Claire Voisin (Paris)

On a question of Borel and Haefliger

Abstract: In 1961, Borel and Haefliger constructed the homology class of a closed analytic subset in a compact complex manifold X. They asked whether, when X is projective, any such class can be written as an integral combination of classes of smooth closed analytic subsets, that is, complex submanifolds. I will describe the answer, which is completely known and depends on the dimension d and codimension c of the considered cycles. In recent work with J. Kollár, I prove that the answer is yes if d c+1, the question is still completely open and can be related to a famous conjecture by Hartshorne (joint work with O. Benoist).

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