Applications of the Kähler-Einstein-Calabi-Yau metric to moduli of K3 surfaces
Applications of the Kähler-Einstein-Calabi-Yau metric to moduli of K3 surfaces
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DOI：
10.1007/bf01390067
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发表时间：
1980-10
影响因子：
3.1
通讯作者：
A. Todorov
中科院分区：
文献类型：
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作者：
A. Todorov
在复曲面理论中最有趣的问题之一是模空间的明确描述，即给定复曲面上所有复杂结构的空间。研究模问题有两种强有力的方法。其中之一是不变量理论开发的模问题的芒福德。另一个是霍奇结构（或积分周期）的变化，其最近的发展是由于格里菲斯。结果表明，Hodge结构的变分方法对于研究K3曲面的模空间是非常有用的。Andreotti、Weil和Turina证明了周期映射是局部同构。在1970年Shafarevich和Piatetski夏皮罗证明了全球Torelli定理代数K3曲面。参见[SP]。Burns和Rapoport以及后来的Looijenga和Peters证明了K~ ihler K_3曲面的整体Torelli定理。参见[LP]和[BR]。这两对作者使用的想法Shafarevich.一个主要问题的模的K3表面是满射的周期地图，即是真的，每一个点的周期域S 0（3，19）/S 0（2）·对应于一个标记K3表面。证明这一事实的代数K 3曲面首先是由库利科夫，后来由Pinkham和Pinkson。在本文中，我们将证明以下两个定理：
One of the most interesting problems in the theory of complex surfaces is that of explicit description of the moduli space, ie the space of all complex structures on a given complex surface. There are two powerful methods for studying the moduli questions. One of them is the Invariant theory developed for moduli questions by Mumford. The other is the variation of Hodge structures (or periods of integrals) whose recent development is due to Griffiths. It turned out that the methods of the variations of Hodge structures is extremely useful for studing the moduli space of K3 surfaces. Andreotti, Weil and Turina proved that the period map is a local isomorphism. In 1970 Shafarevich and Piatetski-Shapiro proved the global Torelli theorem for algebraic K3 surfaces. See [SP]. Burns and Rapoport and later Looijenga and Peters, proved the Global Torelli theorem for K~ ihler K3 surfaces. See [LP] and [BR]. Both pairs of authors used the ideas of Shafarevich.One of the main problems of the moduli of K3 surfaces was that of the epimorphism of the period map, ie is it true that every point of the period domain S0 (3, 19)/S0 (2)• corresponds to a marked K3 surface. The proof of this fact for algebraic K 3 surfaces was first given by Kulikov and later by Pinkham and Persson. In this article we will prove the following two theorems: