Donaldson Thomas invariant of P^1 scroll

TL;DR

This paper investigates the Donaldson-Thomas invariants of P^1 scrolls over complex surfaces with a global two-form, revealing conditions under which these invariants vanish or depend solely on local geometry.

Contribution

It establishes the vanishing of Donaldson-Thomas invariants for certain classes and shows their dependence on local neighborhoods in specific cases.

Findings

Invariants are zero if the class is not a multiple of [C]

Invariants depend only on the analytic neighborhood of L when insertions are above C

Results connect global invariants to local geometric data

Abstract

Let X be a P^1 scroll (a compactification of a line bundle L) over a complex surafce S and assume S has a global two form with zero loci a smooth curve C. The Donaldson Thomas invariants of X is shown to be zero if the curve class has is component on S not a multiple of [C]. For nonzero case, when the prime field insertion are above C, the invariant is shown to depend only on the analytic neighborhood of L in X.