The Calabi conjecture and K-stability
The Calabi conjecture and K-stability
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DOI：
10.1093/imrn/rnr107
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发表时间：
2010-10
影响因子：
1
通讯作者：
Y. Odaka
中科院分区：
文献类型：
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作者：
Y. Odaka
用代数方法证明了极化Calabi-Yau簇和正则极化的弱奇点簇的K-稳定性。特别是，“稳定品种”介绍的Kollar-Shepherd-巴伦[“三重和变形的表面奇点。”Inventiones Mathematicae 91（1988）：299-338]和Alexeev [“Moduli spaces M g，n（W）for surfaces.”Proceeding of“Higher-dimensional complex varieties（特伦托，1994）"，de Gruyter，柏林，1996，1-22]证明了构成紧模空间的高维复簇是K-稳定的，尽管众所周知它们不一定是渐近（半）稳定的。因此，我们有轨道反例的民间传说猜想“K-稳定性意味着渐近稳定”。他们有Kahler-Einstein（orbifold）度量，所以唐纳森[“标量曲率和射影嵌入。I.”Journal of Differential Geometry 59，no. 3（2001）：479-522]并不适用于orbifolds。
We algebraically prove K-stability of polarized Calabi–Yau varieties and canonically polarized varieties with mild singularities. In particular, the “stable varieties” introduced by Kollar–Shepherd-Barron [“Threefolds and deformation of surface singularities.” Inventiones Mathematicae 91 (1988): 299–338] and Alexeev [“Moduli spaces M g,n (W) for surfaces.” Proceeding of “Higher-dimensional complex varieties (Trento, 1994)”, de Gruyter, Berlin, 1996, 1–22], which form compact moduli space, are proven to be K-stable although it is well known that they are not necessarily asymptotically (semi) stable. As a consequence, we have orbifold counterexamples to the folklore conjecture “K-stability implies asymptotic stability”. They have Kahler–Einstein (orbifold) metrics, so the result of Donaldson [“Scalar curvature and projective embeddings. I.” Journal of Differential Geometry 59, no. 3 (2001): 479–522] does not hold for orbifolds.