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 $\leq 2$ 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.

Stability conditions on Kuznetsov components of Gushel–Mukai threefolds and Serre functor

“…Corollary 4.5 has been recently proved in [18, Lemmas 4.22, 4.23, 4.24]. Here, we give an alternative proof making use of the following result obtained from [14]. Theorem Let $$\mathcal {T}$$ be a $$\mathbb {C}$$‐linear triangulated category of finite type whose Serre functor satisfies $$S_{\mathcal {T}}^2=[4]$$ and whose numerical Grothendieck group $$\mathcal {N}(\mathcal {T})$$ has rank 2.…”

“…Here, we give an alternative proof making use of the following result obtained from [14]. Theorem 4.6 [14], Theorem 3.2, Lemma 3.6. Let  be a ℂ-linear triangulated category of finite type whose Serre functor satisfies 𝑆 2  = [4] and whose numerical Grothendieck group  ( ) has rank 2.…”

“…\end{equation*}$$ Via $$\Phi _d$$, the stability conditions $$\sigma (s,q)$$ on $$\mathop {\mathrm{Ku}}\nolimits (X_{4d+2})$$ define stability conditions on $$\mathop {\mathrm{Ku}}\nolimits (Y_d)$$. By Theorem 3.18 and [14, Theorem 3.2], [18, Theorem 4.25], these stability conditions are in the same orbit with respect to the $$\widetilde{\text{GL}}^+_2(\operatorname{\mathbb {R}})$$‐action of those constructed in [4] on $$\mathop {\mathrm{Ku}}\nolimits (Y_d)$$. An interesting problem would be to understand whether there is a unique orbit of stability conditions on $$\mathop {\mathrm{Ku}}\nolimits (Y_d)$$.…”

“…In this section, we drop the superfluous subscript and write $$\mathop {\mathrm{Ku}}\nolimits (X)= \mathop {\mathrm{Ku}}\nolimits (X)_i$$ for any given $$i=1,2,3$$ to simplify the notation, as the results contained herein hold for all such choices. We introduce the following definition (see [14, Definition 3.1]). Definition A stability condition σ on $$\mathop {\mathrm{Ku}}\nolimits (X)$$ is Serre‐invariant, or $$S_{\mathop {\mathrm{Ku}}\nolimits (X)}$$‐invariant, if $$S_{\mathop {\mathrm{Ku}}\nolimits (X)} \cdot \sigma = \sigma \cdot \widetilde{g}$$ for some $$\widetilde{g} \in \widetilde{\text{GL}}^+_2(\operatorname{\mathbb {R}})$$.…”

“…In fact, the property of Serre‐invariance is very helpful in the study of the properties of moduli spaces and the stability of objects, see, for instance, [18, 32] for many recent applications. In [14] the notion of Serre‐invariance is applied to show that the moduli space of stable Ulrich bundles of rank $$d \ge 2$$ on a cubic threefold is irreducible. On the other hand, not all triangulated subcategories of the bounded derived category of a smooth projective variety admit Serre‐invariant stability conditions.…”