Derived equivalences over base schemes and support of complexes

TL;DR

This paper investigates conditions under which derived equivalences between smooth projective varieties over a base scheme are induced by complexes supported on fiber products over the base, with applications to canonical and Albanese fibrations.

Contribution

It establishes criteria linking derived equivalences to complexes supported on fibered products, extending understanding of how equivalences relate to fibrations over a base scheme.

Findings

Derived equivalences can be induced by complexes supported on fibered products.

Conditions are formulated for derived equivalences to respect fiber structures.

Applications include results on canonical and Albanese fibrations.

Abstract

Let $X$ and $Y$ be smooth projective varieties over a field $k$ admitting morphisms $f:X \to T$ and $g:Y \to T$ to a third variety $T$. We formulate conditions on a derived equivalence $\Phi:D(X) \to D(Y)$ ensuring that $\Phi$ is induced by a complex $P \in D(X \times_T Y )$, defining derived equivalences between the fibers of $f$ and $g$. We apply our results to the canonical fibration and albanese fibration.