The structure and classification of complex analytic compactifications of C^n with the second Betti number equal to one

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We investigate mainly(1) the Fano compactifications of C^3 with hypersurface terminal singularities and with second Betti number equal to one and(2) the structure of the non-projective compactifications (X,Y) of C^3 with second Betti number equal to one.Concerning to the investigation (1) I succeeded in constructing the Fano compactifications of C^3 with hypersurface terminal singularities and second Betti number equal to one, which are essentially new.Concerning to the investigation (2) it is easy to see that the canonical divisor can be written as follow: K_X=-rY (r=1,2). When Y is nef, the structure of (X,Y) is completely determined by myself. Thus the problem is the cases where Y is not-nef. Unfortunately the structure is not known well in this case. However I can obtain some partial results. For example, the boundary Y is birational equivalent to a rational surface or a ruled surface.Furthermore, we find that some technique developed in the study of compactifications of C^3 can be applied to the classification of the non-normal del Pezzo surfaces, and I succeeded in its classification. Moreover I prove that -3K_S is always very ample, where we denote by -K_S the anti-canonical divisor of the non-normal del Pezzo surface S. This result is an affirmative answer to the question by Miyanishi. The paper is accepted in Math. Nachrichten (August, 2002).