Stability and Fourier-Mukai Transforms on Higher Dimensional Elliptic Fibrations

TL;DR

This paper extends the study of stability conditions and Fourier-Mukai transforms to higher-dimensional elliptic fibrations, establishing key properties of moduli spaces and their transformations.

Contribution

It generalizes previous results by analyzing stability and Fourier-Mukai transforms on elliptic fibrations with arbitrary base dimensions.

Findings

Universal closedness for moduli of semistable objects established.

Openness of polynomial stability demonstrated.

Criteria for transforming semistable complexes to torsion-free sheaves provided.

Abstract

We consider elliptic fibrations with arbitrary base dimensions, and generalise previous work by the second author. In particular, we check universal closedness for the moduli of semistable objects with respect to a polynomial stability that reduces to PT-stability on threefolds. We also show openness of this polynomial stability. On the other hand, we write down criteria under which certain 2-term polynomial semistable complexes are mapped to torsion-free semistable sheaves under a Fourier-Mukai transform. As an application, we construct an open immersion from a moduli of complexes to a moduli of Gieseker stable sheaves on higher dimensional elliptic fibrations.