Serre-invariant stability conditions and Ulrich bundles on cubic threefolds

TL;DR

This paper establishes a criterion for the uniqueness of Serre-invariant stability conditions in fractional Calabi-Yau categories of dimension ≤2, applies it to the Kuznetsov component of cubic threefolds, and proves the irreducibility of certain Ulrich bundle moduli spaces.

Contribution

It provides a general criterion for the uniqueness of Serre-invariant stability conditions and applies it to cubic threefolds, leading to new results on Ulrich bundle moduli spaces.

Findings

All known stability conditions on Ku(X) are Serre-invariant.

The moduli space of rank ≥ 2 Ulrich bundles on X is irreducible.

The criterion ensures a unique Serre-invariant stability condition up to a specific group action.

Abstract

We prove a general criterion which ensures that a fractional Calabi--Yau category of dimension $\leq 2$ admits a unique Serre-invariant stability condition, up to the action of the universal cover of $\text{GL}^+_2(\mathbb{R})$. We apply this result to the Kuznetsov component $\text{Ku}(X)$ of a cubic threefold $X$. In particular, we show that all the known stability conditions on $\text{Ku}(X)$ are invariant with respect to the action of the Serre functor and thus lie in the same orbit with respect to the action of the universal cover of $\text{GL}^+_2(\mathbb{R})$. As an application, we show that the moduli space of Ulrich bundles of rank $\geq 2$ on $X$ is irreducible, answering a question asked by Lahoz, Macr\`i and Stellari.