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.
