Categorical representability and intermediate Jacobians of Fano threefolds (Q2869236)

The authors investigate relations between the birational geometry of a smooth projective variety and the properties of its derived category. In particular they define the notion of categorical representability in (co)dimension \(m\) for a smooth \(n\)-dimensional projective variety \(X\) by requiring that the derived category \(D^b(X)\) admits a semiorthogonal decomposition by categories which can be fully faithfully embedded into derived categories of smooth projective varieties of dimension bounded by \(m\) (resp. \(n-m\)). This notion is expected to play an important role in birational geometry. It is show that there is a connection with notions of representability of the groups \(A^i_{\mathbb Z}\) of algebraically trivial cycles of codimension \(i\) on \(X\). The group \(A^i_{\mathbb Q}\) is rationally representable if there exists a curve \(\Gamma\) and a cycle \(z\in\mathrm{CH}^i_{\mathbb Q}(X\times \Gamma )\) inducing a surjective map \(z_*: A^i_{\mathbb Q}(\Gamma) \to A^i_{\mathbb Q}(X)\). The authors prove that if \(X\) is a smooth projective threefold, \(h^1(X)=h^5(X)=0\) and \(A^i_{\mathbb Q}\) is rationally representable for all \(i\), then, given any smooth projective curve \(\Gamma\) and a fully faithful functor \(D^b(\Gamma )\to D^b(X)\), we have that \(J(\Gamma )\) is isogenous to a subvariety of the intermediate Jacobian \(J(X)\). It follows that if \(X\) is categorically representable in dimension \(1\) then \(J(X)\) is isogenous to a direct sum of Jacobians of curves.NEWLINENEWLINEFor the entire collection see [Zbl 1256.14001].