Hodge theory of singular algebraic curves
Hodge theory of singular algebraic curves
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奇异代数曲线的 Hodge 理论
DOI：
10.1090/s0002-9939-1990-1002162-8
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发表时间：
1990
影响因子：
0.8
通讯作者：
M. Nagase
中科院分区：
文献类型：
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作者：
M. Nagase
研究了去掉奇异性后的奇异代数曲线上的Laplacian的谱和L2-上同调的结构，该曲线是不完全Kahler流形。1.设C是一条复代数曲线，嵌入在。射影空间PN（C），S是它的奇点集。设C* = CS，用g表示PN（C）的Fubini-Study度量的限制（对C*）。作者希望发展不完全Kahler流形C ~* = C ~*（g）的整体分析（或谱几何）.首先证明C ~* 与具有J. Cheeger（[1]，[3]）意义下的锥状奇点的真实的二维不完全Kahler流形拟等距.取一个非奇异模型7 t：C -？C，并且设定C1 * = C7- 1（S）。然后，由于映射7给出等距C*（7i*g）-C（g），让我们研究7 l（s）附近的7 Tg。通过标准的论证（参见，例如，[12，引理1.6.1]），对于点p E或-1（S），存在局部坐标邻域（U，u），其中u = 0在p处，并且仿射坐标（z1，.，ZN）的P（C）与（zI，.，zN）=（OI.，0），使得z1（u）=um，z2（u）= um ~ 2f 2（u），. . ., z1（u）= Um'f1（U），z1+ I（u）= 0，... ., ZN（u）= 0，其中11 u12（n，1）du 0 du是U12（n，1）du 0 du的拟等距度量。因此，我们有以下引理。编辑于1988年5月27日收到，并于1988年12月16日和1989年4月6日以修订形式收到。1980年数学学科分类（1985年修订）。初级58 A14、58 C40。(? 1990年美国数学学会0002-9939/90 $1.00 + $.25每页
The spectrum of the Laplacian and the structure of the L2-cohomology are studied on a singular algebraic curve with its singularity removed, which is hence an incomplete Kahler manifold. 1. STATEMENTS OF THE RESULTS Let C be a complex algebraic curve embedded in the. projective space PN(C) and S be its singularity set. Put C* = C S and denote by g the restriction (to C* ) of the Fubini-Study metric of PN(C) . The author wishes to develop the global analysis (or, the spectral geometry) on the incomplete Kahler manifold C* = C*(g). First let us show that C* is quasi-isometric to a real 2-dimensional incomplete Kahler manifold with cone-like singularities in the sense of J. Cheeger ([1], [3]). Take a nonsingular model 7t: C -? C and set Ci* = C7-l(S). Then, since the map 7 gives an isometry C*(7i*g) -C(g), let us investigate 7T g near 7 l(s). By a standard argument (see, for example, [12, Lemma 1.6.1]), for a point p E or-1(S), there exists a local coordinate neighborhood (U,u) with u = 0 at p and the affine coordinates (z1, ... ,ZN) of P (C) with (zI,... ,zN)=(OI...,O) at n(p) suchthat zI(u)=um,z2(u)= um2f2(u), . . ., z1(u) = Um'f1(U), z1+ I(u) = O,.. ., ZN(u) = 0, where 1 1 ul2(ml)miJ + uJI_2}, the metric is quasi-isometric to U12(n,1) du 0 du . Therefore, we have the following lemma. Received by the editors May 27, 1988 and, in revised forms, December 16, 1988 and April 6, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 58A14, 58C40. (? 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page