*$-fixed locus $M$, the authors define five notions of virtual signed Euler characteristic of $M$, proving that four of them—Graber-Pandharipande, Behrend Kai-weighted, Kiem-Li cosection, and topological Euler characteristic with sign $(-1)^{\operatorname{vd}}$—are equal, while the Ciocan-Fontanine-Kapranov/Fantechi-Göttsche invariant is equal to the Graber-Pandharipande version, though only the former two are deformation-invariant.

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.

Motivation & Objective

• To define multiple notions of virtual signed Euler characteristic for a space $M$ with perfect obstruction theory.
• To construct a canonical space $N$ with symmetric obstruction theory that is a cone over $M$, containing $M$ as its zero section.
• To relate invariants of $N$ to invariants of $M$ via $\mathbb{C}^*$-localization.
• To compare and unify different localization techniques: Graber-Pandharipande, Behrend Kai, Kiem-Li cosection, and topological Euler characteristic.
• To clarify which invariants are deformation-invariant and which are not, resolving ambiguities in the literature.

Proposed method

• Construct $N$ as the $(-1)$-shifted cotangent bundle of the derived space underlying $M$, or equivalently as the cone $\operatorname{Spec}\operatorname{Sym}^\bullet \operatorname{Ob}_M \to M$.
• Show that $N$ is locally the critical locus of a function $\widetilde{s}$ on the total space of the dual of the obstruction bundle, via local model $\operatorname{Crit}(\widetilde{s}) \subset \mathrm{Tot}(E^*)$.
• Use the $\mathbb{C}^*$-action on $N$ (scaling fibers) to define localization of virtual cycles and Behrend functions to the fixed locus $M$.
• Apply Graber-Pandharipande virtual Atiyah-Bott localization to the virtual cycle of $N$ to obtain a virtual cycle on $M$.
• Use Behrend’s formula $\chi^N(p) = (-1)^{\dim \mathrm{Tot}(E^*)} (1 - e(F_p))$ to compute the Behrend function on $N$, and show it restricts to $(-1)^{\operatorname{vd}}$ on $M$.
• Use Kiem-Li cosection theory: the $\mathbb{C}^*$-action induces an Euler vector field, which gives a cosection of the obstruction sheaf, enabling localization of the virtual cycle of $N$ to $M$.

Experimental results

Research questions

• RQ1 How do different localization techniques for virtual cycles on a noncompact space $N$ relate when restricted to its fixed locus $M$?
• RQ2 Are the virtual signed Euler characteristics defined via Graber-Pandharipande, Behrend Kai-weighting, Kiem-Li cosection, and topological Euler characteristic with sign $(-1)^{\operatorname{vd}}$ equivalent?
• RQ3 Why do some invariants (e.g., Ciocan-Fontanine-Kapranov) remain deformation-invariant while others (e.g., Behrend Kai) do not?
• RQ4 What is the precise value of the Behrend function $\chi^N$ on $M$ when $N$ is the cone over $M$ with symmetric obstruction theory?
• RQ5 Can the $\mathbb{C}^*$-action on $N$ be used to uniformly compute Milnor fibers and Euler characteristics across noncompact fibers?

Key findings

• The Behrend function $\chi^N$ on $N$ restricts to the constant $(-1)^{\operatorname{vd}}$ on $M$, i.e., $\chi^N|_M \equiv (-1)^{\operatorname{vd}}$.
• The Behrend Kai-weighted Euler characteristic of $N$ localizes to $e(M, \chi^N|_M) = (-1)^{\operatorname{vd}} e(M)$, which matches the topological Euler characteristic of $M$ with sign $(-1)^{\operatorname{vd}}$.
• The Kiem-Li cosection localization of the virtual cycle of $N$ to $M$ yields the same result as the Behrend Kai-weighted Euler characteristic, both equaling $(-1)^{\operatorname{vd}} e(M)$.
• The Graber-Pandharipande virtual Atiyah-Bott localization of $[N]^{\operatorname{vir}}$ to $M$ agrees with the Ciocan-Fontanine-Kapranov/Fantechi-Göttsche signed virtual Euler characteristic of $M$, i.e., $\int_{[M]^{\operatorname{vir}}} c_{\operatorname{vd}}(E^\bullet)$.
• The four invariants—Behrend Kai, Kiem-Li cosection, topological Euler characteristic with sign, and Graber-Pandharipande localization—are equal, while the Ciocan-Fontanine-Kapranov invariant equals the Graber-Pandharipande one, but only the latter two are deformation-invariant.