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

Let $X$ and $Y$ be complex smooth projective varieties, and $\mathcal D^b(X)$ and $\mathcal D^b(Y)$ the associated bounded derived categories of coherent sheaves. Assume the existence of a triangulated category $\mathcal T$ which is admissible both in $\mathcal D^b(X)$ as in $\mathcal D^b(Y)$. 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 $\mathcal T^\perp$ of $\mathcal T\subset\mathcal D^b(X)$ has a trivial Jacobian (e.g., when $\mathcal T^\perp$ is generated by exceptional objects), the morphism $\tau$ is split injective. When this also holds for the orthogonal complement $\mathcal T^\perp$ of $\mathcal T\subset\mathcal D^b(Y)$, $\tau$ 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.
As an application, we obtain categorical Torelli theorems, an incompatibility between two conjectures of Kuznetsov (one concerning functors of Fourier–Mukai type and another one concerning Fano threefolds), and also several new results on quadric fibrations and intersections of quadrics.