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

Abstract

We show that the stability conditions on the Kuznetsov component of a Gushel–Mukai threefold, constructed by Bayer, Lahoz, Macrì and Stellari, are preserved by the Serre functor, up to the action of the universal cover of GL2+(prefixdouble-struckR)$\text{GL}^+_2(\operatorname{\mathbb {R}})$. As application, we construct stability conditions on the Kuznetsov component of special Gushel–Mukai fourfolds.

Categorical Torelli theorems for Gushel–Mukai threefolds

“…Now for the last statement, note that  𝑋 (−𝐻) [ When 𝑎 = −3, we have ch ⩽2 (𝐵) = ch ⩽2 ( ∨ ). Then since ch ⩽2 (𝐵) is on the boundary of Lemma 4.6, by a standard argument, we know that 𝐵 is 𝜎 𝛼,𝛽 -semistable for every 𝛼 > 0 and 𝛽 < 0, as explained in [48,Proposition 3.2]. Thus, by Lemma 4.5, 𝐵 is a 𝜇-semistable sheaf.…”

“…By [4], 𝜎(𝛼, 𝛽) is a stability condition on 𝑢(𝑌 𝑑 ) and 𝑢(𝑋 4𝑑+2 ) for suitable (𝛼, 𝛽). Moreover, according to [48,51], these stability conditions are all Serre-invariant.…”

“…In [48] and [51], the authors prove that stability conditions on Kuznetsov components of every del Pezzo threefold 𝑌 𝑑 of degree 𝑑 ⩾ 1 and every index 1 prime Fano threefold of genus g ⩾ 6 are Serre-invariant. In [17], the authors prove the uniqueness of Serre-invariant stability conditions for a general triangulated category satisfying a list of very natural assumptions, which include Kuznetsov components of a series of prime Fano threefolds.…”

Stability conditions and moduli spaces for Kuznetsov components of Gushel–Mukai varieties

“…By [PR21, Theorem 1.1] the stability condition τ X is Serre invariant. By [JLLZ21a, Lemmas 4.27, 4.28, 4.29] (see also [PR21,Corollary 4.5]) there is a unique GL + (2, R)-orbit of Serre invariant stability conditions on Ku(X). This implies the statement.…”