Preservation of semistability under Fourier-Mukai transforms

TL;DR

This paper investigates how Gieseker semistability of sheaves on a trivial elliptic fibration is preserved or related under Fourier-Mukai transforms, especially with fiber-like polarizations and specific Chern class restrictions.

Contribution

It demonstrates preservation of Gieseker semistability under Fourier-Mukai transforms for certain polarizations and Chern classes, and relates it to a slope-like semistability in more general cases.

Findings

Gieseker semistability is preserved under Fourier-Mukai transforms with fiber-like polarizations.

A correspondence between Gieseker semistability and a slope-like semistability is established for general Chern classes.

The study provides conditions under which stability notions are compatible with Fourier-Mukai autoequivalences.

Abstract

For a trivial elliptic fibration $X=C \times S$ with $C$ an elliptic curve and $S$ a projective K3 surface of Picard rank $1$, we study how various notions of stability behave under the Fourier-Mukai autoequivalence $\Phi$ on $D^b(X)$, where $\Phi$ is induced by the classical Fourier-Mukai autoequivalence on $D^b(C)$. We show that, under some restrictions on Chern classes, Gieseker semistability on coherent sheaves is preserved under $\Phi$ when the polarisation is `fiber-like'. Moreover, for more general choices of Chern classes, Gieseker semistability under a `fiber-like' polarisation corresponds to a notion of $\mu_\ast$-semistability defined by a `slope-like' function $\mu_\ast$.