Connected components of moduli spaces
Connected components of moduli spaces
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模空间的连通分量
DOI：
10.4310/jdg/1214440554
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发表时间：
1986
影响因子：
2.5
通讯作者：
F. Catanese
中科院分区：
文献类型：
--
作者：
F. Catanese
设S是一般类型的极小曲面（C上的完备光滑曲面），且设Ji = Ji（S）（分别，Jt）是有向拓扑上复杂结构的粗模空间（分别为，微分）S.根据Gieseker定理[5]，J i（S）是拟投射簇，并且其不可约分支的数目v（S）由两个（拓扑）不变量K = Kg，χ = χ（Θs）的函数vo（K，χ）有界。设λ（S）是Jt（S）的连通分支数，本文回答了文[1]中提出的一个问题，证明了λ（S）可以是任意大的.正如[1]中，我们将不断地提到，我们再次限制我们的注意力到bidouble（即，Galois with group（Z/2））covers of Q = P 1 X P 1：indeed，（cf. [2])我们猜想一个更强的结果成立，即J的许多不同的不可约分支（我们由此得到的）实际上是Jt的连通分支。证明的思想相当简单：如果S和S'是彼此的变形，则存在一个同构f：S -> S”使得f*（Ksr）= Ks e # 2（S，Z），特别地，如果r（S）= max{r e N\（l/r）Ks e i/（S，Z）}，则r（S）= r（S'）。鉴于唐纳森最近的结果[3]，整数r（S）可能是这些曲面的可微结构的不变量;目前还不清楚模空间Jt（S）是否具有更好的性质。然而，当复维数至少为3时，似乎（参见[6]，[7]），类似的现象，高断开也应该出现J（。
Let S be a minimal surface of general type (complete and smooth over C), and let Jί = Jί{S) (resp., Jt) be the coarse moduli space of complex structures on the oriented topological (resp., differential) 4-manifold underlying S. By Gieseker's theorem [5], Jί(S) is a quasiprojective variety, and the number v{S) of its irreducible components is bounded by a function vo{K , χ) of the two (topological) invariants K = Kg, χ = χ(Θs). Let λ(S) be the number of connected components of Jt{S)\ this short note answers a question raised in a previous paper [1], showing that the above number λ(S) can be arbitrarily large. As in [1], to which we shall constantly refer, again we restrict our attention to bidouble (i.e., Galois with group (Z/2)) covers of Q = P 1 X P 1 : indeed, (cf. [2]) we conjecture a stronger result to hold true, namely that many of the different irreducible components of J( we thus obtain are in fact connected components of Jt. The idea of proof is rather simple: if S and S' are deformations of each other, then there exists a diffeomorphism /: S -> S" such that f*(Ksr) = Ks e # 2 (S,Z), and, in particular, if r(S) = max{r e N\(l/r)Ks e i/ (S,Z)}, then r(S) = r(S'). In view of Donaldson's recent result [3], it is possible that the integer r(S) could be an invariant of the differentiable structure for these surfaces; it is not clear at the moment whether nicer properties are enjoyed by the moduli spaces Jt{S). Nevertheless, when the complex dimension is at least 3, it seems (cf. [6], [7]) that similar phenomena of high disconnectedness should appear also for J(.