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A generalization of the Ross-Thomas slope theory. (English) Zbl 1328.14073
In this very nice paper, it is proved various results about algebro-geometric stability (K-stability in the sense of Tian-Donaldson-Stoppa) of certain polarized varieties that were expected for a long time. More precisely, it is firstly proved that a semi log canonical and canonically polarized curve is \(K\)-stable and a semi log canonical variety \(X\) with \(K_X\) trivial is \(K\)-semistable. We refer to [J. Kollár and N. I. Shepherd-Barron, Invent. Math. 91, No. 2, 299–338 (1988; Zbl 0642.14008)] for the notion of semi log canonical singularity. This result was expected for two reasons. On one hand it has been constructed non smooth Kähler-Einstein metrics on such varieties. On another hand, in the smooth case, it is known that the existence of a constant scalar curvature Kähler metric implies \(K\)-polystability.
A key ingredient of the proof is a formula for Donaldson-Futaki invariants for certain special semi test-configurations (the word semi-relative means that the line bundle on the test configuration is considered to be only semi-ample, the case of relative ample line bundle was independentely treated by X.Wang [Math. Res. Lett. 19, No. 4, 909–926 (2012; Zbl 1408.14147)]). These test configurations generalize the test-configurations studied by J. Ross and R. Thomas in their theory of slope stability for manifolds [J. Algebr. Geom. 16, No. 2, 201–255 (2007; Zbl 1200.14095)]. The proof of the formula is based on the original work of D. Mumford on Geometric Invariant Theory. We expect that this formula will find plenty of other applications in the future.
The other key ingredient is a theorem that shows that to test \(K\)-stability it is sufficient to consider the special test configurations that were used in the previous result. This is also an important fact.

MSC:
14L24 Geometric invariant theory
14J17 Singularities of surfaces or higher-dimensional varieties
32Q15 Kähler manifolds
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Full Text: Euclid arXiv
References:
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