Preservation of semistability under Fourier-Mukai transforms

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 ${\mu }_{\ast }$.

36 pages. Comments are welcome

References in corpus (3)

• The Space of Stability Conditions on Abelian Threefolds, and on some Calabi-Yau Threefolds
• T-structures on elliptic fibrations
• Some examples of tilt-stable objects on threefolds