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Let X and Y be complex smooth projective varieties, and D^b(X) and D^b(Y) theassociated bounded derived categories of coherent sheaves. Assume the existenceof 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, weconstruct out of this categorical data a morphism t of abelian varieties (up toisogeny) from the product of the intermediate algebraic Jacobians of X to theproduct of the intermediate algebraic Jacobians of Y. When the orthogonalcomplement of T in D^b(X) has a trivial Jacobian (e.g. when it is generated byexceptional objects), the morphism t is split injective. When this also holdsfor the orthogonal complement of T in D^b(Y), t becomes an isomorphism.Furthermore, in the case where X and Y have a single intermediate algebraicJacobian carrying a principal polarization, we prove that the morphism tpreserves this extra structure. As an application, we obtain categoricalTorelli theorems, an incompatibility between two conjectures of Kuznetsov (oneconcerning Fourier-Mukai functors and another one concerning Fano threefolds),a new proof of a classical theorem of Clemens-Griffiths on blow-ups ofthreefolds, and several new results on quadric fibrations and intersection ofquadrics.

From semi-orthogonal decompositions to polarized intermediate Jacobians via Jacobians of noncommutative motives