Jacobians of noncommutative motives

TL;DR

This paper extends the classical theory of Jacobians to noncommutative motives, constructing a functor that links noncommutative Chow motives to abelian varieties, capturing geometric and cohomological properties.

Contribution

It introduces a new Jacobian functor for noncommutative motives, generalizing classical Jacobians to the noncommutative setting with explicit properties.

Findings

The Jacobian functor J(-) aligns with classical Jacobians for smooth projective schemes.

J(N) captures the algebraic curves within the noncommutative motive via cyclic homology.

The functor provides a bridge between noncommutative motives and abelian varieties.

Abstract

In this article one extends the classical theory of (intermediate) Jacobians to the "noncommutative world". Concretely, one constructs a Q-linear additive Jacobian functor J(-) 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 periodic cyclic homology of N which is generated by algebraic curves; (ii) the abelian variety J(perf(X)) (associated to the derived dg category perf(X) of a smooth projective scheme X) identifies with the union of all the intermediate algebraic Jacobians of X.