Jacobians of noncommutative motives (Q2876653)

Let \(k\) be a field of characteristic zero. Let \(\text{NChow}(k)_{\mathbb{Q}}\) denote the category of noncommutative Chow motives with rational coefficients, the definition of which is recalled in Section 2.3 of the present paper. As is proved in [\textit{M. Marcolli} and \textit{G. Tabuada}, ``Noncommutative numerical motives, Tannakian structures, and motivic Galois groups'', \url{arXiv:1110.2438}], periodic cyclic homology gives rise to a well-defined \(\otimes\)-functor with values in the category of finite dimensional super \(k\)-vector spaces \(HP^{\pm}:\text{NChow}(k)_{\mathbb{Q}}\rightarrow \text{sVect}(k)\). The first main result in this paper shows the following: There is a well-defined \(\mathbb{Q}\)-linear additive Jacobian functor \(\mathbf{J}:\text{NChow}(k)_{\mathbb{Q}}\rightarrow \text{Ab}(k)_{\mathbb{Q}}\) with values in the category of abelian varieties up to isogeny such that the first deRham cohomology group of \(\mathbf{J}(N)\) with \(N\in \text{NChow}(k)_{\mathbb{Q}}\) agrees with the subspace of \(HP^-(N)\) which is generated by algebraic curves. The second main result concerns with the decomposition of the abelian variety \(\mathbf{J}(\text{perf}_{\text{dg}}(X))\) associated to the derived dg category \(\text{perf}_{\text{dg}}(X)\) of a smooth projective \(k\)-scheme \(X\) of dimension \(d\) with \(k\) an algebraically closed subfield of \(\mathbb{C}\). Under a certain natural assumption, they show that there is an isogeny decomposition \(\mathbf{J}(\text{perf}_{\text{dg}}(X))\sim\Pi_{i=0}^{d-1}J_i^a(X)\), where \(J_i^a(X)\) denotes the image of the Abel-Jacobi map in the \(i\)-th intermediate Jacobian \(J_i(X)\) of \(X\).