Authors: Soheyla Feyzbakhsh ; Laura Pertusi

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, Macrì and Stellari.

Comment: 32 pages

• 14F08 - Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
• 14J30 - \(3\)-folds
• 14J45 - Fano varieties
• 14J60 - Vector bundles on surfaces and higher-dimensional varieties, and their moduli
• 18G80 - Derived categories, triangulated categories

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