Geometric stability conditions under autoequivalences and applications: Elliptic Surfaces

TL;DR

This paper investigates how autoequivalences like the Fourier-Mukai transform affect stability conditions on elliptic surfaces, leading to new insights into moduli space projectivity and stability preservation.

Contribution

It describes the action of the relative Fourier-Mukai transform on stability conditions and proves stability preservation and projectivity results for moduli spaces on elliptic surfaces.

Findings

Fourier-Mukai transform action on stability conditions is explicitly described.

Gieseker stability is preserved under derived duals in certain chambers.

Projectivity of some Bridgeland moduli spaces is established.

Abstract

On a Weierstrass elliptic surface, we describe the action of the relative Fourier-Mukai transform on the geometric chamber of $\mathrm{Stab}(X)$, and in the K3 case we also study the action on one of its boundary components. Using new estimates for the Gieseker chamber we prove that Gieseker stability for polarizations on certain Friedman chamber is preserved by the derived dual of the relative Fourier-Mukai transform. As an application of our description of the action, we also prove projectivity for some moduli spaces of Bridgeland semistable objects.