Moscow Mathematical Journal

Volume 14, Issue 3, July–September 2014 pp. 577–594.

Jacobians of Noncommutative Motives

Authors: M. Marcolli (1) and G. Tabuada (2)
Author institution:(1) Mathematics Department, Mail Code 253-37, Caltech, 1200 E. California Blvd. Pasadena, CA 91125, USA
(2) Department of Mathematics, MIT, Cambridge, MA 02139, USA and
Departamento de Matemática e CMA, FCT-UNL, Quinta da Torre, 2829-516 Caparica, Portugal

Summary:

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.

2010 Mathematics Subject Classification. 14C15, 14H40, 14K02, 14K30, 18D20.

Keywords: Jacobians, abelian varieties, isogeny, noncommutative motives.

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