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

(1)设Y是具有普通奇点的代数超曲面，即，在4维复射影空间中，定义了n个普通点、普通尖点和驻点，并设X为它们的正规化。对于这样的Y和X，我们根据Y及其奇异轨迹的数值特征，证明了描述X的化学数c_1（X）^3，c_1（X），c_2（X），c_3（X）的数值公式。作为应用，我们导出了系数在X上全纯向量场层中的X的Euler-Poincare特征线的数值公式。(2)我们给出了一个超曲面的例子，它的普通n重点（2 <$n <$4）、普通尖点和退化普通三重点只作为奇点，而它的正规化只以孤立有理四重点作为奇点。根据Schlessinger准则，这些孤立奇点在小变形下是刚性的。(3)对于嵌入在射影空间中的（n+1）维复代数流形X，以及它的充分充裕的非奇异超平面截面Y，我们证明了以下结果：（a）F^kH^p（X-Y，C）[相似或相等]I^p _k（X，（p+1）Y），F^kH^p（X，C）_0[相似或相等]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）[相似或相等]I^p _k（X，（p+1）Y）_0，F^kGr^<w[q]> _<q+1>H^p（X-Y，C）[相似或相等] 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 [相似或相等]Res（I^<n+1> _<k+1>（X，（n+2）Y））[对称性]r*（I^n _k（X，（n+1）Y '）_0，其中H^p（X，C）_0和H^p（Y，C）_0分别表示X和Y的p阶上同调，I^p _k（X，（n +1）Y'）_0，（p+1）Y）表示闭有理微分形式的De Rham上同调，它沿沿着Y至多有p-k+1阶极点，I^p _k（X，（p+1）Y）_0是由第二类闭有理微分形式生成的I^p _k（X，（p+1）Y）的子空间，Y'是X的一个足够充分的超平面截面，与X横向相交。

(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.