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

TL;DR

This paper develops a categorical approach to relate the intermediate Jacobians of complex varieties using noncommutative motives, leading to Torelli theorems and new insights into algebraic geometry structures.

Contribution

It introduces a method to construct morphisms between Jacobians from categorical data, connecting derived categories and classical geometric invariants.

Findings

Constructed Jacobian morphisms from categorical data.

Proved conditions for morphisms to be isomorphisms or split injections.

Applied results to Torelli theorems and classical algebraic geometry problems.

Abstract

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 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 D^b(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 D^b(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.