Ambient deformations for exceptional sets in two-manifolds
Ambient deformations for exceptional sets in two-manifolds
复制标题
两个歧管中特殊组的环境变形
DOI：
10.1007/bf02139700
复制
发表时间：
1979
影响因子：
3.1
通讯作者：
Henry B. Laufer
中科院分区：
文献类型：
--
作者：
Henry B. Laufer
设M为具有连通异常集a的严格伪凸二维流形，设0):，//~ Q为M的一般变形，其中忽略M边界附近发生的情况[27，定理1，p. 217],[29]。设re: M~ V是A在M中的吹落，p= zr (A)，那么~ z就是Stein法向二维空间V的一个分辨率。设r:~”~ U是（V, p）的一般变形。设T是Q的约简子空间，其中dimHl （Mq. 6）是常数。这是Q的最大简化子空间，在此空间上oJ可以同时被简化[51,39]。诱导映射T—~ U在其图像上是有限的[3,4]。因此，通过研究T和M， v，我们可以得到关于p附近的一些奇异点的信息。设q是Mq中的例外集。设aq是a0的不可约分量。然后(Proposition2。3), Aq. i在~/中与a上的一个环Di> 0有同源性，具有)~(D3=- 89i9d ~+ D~。K) = < 1。这样的Di是有限的。反之，设D b> 0是a上的一个循环。Kodaira关于提升子流形[18]的理论可以加以修改，使其适用于D。解析范畴的细节在[30]中进行。Wahl[53]在代数范畴中有独立的处理。设3为D的无穷小变形的胚芽束，设HI （D, j）—0。当原始奇点p为有理[2]或最小椭圆[25]，基本周期Z几乎减小时，上述一般结果很好地适用于描述附近的异常集Aq， qGr（定义3）。2)，除非奇异有理ab i与Ai外，ie的系数均为l。~ = 2。此类奇点的例子包括有理双点[2]、有理三点[2]、商奇点[5]以及单峰和双峰奇点[1]，见表4。1和4.2。对于这样的奇点，与不可约分量Aq， i， q~ T相对应的环D除了满足z (D)=< 1外，还满足D< z （Lemma3）。1, Propositions4。且H~(D,， Y)= 0（命题3.3和命题4.6）。我们说（定义3.5）a
Let M be a strictly pseudoconvex two-dimensional manifold with connected exceptional set A. Let o):,//~ Q be the versal deformation of M, where we ignore what happens near the boundary of M [27, Theorem 1, p. 217],[29]. Let re: M~ V be the blow-down of A in M, p= zr (A).~ z is then a resolution of the Stein normal two-dimensional space V. Let r:~"~ U be the versal deformation of (V, p). Let Mq=(n 1 (@ Let T be the reduced subspace of Q on which dimHl (Mq. 6) is constant. This is the maximal reduced subspace of Q over which oJ can be simultaneously blown-down [51, 39]. The induced map T--~ U is finite onto its image [3, 4]. Thus by studying T and the M, v we may obtain information about some of the singularities near to p.Specifically. Let Aq be the exceptional set in Mq. Let Aq. i be an irreducible component of A 0. Then (Proposition2. 3), Aq. i is homologous in,~/to a cycle Di> 0 on A with)~(D3=-89 i 9 D~+ D~. K)=< 1. There are only finitely many such Di. Conversely, let D> 0 be a cycle on A. Kodaira's theory for lifting submanifolds [18] may be modified so as to apply to D. Details in the analytic category are carried out in [30]. Wahl [53] has an independent treatment in the algebraic category. Let 3-be the sheaf of germs of infinitesimal deformations of D. Suppose that HI (D, j)---0. Then there are only first obstructions to lifting D. The above general results apply well to describe nearby exceptional sets Aq, qGr, in case the original singularity p is rational [2] or minimally elliptic [25] and the fundamental cycle Z is almost reduced (Definition3. 2), ie has coefficients of l except for non-singular rational A i with Ai. A~=-2. Examples of such singularities include the rational double points [2], the rational triple points [2], the quotient singularities [5], and the unimodal and bimodal singularities [1], Tables4. 1 and 4.2. For such singularities, the cycles D which are homologous to irreducible components Aq, i, q~ T, besides having z (D)=< 1, additionally satisfy D< Z (Lemma3. 1, Propositions4. 4 and 4.5), and have H~(D,, Y)= 0 (Proposition 3.3 and Proposition 4.6). We say (Definition 3.5) that a
DOI：
--
发表时间：
2005
期刊：
影响因子：
--
作者：
T. Kobayashi;T. Oshima
通讯作者：
T. Oshima