KAHLER MANIFOLDS · Several problems on Kahler manifolds

Periodic Report Summary - KAHLER MANIFOLDS (Several problems on Kahler manifolds)

The overarching theme of the first part of my proposal is the use of the result of Lawson and Harvey on the intrinsic characterisation of Kahler manifolds, to obtain new results on Kahler and non-Kahler manifolds. I mentioned the problem of deformations of Kahler manifolds as a possible application of this theorem. I will mention bellow the three most important results that I obtained during the first reporting period. I. Obstruction to the existence of Kahler metrics. By improving on some of my previous work, I managed to obtain a general existence theorem for compact complex manifolds. It says that given a compact complex manifold of dimension n, it is either Kahler, or else there exists a nef pluriharmonic current of bidegree (n-1,n-1) which is the (n-1,n-1) component of a boundary, or a closed positive current of bidegree (n-1,n-1) which is exact. In other words, the obstruction to the existence of a Kahler metric on a given manifold is nef or closed. This is a refinement of the result of Harvey and Lawson. II. Deformation of manifolds. There are several problems related to the deformation of Kahler manifolds. One of them (which was also mentioned in my proposal) concerns the non-Kahler locus of a deformation. It is conjectured that it is contained in a countable union of analytic subsets. To this end, I proved the following particular case of this conjecture: consider a family of compact complex three-folds, which is locally in the Fujiki class C (i.e. it is locally birational to a Kahler family) over a base of dimension 1. Then the non-Kahler locus (i.e. the set of points in the base for which the corresponding manifold is not Kahler) is a countable union of points. This result is closely related to Hironaka's example of a family of three-folds which degenerate to a non-Kahler manifold, since all of the hypotheses in our theorem are satisfied by Hironaka's example. The proof uses the fact that on a non-Kahler three-fold in the Fujiki class C there exist a curve which is part of a positive d-exact current. Then we move this curve to nearby manifolds, and then show that it is part of an obstruction current. We also use a special feature of the three-folds: given a surface in a smooth three-fold, the singularities of the surface can be resolved by a sequence of blow-ups with smooth centers. III. The Kahler rank of compact complex surfaces. It is known that a compact complex surface is Kahler if and only if the first Betti number is even. When the first Betti number is odd, Harvey and Lawson defined the Kahler rank to be 1 if the surface carries a positive form which is closed and 0 otherwise. The Kahler rank is 2 if the surface is Kahler. Harvey and Lawson conjectured that the Kahler rank of a compact complex surface is a birational invariant, and we proved that this is indeed the case. The proof comes down to solving a non-linear system of differential equations of order 2. Then, in an attempt to compute the Kahler rank in terms of the Betti numbers of the surface, I obtained a partial result in this direction: if the closed positive form on compact complex surface of Kahler rank 1 satisfies a certain property, then the surface is a Hopf surface. The remaining case (which is still open), should give an Inoue surface. The final result should say that a minimal surface with first Betti number equal to 1 and a strictly positive second Betti number has 0 Kahler rank. This should shed some more light on the mysterious class VII surfaces.

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