A categorical invariant for cubic threefolds

Stability conditions on Kuznetsov components

“…The most interesting cases of index one are those of even genus g X = 1 2 H 3 +1, for which Mukai [Muk92] constructed an exceptional rank two vector bundle E 2 of slope − 1 2 ; in these cases our Theorem refers to the semiorthogonal decomposition D b (X) = Ku(X), E 2 , O X . The result is straightforward from previous descriptions of Ku(X) for g X ∈ {10, 12}, due to [BMMS12] for g X = 8, and new for g X = 6.…”

“…Their deformation type is determined by d = H 3 ∈ {1, 2, 3, 4, 5}. The result is straightforward from prior descriptions of Ku(X) for d ≥ 4, due to [BMMS12] for cubic threefolds (d = 3) and new for d ∈ {1, 2}. The most interesting cases of index one are those of even genus g X = 1 2 H 3 +1, for which Mukai [Muk92] constructed an exceptional rank two vector bundle E 2 of slope − 1 2 ; in these cases our Theorem refers to the semiorthogonal decomposition D b (X) = Ku(X), E 2 , O X .…”

Categorical representability and intermediate Jacobians of Fano threefolds

“…Notice that if X is a smooth cubic threefold, the equivalence class of a notable admissible subcategory A X (the orthogonal complement of {O X , O X (1)}) corresponds to the isomorphism class of J(X) as principally polarized abelian variety [BMMS09]; the proof is based on the reconstruction of the Fano variety and the techniques used there are far away from the subject of this paper.…”

Stability conditions on Kuznetsov components

“…Their deformation type is determined by d = H 3 ∈ {1, 2, 3, 4, 5}. The result is straightforward from prior descriptions of Ku(X) for d ≥ 4 in [Orl91,BO95], due to [BMMS12] for cubic threefolds (d = 3) and new for d ∈ {1, 2}. The most interesting cases of index one are those of even genus g X = 1 2 H 3 + 1, for which Mukai [Muk92] constructed an exceptional rank two vector bundle E 2 of slope − 1 2 ; in these cases our Theorem refers to the semiorthogonal decomposition D b (X) = Ku(X), E 2 , O X .…”

“…In the case of cubic fourfolds, as mentioned above, they conjecturally determine rationality of X. Finally, they are naturally related to Torelli type questions: on the one hand, they still encode much of the cohomological information of X; on the other hand, one can hope to recover X from Ku(X) (in some cases when equipped with some additional data); see [BMMS12] for such a result for cubic threefolds, and [HR19] for many hypersurfaces, including cubic fourfolds.…”