Chow groups of intersections of quadrics via homological projective duality and (Jacobians of) non-commutative motives (Q2821880)

Let \(X\) be a smooth projective subscheme of \(\mathbb P^n_k\), \(k\) a field. If \(X\) is a complete intersection of multidegree \(d_1\geq\cdots\geq d_r\) there is a numerical invariant \(\kappa=[\frac{n-\sum_{j=2}^rd_j}{d_1}]\) where \([q]\) denotes the integer part of \(q\in\mathbb Q\). Studying the various Weil cohomology theories of \(X\), \textit{K. H. Paranjape} [Ann. Math. (2) 139, No. 3, 641--660 (1994; Zbl 0828.14003)] gave a conjecture of Beilinson-Bloch type stating that for every \(i<\kappa\) there is an isomorphism \(\text{CH}_i(X)_{\mathbb Q}\simeq\mathbb Q.\) \textit{A. Otwinowska} [C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 2, 141--146 (1999; Zbl 0961.14004)] proved this conjecture in the case where \(X\) is a complete intersection of quadrics.NEWLINENEWLINEThe first main result of this article is that the conjecture holds ifNEWLINENEWLINE(i) \(X\) is a smooth complete intersection of two quadricsNEWLINENEWLINE(ii) \(X\) is a smooth complete intersection of three odd-dimensional quadrics. \noindent Otwinowska's proof is based on a recursive geometric argument. Also, Esnault, Levine, and Vieweg [\textit{H. Esnault} et al., Duke Math. J. 87, No. 1, 29--58 (1997; Zbl 0916.14001)] proved the conjecture geometrically for very small values of \(i\) by a generalization of Roitman's techniques.NEWLINENEWLINEThe techniques applied in this article is quite different from the previous. It is categorical, and uses recent technology such as Kuznetsov's homological projective duality and Jacobians of non-commutative motives. These techniques applies geometric information contained in the derived category of \(X\) and its non-commutative motive.NEWLINENEWLINEFor \(k\subseteq\mathbb C\) algebraically closed, the intermediate Jacobians \(J^i(X),\;0<i<\dim(X)-1\) are not necessarily algebraic, in contrast to the Picard variety \(J^0(X)=\text{Pic}^0(X)\) and the Albanese variety \(J^{\dim(X)-1}(X)=\text{Alb}(X)\). However, they contain an algebraic torus \(J_a(X)\subseteq J^i(X)\) defined by the image of the Abel-Jacobi map \(\text{AJ}^i:A^{i+1}(X)_{\mathbb Z}\twoheadrightarrow J^i(X),\;0\leq i\leq\dim(X)-1.\) Here \(A^{i+1}(X)_{\mathbb Z}\) stands for the group of algebraic trivial cycles of codimension \(i+1\).NEWLINENEWLINEThe authors consider intersections of quadrics such that \(\kappa=[\dim(X)/2]\) and \(\dim(X)\) is odd, i.e. \(\dim(X)=2\kappa+1\) Then:NEWLINENEWLINE (i) When \(X\) is a smooth complete intersection of two even-dimensional quadrics, \textit{M. Reid} [The complete intersection of two or more quadrics. Ph.~D. Thesis, Cambridge, Univ. of Cambridge (1972, \url{http://homepages.warwick.ac.uk/~masda/3folds/qu.pdf}] proved that \(J^\kappa_a(X)\) is isomorphic to the Jacobian \(J(C)\) of the hyperelliptic curve naturally associated with \(X\),NEWLINENEWLINE (ii) When \(X\) is a smooth complete intersection of three odd-dimensional quadrics, \textit{A. Beauville} [Ann. Sci. Éc. Norm. Supér. (4) 10, 309--391 (1977; Zbl 0368.14018)] proved that \(J^\kappa_a(X)\) is isomorphic to the Prym variety \(\text{Prym}(\tilde C/C)\) naturally associated with \(X\).NEWLINENEWLINEWhen \(X\) is a complete intersection of either two quadrics, or three odd-dimensional quadrics, the authors use the Veronese embedding and applies the Lefschetz' theorem to conclude that \(H^{p,q}(X)=0\) when \(p\neq q\) and \(p+q\neq\kappa\). Then it follows that the Jacobians \(J^\kappa_a(X)\) are the only trivial ones. An alternative proof of this fact is the second main result of the article. That is (verbatim):NEWLINENEWLINE(i) When \(X\) is a smooth complete intersections of two odd-dimensional quadrics, \(J^i_a(X)=0\) for every \(i\).NEWLINENEWLINE(ii) When \(X\) is a smooth complete intersection of two or three quadrics such that \(\kappa=[\dim(X)/2]\) with \(\dim(X)\) odd, \(J^i_a(X)=0\) for every \(i\neq\kappa\).NEWLINENEWLINEThe article contains two main applications of the main results: The first considers two cases: (i) When \(X\) is a smooth complete intersection of two odd-dimensional quadrics, and (ii) when \(X\) is a smooth complete intersection of two or three quadrics such that \(\kappa=[\dim(X)/2]\) with \(\dim(X)\) odd. In both cases, an explicit motivic decomposition \(M^i(X)_{\mathbb Q}\) is given. and it is proved that the rational Chow motive \(M(X)_{\mathbb Q}\) is Kimura-finite.NEWLINENEWLINEThe second main application combines the first main result with \textit{C. Vial}'s work on fibrations [Doc. Math., J. DMV 18, 1521--1553 (2013; Zbl 1349.14027)], and proves the following (verbatim):NEWLINENEWLINETheorem 2.2. Let \(f:Y\rightarrow B\) be a smooth dominant flat morphism between smooth projective \(k\)-schemes. Assume that the fibres of \(f\) are either complete intersections of two quadrics or complete intersections of three odd-dimensional quadrics. Then the following assertions hold.NEWLINENEWLINE (i) If \(\dim(B)\leq 1\), then the rational Chow motive \(M(Y)_{\mathbb Q}\) is Kimura finite and \(Y\) satisfies Murre's conjecture.NEWLINENEWLINE(ii) If \(\dim(B)\leq 2\), then \(Y\) satisfies Grothendieck's standard conjecture of Lefschetz type.NEWLINENEWLINE (iii) If \(\dim(B)\leq 3\), then \(Y\) satisfies Hodges's conjecture.NEWLINENEWLINEThe article depends heavily on the language of dg categories and motivic theory, and the preliminary sections with the aim of explaining these topics is not (and cannot be) complete for understanding the following proofs.NEWLINENEWLINEHowever, the article has nice results and uses a lot of innovative techniques and new results to prove the interesting theorems.