Uniform stability of twisted constant scalar curvature K\"ahler metrics
Uniform stability of twisted constant scalar curvature K\"ahler metrics
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DOI：
10.1093/imrn/rnv291
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发表时间：
2014-12
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通讯作者：
R. Dervan
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文献类型：
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作者：
R. Dervan
我们在测试配置空间上引入一个范数，我们称之为最小范数。我们猜想关于这个范数的一致K-稳定性等价于常数量曲率K\“ahler度量的存在性.这种一致K稳定性的概念类似于马渊泛函的稳定性。我们证明了测试配置的平凡性，证明了测试配置的最小范数为零当且仅当它的L^2范数为零，当且仅当它几乎是平凡的。证明了当扭曲足够大时，扭曲的常数量曲率K\“ahler度量的存在蕴涵着关于最小范数的一致扭曲K-稳定性.我们给出了代数几何证明的一致K-稳定性的一般类型和Calabi-Yau的情况下，以及在Fano的情况下，α不变的条件。我们的结果保持线丛足够接近（反）正则线丛，也在扭曲的设置。我们表明，日志K-稳定性意味着扭曲的K-稳定性，以及扭曲的K-半稳定性的品种意味着品种有温和的奇异性。
We introduce a norm on the space of test configurations, which we call the minimum norm. We conjecture that uniform K-stability with respect to this norm is equivalent to the existence of a constant scalar curvature K\"ahler metric. This notion of uniform K-stability is analogous to coercivity of the Mabuchi functional. We characterise the triviality of test configurations, by showing that a test configuration has zero minimum norm if and only if it has zero $L^2$-norm, if and only if it is almost trivial. We prove that the existence of a twisted constant scalar curvature K\"ahler metric implies uniform twisted K-stability with respect to the minimum norm, when the twisting is ample. We give algebro-geometric proofs of uniform K-stability in the general type and Calabi-Yau cases, as well as in the Fano case under an alpha invariant condition. Our results hold for line bundles sufficiently close to the (anti)-canonical line bundle, and also in the twisted setting. We show that log K-stability implies twisted K-stability, and also that twisted K-semistability of a variety implies that the variety has mild singularities.