Local Contributions to Donaldson–Thomas Invariants

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.