## Periodic Report Summary 2 - KAHLER MANIFOLDS (Several problems on Kahler manifolds)

The main theme of my project is the use of Lawson and Harvey theorem on the existence of an obstruction current on a non-Kähler manifold, and the work of Demailly and Paun on the characterisation of the Kähler cone, to study Kähler and non-Kähler manifolds. The work carried during the second period is mostly a continuation of the work carried during the first period.

(I) The Kähler rank of compact complex surfaces: Harvey and Lawson introduced the Kähler rank of a surface in an attempt to measure how far a surface is from being Kähler: the rank is 2 if the surface is Kähler, the rank is 1 if it is non-Kähler, but it supports a closed positive (1,1) form, and it is 0 otherwise. Harvey and Lawson conjectured that the Kähler rank is a bimeromorphic invariant. The first approach was local in nature, and we solved a system of differential equations. The second approach was global in nature, where we used the global geometry of a class VII surfaces. Moreover, we also obtained a partial classification of the surfaces of Kähler rank equal to 1: they are either bimeromorphic to a class of Hopf surfaces, or else they support a very special foliation. Then, Brunella (in a separate paper) showed that a surface supporting such a very special foliation is a modification of an Inoue surface. Hence, now we have a complete classification of the surfaces according to their Kähler rank: it is Kähler if the first Betti number is even; if Betti one is odd and greater than three, the rank is 1; if Betti one is 1, and the second Betti number is zero, the rank is either 0 or 1 (Inoue and some Hopf surfaces have rank 1, the other Hopf surfaces have rank 0); if Betti one is 1, and Betti two is strictly positive, the Kähler rank is 0. This result could have a major impact on the study of surfaces of class VII. For instance, the main problem with these surfaces is to show the existence of sufficiently many compact curves on them. Our result states that the Kähler rank of a class VII surface is 0; in other words, an obstruction current (as in Harvey and Lawson) has to have singularities, and the next step would be to try to show that these singularities are in fact analytic curves.

(II) Deformation of manifolds in the Fujiki class C: I showed that for a family of 3-folds X->S which is locally Fujiki, the non-Kähler locus (i.e. the points in S where the fibre is not Kähler) is a countable union of analytic subsets of S. (The result mentioned in the first periodic report was over a one-dimensional base S). For this, we needed to study the space of cycles over a manifold in the Fujiki class C; it was shown by Fujiki that each irreducible component of the space of cycles is compact, and we use this result to obtain the analytic subsets of the base S. Another essential result which we used was a theorem we obtained in a previous paper on the existence of a curve which is part of an exact positive current. Then, in the projective case, we proved the following theorem: let X->S be a family of moishezohn manifolds of dimension 3. Then the non-Kähler locus is s countable union of analytic sub-varieties of S.

(I) The Kähler rank of compact complex surfaces: Harvey and Lawson introduced the Kähler rank of a surface in an attempt to measure how far a surface is from being Kähler: the rank is 2 if the surface is Kähler, the rank is 1 if it is non-Kähler, but it supports a closed positive (1,1) form, and it is 0 otherwise. Harvey and Lawson conjectured that the Kähler rank is a bimeromorphic invariant. The first approach was local in nature, and we solved a system of differential equations. The second approach was global in nature, where we used the global geometry of a class VII surfaces. Moreover, we also obtained a partial classification of the surfaces of Kähler rank equal to 1: they are either bimeromorphic to a class of Hopf surfaces, or else they support a very special foliation. Then, Brunella (in a separate paper) showed that a surface supporting such a very special foliation is a modification of an Inoue surface. Hence, now we have a complete classification of the surfaces according to their Kähler rank: it is Kähler if the first Betti number is even; if Betti one is odd and greater than three, the rank is 1; if Betti one is 1, and the second Betti number is zero, the rank is either 0 or 1 (Inoue and some Hopf surfaces have rank 1, the other Hopf surfaces have rank 0); if Betti one is 1, and Betti two is strictly positive, the Kähler rank is 0. This result could have a major impact on the study of surfaces of class VII. For instance, the main problem with these surfaces is to show the existence of sufficiently many compact curves on them. Our result states that the Kähler rank of a class VII surface is 0; in other words, an obstruction current (as in Harvey and Lawson) has to have singularities, and the next step would be to try to show that these singularities are in fact analytic curves.

(II) Deformation of manifolds in the Fujiki class C: I showed that for a family of 3-folds X->S which is locally Fujiki, the non-Kähler locus (i.e. the points in S where the fibre is not Kähler) is a countable union of analytic subsets of S. (The result mentioned in the first periodic report was over a one-dimensional base S). For this, we needed to study the space of cycles over a manifold in the Fujiki class C; it was shown by Fujiki that each irreducible component of the space of cycles is compact, and we use this result to obtain the analytic subsets of the base S. Another essential result which we used was a theorem we obtained in a previous paper on the existence of a curve which is part of an exact positive current. Then, in the projective case, we proved the following theorem: let X->S be a family of moishezohn manifolds of dimension 3. Then the non-Kähler locus is s countable union of analytic sub-varieties of S.