The $K$-energy on hypersurfaces and stability
The $K$-energy on hypersurfaces and stability
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DOI：
10.4310/cag.1994.v2.n2.a4
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发表时间：
1994
影响因子：
0.7
通讯作者：
G. Tian
中科院分区：
文献类型：
--
作者：
G. Tian
极化射影簇的稳定性的概念是由D.Mumford为了研究射影簇的模问题而提出的。Mumford证明了光滑代数曲线的稳定性，D.Gieseker证明了代数曲面的稳定性，Viehweg证明了代数流形的稳定性。然而，即使给定的极化簇是射影空间中的奇异超曲面，如何检验它的稳定性似乎仍然是一个具有挑战性的问题。本文的目的是给出一个超曲面稳定或半稳定的充要条件。对于Kahler流形，用T.Mubachi提出的广义K-能量的适定性或下有界性给出了这一条件。特别地，我们将证明任何超曲面是半稳定的，如果它至少有两个余维奇点，并且允许Kahler-Einstein或双曲度规。我们记I？n^表示Cn+2上所有d次齐次多项式的空间，B表示射影空间Prn^d.B中的任一点[/]决定CP中唯一的d次超曲面S/.特殊线性群G-Sx(n+2，C)通过对G中的任一a赋值/to/o<；j~来诱导向量空间Rn^d上的作用，如果G中的轨道Gf是闭的，且/在G中的稳定器是有限的，则称E/是稳定的；如果Rn^d中的零不包含在Gf的轨道闭包中，则称S^是半稳定的.众所周知，任何光滑的超曲面都是稳定的。但是，如果E/只有一个隔离的
The notion of stability for a polarized projective variety was introduced by D. Mumford for the study of the moduli problem of projective varieties. The stability has been verified by Mumford for smooth algebraic curves, D. Gieseker for algebraic surfaces and Viehweg for algebraic manifolds, which are polarized by m-pluri-canonical bundles for m sufficiently large ([Md], [Gi], [Vi]). However, it still seems to be a challenging problem to check the stability for a given polarized variety, even if the variety is a singular hypersurface in some projective space. The purpose of this paper is to give a sufficient and intrinsic condition for a hypersurface to be stable or semistable. The condition is given in terms of the properness or lower buundedness of a generalized K-energy, which was introduced by T. Mubachi for Kahler manifolds. In particular, we will prove that any hypersurface is semistable if it has only orbifold singularities of codimension at least two and admits a Kahler-Einstein orbifold metric. We denote by i?n^ the space of all homogeneous polynomials on C n+2 of degree d, and B the projective space PRn^d. Any point [/] in B determinates a unique hypersurface S/ in CP of degree d. The special linear group G — SX(n + 2, C) induces an action on the vector space Rn^d by assigning / to / o <j~ for any a in G. Then we say that E/ is stable if the orbit Gf is closed and the stablier of / in G is finite; we say that S^ is semistable if the zero in Rn^d is not contained in the closure of the orbit Gf. It is well-known that any smooth hypersurface E/ is stable. However, if E/ has only one isolated