2013/05/21 by Bernardara, Marcello, Tabuada, Goncalo
#1422 #14C34 #14E08 #14J30 #14J45 #14K30 #18E30 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT)
paper · doi:10.48550/arxiv.1305.4687
Let X and Y be complex smooth projective varieties, and Db(X) and Db(Y) the associated bounded derived categories of coherent sheaves. Assume the existence of a triangulated category T which is admissible both in Db(X) as in Db(Y). Making use of the recent theory of Jacobians of noncommutative motives, we construct out of this categorical data a morphism t 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. When the orthogonal complement of T in Db(X) has a trivial Jacobian (e.g. when it is generated by exceptional objects), the morphism t is split injective. When this also holds for the orthogonal complement of T in Db(Y), t becomes an isomorphism. Furthermore, in the case where X and Y have a single intermediate algebraic Jacobian carrying a principal polarization, we prove that the morphism t preserves this extra structure. As an application, we obtain categorical Torelli theorems, an incompatibility between two conjectures of Kuznetsov (one concerning Fourier-Mukai functors and another one concerning Fano threefolds), a new proof of a classical theorem of Clemens-Griffiths on blow-ups of threefolds, and several new results on quadric fibrations and intersection of quadrics.