A topological and analytical study on three dimensional singular complex projective hypersurfaces

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(1) Let Y be an algebraic hypersurface with ordinary singularities, i.e., ordinary n-ple points, ordinary cuspidal points and stationary points, in the 4-dimensional complex projective space, and let X be its normalization. For such Y and X, we have proved numerical formulas which describe the Chem numbers c_1(X)^3, c_1(X)c_2(X), c_3(X) of X in terms of numerical characteristics of Y and its singular locus. As an application, we have derived a numerical formula which gives the Euler-Poincare characteristic of X with coefficients in the sheaf of holomorphic vector fields on X.(2) We have given an example of hypersurfaces which have ordinary n-ple points (2≦n≦4), ordinary cuspidal points and degenerate ordinary triple points only as singularities, and whose normalizations have isolated rational quadruple points only as singularities. From Schlessinger's criterion, it follows that these isolated singular points are rigid under small deformations.(3) For a (n+1)-dimensional complex algebraic manifold X, embedded in a projective space, and its non-singular hyperplane section Y which is sufficiently ample, we have proved the following:(a) F^kH^p(X-Y,C)【similar or equal】I^p _k(X,(p+1)Y), F^kH^p(X,C)_0【similar or equal】I^p _k(X,(p+1)Y)_0 (0≦k≦p, 0≦p≦n+1),(b) F^kGr^<w[q]> _qH^p(X-Y,C)【similar or equal】I^p _k(X,(p+1)Y)_0, F^kGr^<w[q]> _<q+1>H^p(X-Y,C)【similar or equal】 I^p _k(X,(p+1)Y)/I^p _k(X,(p+1)Y)_0 (0≦k≦p, 0≦p≦n+1),(c) F^kH^n(Y,C)_0 【similar or equal】Res(I^<n+1> _<k+1>(X,(n+2)Y))【symmetry】r*(I^n _k(X,(n+1)Y')_0,where H^p(X,C)_0 and H^p(Y,C)_0 denote the p-th cohomology of X and Y, respectively, and I^p _k(X,(p+1)Y) denotes the De Rham cohomology of closed rational differential forms which has poles of order p-k+1 at most along Y, I^p _k(X,(p+1)Y)_0 the subspace of I^p _k(X,(p+1)Y) generated by closed rational differential forms of the second kind, and Y' a sufficiently ample hyperplane section of X, intersecting with X transversely.