Komplexe Analysis
Authors: Demailly Jean-Pierre , Hulek Klaus , Mok Ngaiming , Peternell Thomas 
Published: December 2009

The workshop
Komplexe Analysis
, organised by Jean-Pierre Demailly (Grenoble), Klaus Hulek (Hannover), Ngaiming Mok (Hong Kong) and Thomas Peternell (Bayreuth) was held August 24th–August 30, 2008. This meeting was well attended with 46 participants from Europe, US, and the Far East. The participants included several leaders in the field as well as many young (non-tenured) researchers.

The aim of the meeting was to present recent important results in several complex variables and complex geometry with particular emphasis on topics linking different areas of the field, as well as to discuss new directions and open problems. Altogether there were nineteen talks of

60

minutes each, a programme which left sufficient time for informal discussions and joint work on research projects.

One of the topics at the center of the conference was the classification theory of higher dimensional varieties. Y. Kawamata lectured on the connections between the minimal model programme and derived categories; A. Corti discussed an approach to the finite generation of the canonical ring without minimal models, but still in connection with the seminal work which was presented by J. McKernan in the last Complex Analysis meeting in Oberwolfach 2006, where the finite generation of the canonical ring of varieties of general type was announced. Extension theorems, non vanishing and positivity result for certain direct image sheaves play a role in the global classification of complex manifolds. This was largely discussed by M. Paun and B. Berndtsson. In their work analytic methods are central, whereas the talks by Kawamata and Corti were more of an algebraic nature. Also very much on the analytic side and connected to Berndtsson's talk, H. Tsuji lectured on generalised Kähler–Einstein metrics. Families of projective manifolds over higher-dimensional base spaces were considered in the talk by S. Kebekus. Direct images of coherent sheaves also play a central role in this context.

About five years ago, Campana introduced new variations on the concept of “orbifolds”; they were already the suject of talks in past sessions and have turned out to be of increasing interest – in the present session, new results on the hyperbolicity of orbifolds were presented in the talk by E. Rousseau.

As to varieties with special geometry, K. Oguiso spoke on non-algebraic hyperkähler manifolds and, with a rather different flavour, F. Catanese on complex and real threefolds fibered by rational curves, with a special emphasis on real algebraic geometry. J. Chen discussed the influence of terminal singularities in three-dimensional geometry, a more algebraic topic. On the analytic side, A. Teleman reported on recent progress in the classification of non-Kähler surfaces in the so called Kodaira class VII, using gauge-theoretical methods, and S. K. Yeung lectured on new results on fake projective planes. Group actions and envelopes of holomorphy were the topics of the talk by X. Zhou. S. Boucksom discussed equidistribution of Fekete points on complex manifolds, in relation with energy functionals for Monge-Ampère operators.

R. Lazarsfeld presented a very interesting new approach to study properties of linear systems and line bundles via convex geometry.

Overall, moduli spaces appeared to be a central theme in the workshop, and were discussed extensively in at least four talks: V. Gritsenko considered moduli spaces of K3-surfaces; S. Grushevsky spoke on intersection numbers of divisor on the moduli space of curves, and K. Ludwig and G. Farkas lectured on the moduli spaces of spin and Prym curves, their singularities, Kodaira dimension and enumerative geometry.