Serre-invariant stability conditions and Ulrich bundles on cubic threefolds

Abstract

We prove a general criterion which ensures that a fractional Calabi--Yau category of dimension $\le 2$ 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 $\ge 2$ on $X$ is irreducible, answering a question asked by Lahoz, Macrì and Stellari.

arXiv comment32 pages

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Cited by in corpus (2)

• Categorical Torelli theorems for Gushel-Mukai threefolds
• On two families of Enriques categories over K3 surfaces