Atiyah class and sheaf counting on local Calabi Yau fourfolds

TL;DR

This paper explores Donaldson-Thomas invariants of sheaves on local Calabi-Yau fourfolds, establishing a reduction to threefold DT theory and predicting modularity for elliptic K3 bases.

Contribution

It demonstrates a reduction of the universal Atiyah class for certain sheaves, linking fourfold DT invariants to threefold and surface sheaf moduli spaces.

Findings

Universal Atiyah class reduces to one in specific cases

Fourfold DT theory relates to reduced threefold DT theory

Predictions made about modularity for elliptic K3 surfaces

Abstract

We discuss Donaldson-Thomas (DT) invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non-compact Calabi Yau fourfold given by the total space of a rank 2 bundle on the surface. We prove that in certain cases, when the rank 2 bundle is chosen appropriately, the universal truncated Atiyah class of these codimension 2 sheaves reduces to one, defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor (threefold) embedded in the ambient fourfold. Such reduction property of universal Atiyah class enables us to relate our fourfold DT theory to a reduced DT theory of a threefold and subsequently then to the moduli spaces of sheaves on the base surface using results in arXiv:1701.08899 and arXiv:1701.08902 and . We finally make predictions about modularity of such fourfold invariants when the base surface is an elliptic K3.