Homogeneous ideals associated to a smooth subvariety (Q2845045)

The Hodge conjecture predicts that each Hodge class is represented by an algebraic cycle. An effective solution would be the construction of an algebraic cycle purely from the Hodge class. In this paper, the author shows that when a subvariety \(Z\) of a smooth projective variety \(X\) is given, then its homogeneous ideal \(I\) (in the homogeneous coordinate ring of \(X\)) can be constructed from purely Hodge theoretical data. Following \textit{A. Otwinowska} [``Sur les lieux de Hodge des hypersurfaces'', \url{arXiv:math/0401092}], the author associates an homogeneous ideal \(E_Y\) to each smooth hypersurface \(Y\) of sufficiently high degree that contains \(Z\). By varying \(Y\) and also its degree, one recovers \(I\) by taking the intersection of the \(E_Y\).