Manifolds with ample canonical class
Manifolds with ample canonical class
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DOI：
10.1007/bf01425543
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发表时间：
1968-06
影响因子：
3.1
通讯作者：
M. Narasimhan;R. Simha
中科院分区：
文献类型：
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作者：
M. Narasimhan;R. Simha
设V是紧致连通的真实的解析流形.本文用微分几何方法证明了V上具有充分标准线丛的复结构的同构类集合具有Hausdorff复空间的自然结构（定理w5）.现在我们简要地概述一下证明。设Vo是一个具有丰富标准丛的紧致连通复流形.设T是局部完备Kuranishi族{V，}，~ r在Vo处的参数化空间，我们可以假定对t~ T，流形V_t的标准类是充分的. I10的自同构群Aut（Vo）是有限的，作用在T上。我们证明了，如果T被限制为很小，T的两个点对应于同构流形当且仅当它们在Aut（Vo）对T的作用的同一轨道上（引理4.1）。复空间是T与Aut（Vo）的商，它被取为在Vo同构类上的局部坐标系;由于Kuranishi族的局部完备性，这些坐标系被修补。上述引理和空间的Hausdorff性质在Bergmann度量的推广下得到了证明。设Vo上的线丛K k是非常充足的，其中K是Vo上的典范线丛.我们在Vo上引入一个依赖于k的自然厄米特度量，如果k= 1，即如果K非常充足，则它简化为Bergmann度量;由于Vo嵌入H~（Vo，K~）的对偶的l维子空间的射影空间P（H~ Kk）*）中，为了定义Vo上的度量，在H~（Kk）上引入一个厄米特形式就足够了。为此，我们首先在Vo上定义一个正体积元。利用这个体积元，我们用一种类似于在模形式上引入Petersson度量的方法，定义了H~ K（k）上的厄米特形式.现在如果Vo和V6是两个具有充足标准类的流形，并且q~：Vo-* V6是同构，则q~是关于对应于大k的度量的等距。这一事实，连同度量对参数的连续依赖性，产生了证明（引理3.2）中所需的某些映射的等度连续性，这特别表明同构流形的极限是同构的，这证明了豪斯多夫性质。
Let V be a compact connected real analytic manifold. We prove in this paper, by differential geometric methods, that the set of isomorphism classes of complex structures on V with ample canonical line bundle has a natural structure of a Hausdorff complex space (Theorem in w 5). We now briefly outline the proof. Let Vo be a compact connected complex manifold with ample canonical bundle. Let T be the parametrising space of the locally complete Kuranishi family {V,},~ r at Vo; we may assume that for t~ T the canonical class of the manifold Vt is ample. The group of automorphisms Aut (Vo) of I1o is finite and acts on T. We prove that, if T is restricted to be small, two points of T correspond to isomorphic manifolds if and only if they are in the same orbit for the action of Aut (Vo) on T (Lemma 4.1). The complex space which is the quotient of T by Aut (Vo) is taken to be a local coordinate system at the isomorphism class of Vo; these coordinate systems patch up due to the local completeness of Kuranishi families. The above lemma and the Hausdorff nature of the space are proved with the help of a generalisation of the Bergmann metric. Suppose that the line bundle K k is very ample on Vo, where K is the canonical line bundle on Vo. We introduce a natural hermitian metric on Vo, depending on k, which reduces to the Bergmann metric if k= l, ie, if K is very ample; since Vo gets imbedded in the projective space P (H~ Kk)*) of l-dimensional subspaces of the dual of H~(Vo, K~), in order to define a metric on Vo it is sufficient to introduce a hermitian form on H~ Kk). To do this, we first define a positive volume element on Vo. Using this volume element, we define a hermitian form on H~ K k) by a device similar to that used in introducing the Petersson metric on modular forms. Now if Vo and V6 are two manifolds with ample canonical class and q~: Vo-* V6 is an isomorphism, then q~ is an isometry with respect to the metrics corresponding to large k. This fact, together with the continuous dependence of the metrics on parameters, yields the equicontinuity of some maps required in the proof (Lemma 3.2) and this shows in particular that limits of isomorphic manifolds are isomorphic, which proves the Hausdorff nature.