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

Jacobians of Noncommutative Motives

In this article one extends the classical theory of (intermediate) Jacobians to the “noncommutative world”. Concretely, one constructs a Q-linear additive Jacobian functor N 7→ J(N) from the category of noncommutative Chow motives to the category of abelian varieties up to isogeny, with the following properties: (i) the first de Rham cohomology group of J(N) agrees with the subspace of the odd ...

From Exceptional Collections to Motivic Decompositions via Noncommutative Motives

Making use of noncommutative motives we relate exceptional collections (and more generally semi-orthogonal decompositions) to motivic decompositions. On one hand we prove that the Chow motive M(X )Q of every smooth and proper Deligne-Mumford stack X , whose bounded derived category D(X ) of coherent schemes admits a full exceptional collection, decomposes into a direct sum of tensor powers of t...

Mixed Intermediate Jacobians

Curves in Jacobians to Non - Jacobians Ii

This is a second paper where we introduce deformation theory methods which can be applied to finding curves in families of principally polarized abelian varieties (ppav) containing jacobians. One of our motivations for finding interesting and computationally tractable curves in ppav is to solve the Hodge conjecture for the primitive cohomology of the theta divisor which we explain below. For ot...

Dynamics of Special Points on Intermediate Jacobians

We prove some general density statements about the subgroup of invertible points on intermediate jacobians; namely those points in the Abel-Jacobi image of nullhomologous algebraic cycles on projective algebraic manifolds.