Title:

Specific complex geometry of certain complex surfaces and threefolds

One of the most important consequences of Yau's proof of the Calabi’s conjecture is the existence of a nontrivial Ricciflat metric on K3 surfaces. For its explicit construction would be of great interest. Since it is not available yet the qualitative description of this metric would also have certain significance. In Chapter 1 we propose an approximation of the K3 KahlerEinsteinCalabiYau metric for Kummer surfaces. It is obtained by gluing 16 pieces of the EguchiHanson metric and 16 pieces of the Euclidean metric. Two estimates on its curvature are proved. Then we discuss the possibility of application of C.Taubes’s iteration scheme for solving antiselfduality equations. The reason is that the curvature of the metric in question is concentrated in small thin regions and it is almost antiselfdual. It can be also used later to deduce stability of Kummer surfaces’ tangent bundle. In Chapter 2 we consider a special case of compact 3folds M which are diffeo morphic to the connected sum of n copies of S3 x S3. If n > 103, there is a complex structure of C1 = 0 on M, which is a nonKahler manifold. We prove that there are no nontrivial fine bundles on M and hence we deduce that its tangent bundle is stable with respect to any Gauduchon metric. By a theorem of Li and Yau we conclude that there is an HermitianEinstein metric on M. Our basic hypothesis is that the HermitianEinstein metric and the Gauduchon metric coincide. This is similar to the previous situation on K3. Then we consider the deformations of this metric, keeping the volume and the complex structure fixed. We seek the place of M in the classification of almost Hermitian manifolds by Gray and Hervella and explore some sorts of conditions which can be imposed on M and which can substitute the Kiihler one. We also show that on Hermitian nonKaliler manifolds with h2'0 = 0 there are no nonzero antisymmetric deformations of the complex structure.
