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 N → 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 periodic cyclic homology of N which is generated by algebraic curves; (ii) the abelian variety J(perf_(dg)(X)) (associated to the derived dg category perf_(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 perf(X) gives rise to a decomposition of the intermediate algebraic Jacobians of X.
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