[Paper Review] Donaldson-Thomas invariants via microlocal geometry

Kai Behrend|ArXiv.org|Jul 26, 2005

Algebraic Geometry and Number TheoryMathematics9 references127 citations

TL;DR

This paper establishes that Donaldson-Thomas type invariants are equal to the weighted Euler characteristic of the moduli space using a canonical constructible function $\nu_X$, which depends only on the scheme structure, not the symmetric obstruction theory. The key contribution is a microlocal formula for $\nu_X(P)$ as a linking number of Lagrangian cycles in the cotangent bundle, unifying virtual counts with singularity invariants and enabling invariants for non-proper moduli spaces.

ABSTRACT

We prove that Donaldson-Thomas type invariants are equal to weighted Euler characteristics of their moduli spaces. In particular, such invariants depend only on the scheme structure of the moduli space, not the symmetric obstruction theory used to define them. We also introduce new invariants generalizing Donaldson-Thomas type invariants to moduli problems with open moduli space. These are useful for computing Donaldson-Thomas type invariants over stratifications.

Motivation & Objective

• To show that Donaldson-Thomas type invariants depend only on the scheme structure of the moduli space, not on the symmetric obstruction theory used.
• To define a generalization of Donaldson-Thomas invariants for non-proper moduli spaces using weighted Euler characteristics of $\nu_X$.
• To provide a microlocal geometric formula for $\nu_X(P)$ in terms of linking numbers of Lagrangian cycles in the cotangent bundle.
• To establish a connection between $\nu_X$ and the Euler characteristic of the Milnor fibre via vanishing cycles.
• To motivate a motivic generalization of Donaldson-Thomas invariants using $\tilde{\mu}(X) = \int_X \nu_X d\mu$.

Proposed method

• Define $\nu_X: X \to \mathbb{Z}$ as the Euler obstruction of the intrinsic normal cone $\mathfrak{c}_X$, which is canonical and scheme-theoretic.
• Use microlocal geometry to interpret $\nu_X(P)$ as the linking number $L_{S_\epsilon}(\Gamma_\eta \cap S_\epsilon, \Delta \cap S_\epsilon)$ for small $\epsilon$ and $\eta$.
• Embed the moduli space $X$ into a smooth ambient scheme $M$, and lift the 1-form $\omega$ defining $X = Z(\omega)$ to a section in $\Omega_M$.
• Construct the Lagrangian cycle $\Gamma_\eta \subset \Omega_M$ as the image of $M \to \Omega_M$ via $\frac{1}{\eta}\omega$, and $\Delta$ as the image of $d\rho$ for the distance function $\rho$.
• Apply intersection theory and specialization to show $\nu_X(P) = I_{\{P\}}([C], [\Delta])$, where $[C]$ is the normal cone of $X$ in $M$.
• Use the fact that $\lim_{\eta \to 0} [\Gamma_\eta] = [C]$ to replace $[C]$ with $\Gamma_\eta$ in the linking number formula.

Experimental results

Research questions

• RQ1 Does the Donaldson-Thomas invariant depend only on the scheme structure of the moduli space, or on the choice of symmetric obstruction theory?
• RQ2 Can Donaldson-Thomas type invariants be generalized to non-proper moduli spaces?
• RQ3 Is there a geometric, microlocal formula for the contribution $\nu_X(P)$ of a point to the virtual count?
• RQ4 How does $\nu_X$ relate to classical invariants like the Milnor number or Euler characteristic of the Milnor fibre?
• RQ5 Can motivic invariants be defined that generalize the virtual count in a way that respects scheme structure and $\nu_X$?

Key findings

• The Donaldson-Thomas type invariant $\#^{\rm vir}(X)$ equals the weighted Euler characteristic $\chi(X, \nu_X)$, proving invariance under symmetric obstruction theory choice.
• $\nu_X(P) = (-1)^{\dim X}$ at smooth points $P \in X$, generalizing the Euler characteristic formula.
• For $X = Z(df)$, the critical locus of a function $f$, $\nu_X(P) = (-1)^{\dim M}(1 - \chi(F_P))$, where $F_P$ is the Milnor fibre.
• $\nu_X(P)$ is computed microlocally as the linking number $L_{S_\epsilon}(\Gamma_\eta \cap S_\epsilon, \Delta \cap S_\epsilon)$, providing a geometric interpretation.
• The normal cone $C_{X/M}$ in $\Omega_M$ is Lagrangian, and its intersection with $\Delta$ gives the correct linking number for $\nu_X(P)$.
• The motivic generalization $\tilde{\mu}(X) = \int_X \nu_X d\mu$ encodes scheme structure more deeply than the usual motive $\mu(X)$, but faces limitations on stacks due to $\chi(GL_n) = 0$.