Local contributions to Donaldson-Thomas invariants

TL;DR

This paper proves a new identity relating Behrend weighted Euler characteristics and ordinary Euler characteristics for Quot schemes of ideal sheaves of smooth curves, providing insights into Donaldson-Thomas invariants and their local contributions.

Contribution

It establishes a fundamental identity connecting Behrend and ordinary Euler characteristics for Quot schemes of smooth curves, advancing understanding of DT invariants in algebraic geometry.

Findings

Proves $ ilde ext{chi}(Q^n_C)=(-1)^n ext{chi}(Q^n_C)$ for Quot schemes.

Shows DT contribution of a smooth rigid curve equals its signed Euler characteristic.

Reformulates the result as a DT/PT wall-crossing formula and a Behrend function identity.

Abstract

Let $C$ be a smooth curve embedded in a smooth quasi-projective threefold $Y$, and let $Q^n_C=\textrm{Quot}_n(\mathscr I_C)$ be the Quot scheme of length $n$ quotients of its ideal sheaf. We show the identity $\tilde\chi(Q^n_C)=(-1)^n\chi(Q^n_C)$, where $\tilde\chi$ is the Behrend weighted Euler characteristic. When $Y$ is a projective Calabi-Yau threefold, this shows that the DT contribution of a smooth rigid curve is the signed Euler characteristic of the moduli space. This can be rephrased as a DT/PT wall-crossing type formula, which can be formulated for arbitrary smooth curves. In general, the formula is shown to be equivalent to a certain Behrend function identity.