Jacobians of noncommutative motives

In this article one extends the classical theory of (intermediate) Jacobians to the “noncommutative world”. Concretely, one constructs a $\mathbb Q$-linear additive Jacobian functor $N\mapsto\boldsymbol J(N)$ 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 $\boldsymbol J(N)$ agrees with the subspace of the odd periodic cyclic homology of $N$ which is generated by algebraic curves; (ii) the abelian variety $\boldsymbol J(\mathrm{perf}_\mathrm{dg}(X))$ (associated to the derived dg category $\mathrm{perf}_\mathrm{dg}(X)$ of a smooth projective $k$-scheme $X$) identifies with the product of all the intermediate algebraic Jacobians of $X$. As an application, every semi-orthogonal decomposition of the derived category $\mathrm{perf}(X)$ gives rise to a decomposition of the intermediate algebraic Jacobians of $X$.