Stability conditions on product varieties

TL;DR

This paper constructs new stability conditions on product varieties, especially on products of curves, by extending existing stability conditions from a base variety using advanced techniques in derived categories.

Contribution

It introduces a method to build stability conditions on product varieties from those on the base variety, including products of curves, using positivity lemmas and weak stability conditions.

Findings

Existence of stability conditions on arbitrary products of curves.

Construction method based on positivity lemma and weak stability conditions.

Extension of stability conditions from a variety to its product with a curve.

Abstract

Given a stability condition on a smooth projective variety $X$, we construct a family of stability conditions on $X\times C$, where $C$ is a smooth projective curve. In particular, this gives the existence of stability conditions on arbitrary products of curves. The proof uses, by following an idea of Toda, the positivity lemma established by Bayer and Macr\`i and weak stability conditions on the Abramovich-Polishchuk heart of a bounded t-structure in $D(X\times C)$.