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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 andN. 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 byX.Wang [Math. Res. Lett. 19, No. 4, 909–926 (2012;Zbl 1408.14147)]). These test configurations generalize the test-configurations studied byJ. Ross andR. 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

Cite

References:

[1]V. Alexeev: Log canonical singularities and complete moduli of stable pairs , 1996).
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[3]\begingroup S.K. Donaldson: Scalar curvature and projective embeddings , I, J. Differential Geom. 59 (2001), 479-522. \endgroup ·Zbl 1052.32017
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[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
[15]C. Li and C. Xu: Special test configurations and K-stability of \(\mathbb{Q}\)-Fano varieties , 2011). arXiv:
[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
[17]T. Mabuchi: K-stability of constant scalar curvature polarization , 2008). arXiv: ·Zbl 1152.32301
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[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 1484.32043
[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
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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