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On the locus of Hodge classes. (English) Zbl 0851.14004

Let \(S\) be a non-singular complex algebraic variety and \({\mathcal V}\) a variation of Hodge structures of weight zero on \(S\) with polarization form \(Q\). Let \(h(-, †)\) be the hermitian form canonically induced by \(Q\), e.g. if \(u\) is a real element of type \((0, 0)\) then \(h(u, u)= Q(u,u)\). For a fixed integer \(K\) let \(S^{(K)}\) be the space of pairs \((s, u)\) with \(s\in S\) and \(u\in {\mathcal V}_s\) integral of type \((0, 0)\) such that \(Q(u, u)\leq K\). The main result of this paper is a proof of the following claim: \(S^{(K)}\) is an algebraic variety, finite over \(S\). It follows immediately that, for a fixed pair \((s, u)\) as above, the germ of an analytic subvariety of \(S\) where \(u\) remains of type \((0, 0)\) is algebraic, because there are images in \(S\) of irreducible components of \(S^{(K)}\). These results give a positive answer to a question of A. Weil : “…whether imposing a certain Hodge class upon a generic member [of a family] amounts to an algebraic condition upon the parameters”. In particular an algebraic family \(f: X\to S\) and a cohomology class \(u\) of Hodge type \((p, p)\) at \(s\) define a complex analytic space as the locus \(T\), in an open, simply connected neighborhood of \(s\), where \(u\) remains of type \((p,p)\). It follows from the Hodge conjecture that the germ of \(T\) at \(s\) is algebraic: these results give an unconditional proof, providing as well some more evidence to a positive answer to the Hodge conjecture.

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14C25 Algebraic cycles
14D07 Variation of Hodge structures (algebro-geometric aspects)
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
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References:

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