# Moscow Mathematical Journal

Volume 16, Issue 2, April–June 2016 pp. 205–235.

From Semi-Orthogonal Decompositions to Polarized Intermediate Jacobians via Jacobians of Noncommutative Motives

**Authors**:
Marcello Bernardara (1) and Gonçalo Tabuada (2)

**Author institution:**(1) Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9, France

(2) Department of Mathematics, MIT, Cambridge, MA 02139, USA, Departamento de Matemática, FCT, UNL, Portugal, Centro de Matemática e Aplicações (CMA), FCT, UNL, Portugal

**Summary: **

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 τ 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^{⊥} of 𝒯⊂𝒟^{b}(*X*) has a trivial Jacobian (e.g., when 𝒯^{⊥} is generated by
exceptional objects), the morphism τ is split injective. When this also
holds for the orthogonal complement 𝒯^{⊥} of 𝒯⊂𝒟^{b}(*Y*), τ becomes an
isomorphism. Furthermore, in the case where *X* and *Y* have a unique
principally polarized intermediate Jacobian, we prove that τ 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.

2010 Math. Subj. Class. 14A22, 14C34, 14E08, 14J30, 14J45, 14K30, 18E30.

**Keywords:**Intermediate Jacobians, polarizations, noncommutative motives, semi-orthogonal decompositions, Torelli theorem, Fano threefolds, quadric fibrations, blow-ups.

Contents Full-Text PDF