Vertical sheaves and Fourier-Mukai transform on elliptic Calabi-Yau threefolds

TL;DR

This paper explores how the Fourier-Mukai transform acts on moduli spaces of vertical torsion sheaves on elliptic Calabi-Yau threefolds, providing insights into their structure and implications for Donaldson-Thomas invariants.

Contribution

It establishes a correspondence between moduli stacks of semistable one-dimensional and two-dimensional sheaves via Fourier-Mukai transform on elliptic Calabi-Yau threefolds.

Findings

Identification of moduli stacks with open and closed substacks

Explicit conjectural formulas for Donaldson-Thomas invariants

Insights into the structure of vertical sheaves on elliptic Calabi-Yau threefolds

Abstract

This paper studies the action of the Fourier-Mukai transform on moduli spaces of vertical torsion sheaves on elliptic Calabi-Yau threefolds in Weierstrass form. Moduli stacks of semistable one dimensional sheaves on such threefolds are identified with open and closed substacksof moduli stacks of vertical semistable two dimensional sheaves on their Fourier-Mukai duals. In particular, this yields explicit conjectural results for Donaldson-Thomas invariants of vertical two dimensional sheaves on K3-fibered elliptic Calabi-Yau threefolds.