Logarithmic deformations of complex projective hypersurfaces with ordinary singularities and their period maps

点击翻译按钮获取中文摘要

1. We have formulated the infinitesimal mixed Torelli problem for a locally trivial analytic family of complex projective surfaces with ordinary singularities, parametrized by a manifold, relativizing the notion of cubic hyper-resolution due to V.Navarro Aznar, F.Guillen et al., and have gave cohomological sufficient conditions for this problem to be affirmatively solved. Furthermore we have constructed a few examples for which these sufficient conditions are satisfied.2. We have also considered the infinitesimal mixed Torelli problem for complex projective threefolds of so-called type (n, r_1, r_2, r_3, r_4). In this procedure we have found a certain weakly normal, non-isolated singularity which is a degenerate one of an ordinary triple point, and is described as (xy)^2+(yz)^2+(zx)^2+wxyz=0 by use of affine coordinates. It has turned out that singularity is a cone over the Steiner surface which is a rational surface with ordinary singularities in P^3 (C). The normalization of it is a cone over P^2 (C) embedded in P^5 (C) by the Veronese map of degree 2, a rational isolated singularity of multiplicity 4, and is rigid under deformation.3. Besides the above results, we have obtained a formula which gives the Euler number of the non-singular normalization of a complex hypersurface with ordinary singularities in P^4 (C), generalizing the classical one for a complex hypersurface with ordinary singularities in P^3 (C) due to Enriques. This work is in preparation to be published.