A categorical invariant for cubic threefolds

Abstract

We prove a categorical version of the Torelli theorem for cubic threefolds. More precisely, we show that the non-trivial part of a semi-orthogonal decomposition of the derived category of a cubic threefold characterizes its isomorphism class.

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.…”

Gepner Type Stability Condition via Orlov/Kuznetsov Equivalence

“…In [BMMS12], motivated by Kuznetsov's work [Kuz08], Bernardara-Macri-Mehrotra-Stellari described the triangulated category D X in terms of sheaves of Clifford algebras on P 2 . Let B 0 (resp.…”

“…All the assumptions in Subsection 6.1 are checked to satisfy in [BMMS12, Section 2.3]. Also there is an equivalence of triangulated categories (denoted by (σ * • Φ ′ ) −1 in [BMMS12])…”

The transcendental lattice of the sextic Fermat surface

“…Hence 1,1,3) , P (1,1,3,1) , P (1,3,1,1) , P (3,1,1,1) , P (5,5,5,3) , P (5,5,3,5) , P (5,3,5,5) , P (3,5,5,5) , Z (1,1,2,2) := P (1,1,2,2) , P (2,2,1,1) , P (5,5,4,4) , P (4,4,5,5) , Z (1,2,1,2) := P (1,2,1,2) , P (2,1,2,1) , P (5,4,5,4) , P (4,5,4,5) , Z (1,2,2,1) := P (1,2,2,1) , P (2,1,1,2) , P (5,4,4,5) , P (4,5,5,4) and…”

“…A derived categorical approach due to Kuznetsov [22] has seen recent activity [1,2,5,24]. Using the theory of semiorthogonal decompositions Kuznetsov constructs a triangulated category A X ⊂ D b (X) and conjectures that it encodes all the information concerning the rationality of X.…”