Research objectives and content
I plan to study compact hyperkahler manifolds by means of complex algebraic geometry. This comprises an investigation of the period map and other basic structures, eg. the ample cone, the Kahler cone, the cohomology ring, etc., naturally associated with such a manifold. The question will be approached by using the deformation theory and, more globally, the moduli space of hyperkahler manifolds.
A thourough analysis of some of the known higher-dimensional examples moduli spaces of bundles and Hilbert schemes of K3 surfaces) should clarify to what extent the theory of K3 surfaces generalizes to higher dimensions. Another interesting class of hyperkahler manifolds is provided by complex integrable systems. The study of these special structures promises to reveal important geometric properties of hyperkahler manifolds in general.
Last but not least, the underlying hyperkahler metric is encoded by the twistor space. I will attempt to give a concrete example of a twistor space and thus of a hyperkahler manifold. Note that explicit examples of hyperkahler metric on compact manifolds are extremely rare.
Training content (objective, benefit and expected impact)
The E. N. S. together with the many nearby institutions (eg., Jussieu, IHES) would provide an excellent training opportunity for algebraic geometers by offering specialized courses and seminars in several branches closely related to algebraic geometry,eg. differential and complex geometry, number theory.
The personal contact to leading experts (A. Beauville, M. Kontsevich, C Voisin) would be very inspiring and helpful for my own research work.