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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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##### References:
 [1] V. Alexeev: Log canonical singularities and complete moduli of stable pairs , 1996). [2] X.X. Chen and G. Tian: Geometry of Kähler metrics and foliations by holomorphic discs , Publ. Math. Inst. Hautes Études Sci. 107 (2008), 1-107. · Zbl 1182.32009 · doi:10.1007/s10240-008-0013-4 [3] \begingroup S.K. Donaldson: Scalar curvature and projective embeddings , I, J. Differential Geom. 59 (2001), 479-522. \endgroup · Zbl 1052.32017 [4] S.K. Donaldson: Scalar curvature and stability of toric varieties , J. Differential Geom. 62 (2002), 289-349. · Zbl 1074.53059 · euclid:jdg/1090950195 [5] S.K. Donaldson: Lower bounds on the Calabi functional , J. Differential Geom. 70 (2005), 453-472. · Zbl 1149.53042 · euclid:jdg/1143642909 [6] W. Fulton: Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 2 , Springer, Berlin, 1984. · Zbl 0541.14005 [7] A. Futaki: An obstruction to the existence of Einstein Kähler metrics , Invent. Math. 73 (1983), 437-443. · Zbl 0506.53030 · doi:10.1007/BF01388438 · eudml:143056 [8] D. Gieseker: Global moduli for surfaces of general type , Invent. Math. 43 (1977), 233-282. · Zbl 0389.14006 · doi:10.1007/BF01390081 · eudml:142515 [9] D. Gieseker: Lectures on Moduli of Curves, Tata Institute of Fundamental Research Lectures on Mathematics and Physics 69 , Springer-Verlag, New York, 1982. · Zbl 0534.14012 [10] R. Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics 52 , Springer, New York, 1977. · Zbl 0367.14001 [11] H. Hironaka: Resolution of singularities of an algebraic variety over a field of characteristic zero , I, Ann. of Math. (2) 79 (1964), 109-203; · Zbl 0122.38603 · doi:10.2307/1970486 [12] H. Hironaka: Resolution of singularities of an algebraic variety over a field of characteristic zero , II, Ann. of Math. (2) 79 (1964), 205-326. · Zbl 0122.38603 · doi:10.2307/1970486 [13] J. Kollár and S. Mori: Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics 134 , Cambridge Univ. Press, Cambridge, 1998. [14] J. Kollár and N.I. Shepherd-Barron: Threefolds and deformations of surface singularities , Invent. Math. 91 (1988), 299-338. · Zbl 0642.14008 · doi:10.1007/BF01389370 · eudml:143542 [15] C. Li and C. Xu: Special test configurations and K-stability of $$\mathbb{Q}$$-Fano varieties , 2011). arXiv: · arxiv.org [16] \begingroup T. Mabuchi: Chow-stability and Hilbert-stability in Mumford’s geometric invariant theory , Osaka J. Math. 45 (2008), 833-846. \endgroup · Zbl 1156.14039 · euclid:ojm/1221656656 [17] T. Mabuchi: K-stability of constant scalar curvature polarization , 2008). arXiv: · Zbl 1152.32301 · arxiv.org [18] T. Mabuchi: A stronger concept of K-stability , 2009). arXiv: · arxiv.org [19] H. Matsumura: Commutative Ring Theory, Cambridge Studies in Advanced Mathematics 8 , Cambridge Univ. Press, Cambridge, 1986. · Zbl 0603.13001 [20] D. Mumford: Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge 34 , Springer, Berlin, 1965. · Zbl 0147.39304 [21] D. Mumford: Stability of Projective Varieties, Enseignement Math., Geneva, 1977. · Zbl 0497.14004 [22] Y. Odaka: The GIT stability of polarized varieties via discrepancy , to appear in Annals of Mathematics. · Zbl 1271.14067 [23] Y. Odaka: The Calabi Conjecture and K-stability, Int. Math. Res. Not. 13 , 2011. · Zbl 06043643 [24] Y. Odaka and Y. Sano: Alpha invariant and K-stability of $$\mathbb{Q}$$-Fano varieties , Adv. Math. 229 (2012), 2818-2834. · Zbl 1243.14037 · doi:10.1016/j.aim.2012.01.017 [25] Y. Odaka: On parametrization, optimization and triviality of test configurations , 2012). arXiv: · Zbl 1331.14046 · arxiv.org [26] D. Panov and J. Ross: Slope stability and exceptional divisors of high genus , Math. Ann. 343 (2009), 79-101. · Zbl 1162.14034 · doi:10.1007/s00208-008-0266-8 [27] J. Ross and R. Thomas: An obstruction to the existence of constant scalar curvature Kähler metrics , J. Differential Geom. 72 (2006), 429-466. · Zbl 1125.53057 [28] J. Ross and R. Thomas: A study of the Hilbert-Mumford criterion for the stability of projective varieties , J. Algebraic Geom. 16 (2007), 201-255. · Zbl 1200.14095 · doi:10.1090/S1056-3911-06-00461-9 [29] J. Stoppa: K-stability of constant scalar curvature Kähler manifolds , Adv. Math. 221 (2009), 1397-1408. · Zbl 1181.53060 · doi:10.1016/j.aim.2009.02.013 [30] G. Tian: Kähler-Einstein metrics with positive scalar curvature , Invent. Math. 130 (1997), 1-37. · Zbl 0892.53027 · doi:10.1007/s002220050176 [31] X. Wang: Heights and GIT weights , to appear in Math. Research Letters. [32] S.T. Yau: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation , I, Comm. Pure Appl. Math. 31 (1978), 339-411. \endthebibliography* · Zbl 0369.53059 · doi:10.1002/cpa.3160310304
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