Local contributions to Donaldson-Thomas invariants

Abstract

Let $C$ be a smooth curve embedded in a smooth quasi-projective threefold $Y$, and let $Q_C^n={\text{Quot}}_n({\mathcal{I}}_C)$ be the Quot scheme of length $n$ quotients of its ideal sheaf. We show the identity $\tilde{\chi }(Q_C^n)=(-1{)}^n\chi (Q_C^n)$, 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.

Generalized the assumptions in Section 3.2

References in corpus (1)

• Zero dimensional Donaldson-Thomas invariants of threefolds

Cited by in corpus (5)

• Higher rank K-theoretic Donaldson-Thomas theory of points
• The DT/PT correspondence for smooth curves
• The local motivic DT/PT correspondence
• The Hilbert scheme of hyperelliptic Jacobians and moduli of Picard sheaves
• Unweighted Donaldson-Thomas theory of the banana 3-fold with section classes