FROM SEMI-ORTHOGONAL DECOMPOSITIONS TO POLARIZED INTERMEDIATE JACOBIANS VIA JACOBIANS OF NONCOMMUTATIVE MOTIVES

摘要

Let X and Y be complex smooth projective varieties, and D-b(X) and D-b(Y) the associated bounded derived categories of coherent sheaves. Assume the existence of a triangulated category T which is admissible both in D-b(X) as in D-b(Y). Making use of the recent theory of Jacobians of noncommutative motives, we construct out of this categorical data a morphism tau of abelian varieties (up to isogeny) from the product of the intermediate algebraic Jacobians of X to the product of the intermediate algebraic Jacobians of Y. Our construction is conditional on a conjecture of Kuznetsov concerning functors of Fourier Mukai type and on a conjecture concerning intersection bilinear pairings (which follows from Grothendieck's standard conjecture of Lefschetz type). We describe several examples where these conjectures hold and also some conditional examples. When the orthogonal complement T-perpendicular to of T subset of D-b (X) has a trivial Jacobian (e.g., when T-perpendicular to is generated by exceptional objects), the morphism tau is split injective. When this also holds for the orthogonal complement T-perpendicular to of T subset of D-b(Y), tau becomes an isomorphism. Furthermore, in the case where X and Y have a unique principally polarized intermediate Jacobian, we prove that tau preserves the principal polarization.