New methods in non-Kählerian geometry – MNGNK

1. Using gauge theory (more precisely Donaldson theory) and moduli spaces of holomorphic bundles, for proving existence of curves on class VII surfaces.
2. Using new theorems of type Grothendieck Riemann-Roch for studying the geometry of the moduli spaces.
3. Using the theory of foliations, holomorphic dynamic, special metrics.

We aim to obtain substantial progress on the classification of complex surfaces.

1. To complete the Enriques Kodaira classifcation of complex surfaces (classical, longstanding problem in modern geometry),

2. Extend our methods to higher dimensions.

Articles, books, conferences.

Every area of modern mathematics has a fundamental classification problem, to which the whole research in that area is directly on indirectly related. The fundamental problem of complex geometry is the classification of compact complex manifolds. Whereas in the classification of projective algebraic complex manifolds (and more generally of Kählerian manifolds) remarkable progress has been made in arbitrary dimension, the non-Kählerian manifolds still pose huge difficulties even in dimension 2.

Our project “New Methods in Non-Kählerian Geometry” is mainly dedicated to this classification problem of complex manifolds in the non-Kählerian framework, and related problems. More specifically, the research topics to be treated during the duration of the project will be:

- Classification of non-Kählerian surfaces,
- Local index theorems and Grothendieck-Riemann-Roch type theorems in non-Kählerian geometry,
- Special metrics on non-Kählerian manifolds,
- Foliations and group-actions on complex manifolds

We explain briefly the importance of these topics and the relations between them.

The methods developed for the algebraic and Kählerian case are not directly applicable to the non-Kählerian framework. On the other hand, thanks to the conjugated efforts of several experts (and in particular of some participants in this project), important results concerning the classification of non-Kählerian surfaces have recently been obtained. These results revived the interest in this theory within the mathematical community and revived the hopes to have soon a complete classification.

In order to extend these new methods to the general case (class VII surfaces with arbitrary second Betti number) we noticed that one needs a very fine version of the famous Grothendieck-Riemann-Roch theorem for holomorphic families, which is not known at this moment. More precisely, one needs a method to compute the Chern character of the total direct image in the Bott-Chern cohomology (or in the Deligne cohomology of the base). The problem appears to be very difficult, even for the best experts in the field, so we included the second topic in our project (which is obviously important independently of the the other three).

One of the hottest topics in modern complex geometry is the theory of “special metrics”. The idea is very natural: endow every compact complex manifold with a Hermitian metric which is optimal (for instance minimizes a certain functional). This idea (which relates complex geometry to global analysis and differential geometry) has been intensively developed in the algebraic and the Kählerian case. For instance, the theorems of Aubin-Yau concerning the existence of Kähler-Einstein metrics can be regarded as the first concretization of this program. But recently new spectacular results (obtained for instance by Donaldson, Tian and many others) showed that this idea can be applied to a much broader class of Kählerian manifolds, using he appropriate class of metrics (for instance extremal, or constant scalar curvature Kählerian metrics). Our team will mainly concentrate on existence and classification of bihermitian and locally conformally Kähler metrics, which are known to exist on certain non-Kählerian surfaces. We will also consider nearly Kähler structures.

An important problem related to the classification of complex surfaces (in particular non-Kählerian surfaces) is the following: which complex surfaces admit holomorphic foliations? For the surfaces for which the answer is positive, one can further ask for the classification of all possible holomorphic foliations. For non-Kählerian surfaces these questions are particularly important, because (as shown by Dlousky-Oeljeklaus-Toma) surfaces which admit certain types of holomorphic foliations can be easily classified.