Komplexe Algebraische Geometrie

Abstract

The Workshop Komplexe algebraische Geometrie, organized by Fabrizio Catanese (Bayreuth), Yujiro Kawamata (Tokyo), Gang Tian (Princeton), and Eckart Viehweg (Essen), drew together 50 participants. There were several young PhD students and other PostDocs in their 20's and early 30's, together with established leaders of the fields related to the thematic title of the workshop. There were 23 talks, each lasting 55 minutes or one hour, and each followed by a lively 10 minutes discussion.

As usual at an Oberwolfach Meeting, the mathematical discussions continued outside the lecture room throughout the day and the night. The Conference fully fulfilled its purported aim, of setting in contact mathematicians with different specializations and non uniform background, of presenting new fashionable topics alongside with new insights on long standing classical open problems, and also cross-fertilizations with other research topics as arithmetic and physics. For the latter, cf. the talk by Bernd Siebert on the new approach to Mirror Symmetry through logarithmic geometry and toric affine Calabi Yau varieties and the one by van Straten on quantization of completely integrable Hamiltonian Systems. For the former, cf. the talks by Winkelmann on integral sets and by McQuillan on the Bloch principle.

A central role was occupied by the new results around the Hodge conjecture presented by Voisin, and the new results by Hacon, McKernan et al which give an essential step towards the final solution of the Minimal Model Program, and were presented here by McKernan with a proposed approach to the Sarkisov program.

There were many expositions dealing with several classical problems and classical and modern theories. It would take too long to dwell on each of the outstanding contributions presented at the Conference. For some topics there were several interrelated talks, for instance we could mention the following classical themes:

1. Abelian Varieties, Jacobian and Prym Varieties and their Moduli spaces (van der Geer, Farkas, Lange, Hulek)
2. Hodge Theory and Variation of Hodge structures (Voisin, Möller, Barja)
3. Fibred varieties (Oguiso, Moeller)
4. Algebraic Surfaces (Mukai, Pardini)
5. Fundamental groups and algebraic fundamental groups (Esnault, Bauer, Pardini).

There were also expositions on many other beautiful topics:

1. Moduli spaces of sheaves on higher dimensional varieties (Lehn)
2. Deformations of special complex maifolds (Rollenske)
3. Varieties of power sums (Takagi)
4. Ball quotients (Müller-Stach)
5. Stacks and Azumaya algebras (Schröer)
6. nefness and vector bundles (Peternell)

The variety of striking results and the very interesting and challenging proposals made the participation in the workshop rather strenuous but certainly highly rewarding. We hope that the quality of the expositions in these abstracts will make them quite useful to the mathematical community.