A remark on primitive cycles and Fourier-Radon transform (Q2906497)

Consider a primitive Hodge cohomology class \(\zeta \in H^{2n}(X)\), where \(X\) is a smooth, irreducible, projective variety of dimension \(2n\) and let \(\mathcal L\) be the hyperplane line-bundle. Thomas has reformulated the Hodge conjecture into the statement that the restriction of \(\zeta\) to some (singular) hypersurface section does not vanish. Green and Griffiths studied the concept of the singularity of a normal function, they predicted that Thomas' form of the Hodge conjecture is equivalent to the existence of singularities for a normal function corresponding to \(\zeta\).NEWLINENEWLINEThis result was obtained by \textit{P. Brosnan, H. Fang, Z. Nie} and \textit{G. Pearlstein} [Invent. Math. 177, No. 3, 599--629 (2009; Zbl 1174.14009)] and, independently, by \textit{M. A. A. de Cataldo} and \textit{L. Migliorini} [Proc. Am. Math. Soc. 137, No. 11, 3593--3600 (2009; Zbl 1174.14011)]. The proof requires the power of the decomposition theorem of Beilinson-Bernstein-Deligne-Gabber for the universal family of sections of \(X\) fibered over \( \mathbb P^{\vee}:= \mathbb P ( H^0 ( X, {\mathcal L}^k))\), \(k \gg 0\).NEWLINENEWLINE Beilinson gives here his own analysis of the group \(N_1 H^{2n} (X)^{\mathrm{prim}}\), which is the intersection of all the kernels of the restriction maps \(H^{2n}(X)^{\mathrm{prim}} \to H^{2n}(Y)\), with \(Y\) a proper closed subvariety in \(X\). In a recent Bourbaki séminaire, F. Charles has commented that Beilinson's method {\textit{ repose sur une utilisation particulièrement élégante }} of the Brylinski's Radon transform associated with the incidence correspondence in \(\mathbb P \times \mathbb P^{\vee}\). The main result proved in this direction is the property that the Radon transform functor preserves primitive cohomology (while reversing its grading).NEWLINENEWLINEFor the entire collection see [Zbl 1242.14001].