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

(1)设Y是4维复射影空间中具有普通奇点的代数超曲面，即普通n点、普通尖点和驻点，X是它的正规化。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，我们得到了X的欧拉-庞加莱特征及其在全纯向量场的鞘中的系数的数值公式。(2)我们给出了一个仅有普通n点(2≦n≦4)、普通尖点和退化普通三点作为奇点的超曲面的例子，并且它们的正规化只将有理四点分离为奇点。(3)对于嵌入在射影空间中的(n+1)维复代数流形X及其足够充分的非奇异超平面截面Y，我们证明了：(A)F^kh^p(X-Y，C)[相似或相等]i^p_k(X，(p+1)Y)，F^kh^p(X，(p+1)Y)_0[相似或相等]i^p_k(X，(p+1)Y)_0(0≦k≦p，0≦p≦n+1)，(B)F^KGR^&lt；W[q]&gt；_qh^p(X-Y，C)[相似或相等]i^p_k(X，(p+1)Y)_0，F^kgr^&lt；w[q]&gt；_&lt；q+1&gt；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^&lt；n+1&gt；_&lt；k+1&gt；(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的第p次上同调，i^p_k(X，(p+1)Y)表示闭有理微分形式的de Rham上同调，其中沿Y至多有p-k+1阶极点，i^p_k(X，(p+1)Y)_0是i^p_k(X，(p+1)Y)_0的子空间(P+1)Y)是由第二类有理闭微分形式生成的，Y‘是X的一个充分的超平面截面，与X横交。

(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.