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

具有普通奇点的复杂射影超曲面的对数变形及其周期图

• 批准号: 11640086
• 负责人: TSUBOI Shoji
• 金额: $ 2.18万
• 依托单位: Kagoshima University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640086/
• 关键词: Ordinary singularity; Normal crossing variety; Logarithmic deformation; Infinitesimal mixed Torelli problem; Kodaira-Spencer map; Chech de Rham cohomology; Cubic hyper-resolution; Rigid singularity; 半安定還元; オイラー数; チャーン数

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.

1.我们已经为具有普通奇点的复射影曲面的局部平凡解析族制定了无穷小混合Torelli问题，由流形参数化，相对化了由于V.Navarro Aznar，F.Guillen等人提出的三次超分辨率的概念，并给出了该问题肯定解的上同调充分条件。此外，我们还构造了几个满足这些充分条件的例子.我们还考虑了所谓（n，r_1，r_2，r_3，r_4）型复射影三重投影的无穷小混合Torelli问题。在这一过程中，我们发现了一个弱正规的非孤立奇点，它是一个普通的三重点的退化奇点，用仿射坐标描述为（xy）^2+（yz）^2+（zx）^2+wxyz=0。事实证明，奇点是施泰纳曲面上的一个圆锥体，施泰纳曲面是P^3（C）中具有普通奇点的有理曲面。它的正规化是P^2（C）上的一个锥，通过2次Veronese映射嵌入到P^5（C）中，是一个重数为4的有理孤立奇点，并且在形变下是刚性的.除上述结果外，我们还得到了P^4（C）中具有常奇性的复超曲面的非奇异正规化的Euler数的一个公式，推广了Enriques关于P^3（C）中具有常奇性的复超曲面的经典公式.这项工作正在准备出版。

项目成果

S.Tsuboi and F.Guillen: "Simultaneous Cubic Hyper-resolutins of Locally Trivial Analytic Families of Complex Projective Varieties and Cohomological Descent."The Reports of the Faculty of Science, Kagoshima University.. No.33. 1-33 (2000)

S.Tsuboi 和 F.Guillen：“复杂射影簇和上同调下降的局部平凡解析族的同时三次超分辨率”。鹿儿岛大学理学院的报告。第 33 期。

K.Miyajima: "A note on the closed rengeness of vector bundle-valued tangential Cauchy-Riemann complex, Analysis and Geometry in Several Complex Variables"Proceedings of the 40th Taniguchi Symposium, ed. G.Komatsu and M.Kuranishi, Trends in Mathematics, Bi

K.Miyajima：“关于向量丛值切向柯西-黎曼复形的闭合重整性的说明，多个复变量中的分析和几何”第 40 届谷口研讨会论文集，编辑。

T.Aikou: "Conformal flatness of complex Finsler structures"Publ.Math.Debrecen. 54/1-2. 165-179 (1999)

T.Aikou：“复杂芬斯勒结构的共形平坦度”Publ.Math.Debrecen。

M.Nakashima: "Explicit A-stable Rational Runge-Kutta methods for parabolic differential equations (II), International Symposium on Applied Mathematics (Dalian, China Aug 14, 2000-Aug 18, 2000)"Proceedings of International Symposium on Applied Mathematics.

M.Nakashima：“抛物型微分方程的显式A-稳定有理龙格-库塔方法（II），国际应用数学研讨会（中国大连，2000年8月14日-2000年8月18日）”国际应用数学研讨会论文集。

S.Tsuboi: "Infinitesimal Parameter Spaces of Locally Trivial Deformations of Compact Complex Surfaces with Ordinary Singularities"Finite or Infinite Dimensional Complex Analysis (Marcel Dekker, Inc.). 523-532 (2000)

S.Tsuboi：“具有普通奇点的紧凑复杂表面的局部平凡变形的无限​​小参数空间”有限或无限维复杂分析（Marcel Dekker，Inc.）。