Bozhkov, Yuri Dimitrov (1992) Specific complex geometry of certain complex surfaces and three-folds. PhD thesis, University of Warwick.

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Abstract

One of the most important consequences of Yau's proof of the Calabi’s conjecture is the existence of a non-trivial Ricci-flat 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 Kahler-Einstein-Calabi-Yau metric for Kummer surfaces. It is obtained by gluing 16 pieces of the Eguchi-Hanson 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 anti-self-duality equations. The reason is that the curvature of the metric in question is concentrated in small thin regions and it is almost anti-self-dual. It can be also used later to deduce stability of Kummer surfaces’ tangent bundle.

In Chapter 2 we consider a special case of compact 3-folds 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 non-Kahler manifold. We prove that there are no non-trivial 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 Hermitian-Einstein metric on M. Our basic hypothesis is that the Hermitian-Einstein 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 non-Kaliler manifolds with h2'0 = 0 there are no non-zero anti-symmetric deformations of the complex structure.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Threefolds (Algebraic geometry), Surfaces, Algebraic, Complex manifolds
Official Date:	June 1992
Dates:	Date Event June 1992 UNSPECIFIED
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Eells, James, 1926-2007 ; Hitchin, N. J. (Nigel J.), 1946-
Sponsors:	Abdus Salam International Centre for Theoretical Physics
Format of File:	pdf
Extent:	iv, 75 leaves
Language:	eng

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