Australian Mathematical Society
Mathematics Department at Macquarie University

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50th Annual Meeting of the Australian Mathematical Society

Plenary Talk in Macquarie Theatre

Monday 25 September 2006 at 18:00


Claire Voisin (Centre National de la Recherche Scientifique)


Hodge Theory, Kähler Geometry and Projective Geometry


Hodge theory and the study of variations of Hodge structures have been developed by Hodge, Griffiths and Deligne in the context of compact Kähler manifolds, including projective complex manifolds. In the last years, I showed however that there is a big gap between Kähler geometry and projective geometry. The first gap concerns analytic geometry: namely, I show that there is no way to extend the Hodge conjecture to the Kähler situation. Furthermore, coherent sheaves do not necessarily admit locally free resolutions. The second gap is topological: indeed, I discovered topological obstructions for compact Kähler manifolds to admit a projective complex structure. These differences appear in higher dimension. For complex surfaces, Kodaira's work gave a completely different picture, as any Kähler complex structure deforms to a projective one.