Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type
Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type
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DOI：
10.1007/bf01449219
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发表时间：
1988-03
影响因子：
1.4
通讯作者：
H. Tsuji
中科院分区：
文献类型：
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作者：
H. Tsuji
在112上的射影代数曲面理论中，极小模型的存在性起着至关重要的作用。在高维情形中，有一个猜想，如果一个在ti上的射影代数流形不是uniruled，则存在一个极小代数簇与它双有理等价，其中极小代数簇意味着一个Q-阶乘射影簇使得1。它至多有终端奇点。2.它的正则因子在数值上是有效的。关于最小模型猜想，参见[7]。1977年，S. T. Yau证明了具有负或零第一Chern类的紧致K/ihler流形允许K/ihler-Einstein度量[9，10]（奥宾对负第一Chern类情形的结果也有贡献[1]）。这个存在性定理有许多富有成效的应用。但不幸的是，大多数t12上的投射代数簇不具有负的或零的第一陈类。因此，它是可取的推广存在的K/ihler-Einstein度量到更广泛的一类射影代数簇。在这种情况下，我们必须考虑奇异的Kfihler-Einstein度量。本文证明了定义在环II上的一般型极小代数簇上的标准Kfihler-Einstein度量的存在性。如果簇的标准丛不充足，则这个度量必须是奇异的。作为Kfihler-Einstein度量集中的一个直接结果，我们可以研究K/ihler-Einstein度量在一般型光滑极小代数簇的光滑族上的退化.这等价于在一族K/ihler-Einstein度量的相对正则模型上研究一族K/ihler-Einstein度量。对于一般类型的光滑极小代数曲面的变形，我们的结果意味着在纤维上的（-2）-有理曲线的C~外，存在族上的K/ihler-Einstein度量.本文假定所有的代数簇都定义在环上。和奇异簇上的Kfihler度量在下列意义下使用。
In the theory of projective algebraic surfaces over 112, the existence of minimal models plays an essential role. In the higher dimensional case, there has been a conjecture that if a projective algebraic manifold over ti; is not uniruled, there exists a minimal algebraic variety birationally equivalent to it, where a minimal algebraic variety means a Q-factorial projective variety such that 1. It has at most terminal singularities. 2. Its canonical divisor is numerically effective. For the Minimal Model Conjecture, see [7]. On the other hand, in 1977, S.-T. Yau proved that a compact K/ihler manifold with the negative or zero first Chern class admits a K/ihler-Einstein metric [9, 10](Aubin contributed also to the result in the case of the negative first Chern class [1]). And there are many fruitful applications of this existence theorem. But unfortunately most of projective algebraic varieties over t12 do not have negative or zero first Chern classes. Hence it is desirable to generalize the existence of K/ihler-Einstein metrics to a broader class of projective algebraic varieties. In this case, we must consider singular Kfihler-Einstein metrics. The purpose of this paper is to prove the existence of a canonical Kfihler-Einstein metric on a minimal algebraic varieties of general type defined over II;. This metric must be singular if the canonical bundle of the variety is not ample. As a direct consequence of the concentration of the Kfihler-Einstein metrics, we can study the degeneration of K/ihler-Einstein metrics on a smooth family of smooth minimal algebraic varieties of general type. This is equivalent to study a family of K/ihler-Einstein metrics on the relative canonical model of the family. In the case of a deformation of smooth minimal algebraic surfaces of general type, our result implies that there exists a K/ihler-Einstein metrics on the family which is C~ outside the (-2)-rational curves on the fibres. In this paper all the algebraic varieties are assumed to be defined over II;. And a Kfihler metric on a singular variety are used in the following sense.