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Abstract:
Let Y be a projective complex variety endowed with an ample line
bundle L. The Noether-Lefschetz locus NL is the subset of |L|
parametrising smooth hypersurfaces of Y which vanishing cohomology has a
non-zero Hodge class. First, I will give an explicit asymptotic
description of the components of small codimension of NL for L
sufficiently ample and will show that for these components the Hodge class
is in the image of the cycle map, as predicted by the Hodge Conjecture. Next,
I will explain a (partly conjectural) generalization of this result and
its links with Nori's connectivity theorem. I will also give an explicit
asymptotic bound for Nori's theorem to hold and will give examples showing
that this bound is optimal.
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