Donaldson-Thomas invariants, linear systems and punctual Hilbert schemes

TL;DR

This paper explores Donaldson-Thomas invariants linked to stable sheaves on threefolds, revealing their connection to Hilbert schemes and modular properties under certain positivity conditions.

Contribution

It establishes a relationship between DT invariants and Carlsson-Okounkov formulas, highlighting modularity in the context of linear systems and punctual Hilbert schemes.

Findings

DT invariants relate to Carlsson-Okounkov formulas

Under positivity conditions, invariants exhibit modular properties

Connections between sheaf stability, Hilbert schemes, and modularity are demonstrated

Abstract

We study certain DT invariants arising from stable coherent sheaves in a nonsingular projective threefold supported on the members of a linear system of a fixed line bundle. When the canonical bundle of the threefold satisfies certain positivity conditions, we relate the DT invariants to Carlsson-Okounkov formulas for the "twisted Euler's number" of the punctual Hilbert schemes of nonsingular surfaces, and conclude they have a modular property.