AI Chat Paper
Note: Please note that the following content is generated by AMiner AI. SciOpen does not take any responsibility related to this content.
{{lang === 'zh_CN' ? '文章概述' : 'Summary'}}
{{lang === 'en_US' ? '中' : 'Eng'}}
Chat more with AI
Article Link
Collect
Show Outline
Outline
Show full outline
Hide outline
Outline
Show full outline
Hide outline
Original Article

Stability of Valuations: Higher Rational Rank

Purdue University, West Lafayette, USA
Beijing International Center for Mathematical Research, Beijing, China
Show Author Information

Abstract

Given a klt singularity x(X,D), we show that a quasi-monomial valuation v with a finitely generated associated graded ring is a minimizer of the normalized volume function vol^(X,D),x, if and only if v induces a degeneration to a K-semistable log Fano cone singularity. Moreover, such a minimizer is unique among all quasi-monomial valuations up to rescaling. As a consequence, we prove that for a klt singularity xX on the Gromov–Hausdorff limit of Kähler–Einstein Fano manifolds, the intermediate K-semistable cone associated with its metric tangent cone is uniquely determined by the algebraic structure of xX, hence confirming a conjecture by Donaldson–Sun.

References

1

Altmann, K., Hausen, J. : Polyhedral divisors and algebraic torus actions. Math. Ann. 224(3), 557–607 (2006)

2

Altmann, K., Ilten, N.O., Petersen, L., Süss, H., Vollmert, R. : The geometry of T-varieties. Contrib. Algebraic Geom. EMS Ser. Congr. Rep. Eur. Math. Soc. Zürich 15, 17–69 (2012)

3

Ambro, F. : The set of toric minimal log discrepancies. Cent. Eur. J. Math. 4(3), 358–370 (2006)

4

Berman, R. : K-polystability of Q-Fano varieties admitting Kähler–Einstein metrics. Invent. Math. 203(3), 973–1025 (2015)

5

Berndtsson, B. : A Brunn–Minkowski type inequality for Fano manifolds and the Bando–Mabuchi uniqueness theorem. Invent. Math. 205(1), 149–200 (2015)

6
Berman, R., Boucksom, S., Eyssidieux, P., Guedj, V., Zeriahi, A. : Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties. J. Reine Angew. Math. arXiv: 1111.7158
7

Birkar, C., Cascini, P., Hacon, C.D., McKernan, J. : Existence of minimal models for varieties of log general type. J. Am. Math. Soc 23, 405–468 (2010)

8

Blum, H. : Existence of valuations with smallest normalized volume. Compos. Math. 154(4), 820–849 (2018)

9
Blum, H., Jonsson, M. : Thresholds, valuations, and K-stability. arXiv: 1706.04548
10

Boucksom, S., Favre, C., Jonsson, M. : A refinement of Izumi's theorem. Valuat. Theory. Interact. EMS Ser. Congr. Rep. Eur. Math. Soc. Zürich 18, 55–81 (2014)

11

Boucksom, S., Hisamoto, T., Jonsson, M. : Uniform K-stability, Duistermaat–Heckman measures and singularities of pairs. Ann. Inst. Fourier (Grenoble) 67(2), 743–841 (2017)

12

Cheeger, J., Colding, T.H. : On the structure of spaces with Ricci curvature bounded below. I. J. Differ. Geom. 46(3), 406–480 (1997)

13

Cheeger, J., Colding, T.H., Tian, G. : On the singularities of spaces with bounded Ricci curvature. Geom. Funct. Anal. 12(5), 873–914 (2002)

14
Chen, X.X., Donaldson, S.K., Sun, S. : Kähler–Einstein metrics on Fano manifolds, I–III. J. Am. Math. Soc. 28, 183–197, 199–234, 235–278 (2015)
15
Collins, T., Székelyhidi, G. : K-semistability for irregular Sasakian manifolds. J. Differ. Geom. arXiv: 1204.2230
16
Collins, T., Székelyhidi, G. : Sasaki-Einstein metrics and K-stability. arXiv: 1512.07213
17

Cutkosky, S.D. : Multiplicities associated to graded families of ideals. Algebra Number Theory 7(9), 2059–2083 (2013)

18

de Fernex, T., Kollár, J., Xu, C. : The dual complex of singularities. Higher dimensional algebraic geometry. Adv. Stud. Pure Math. 74, 103–130 (2017)

19

Donaldson, S. : Scalar curvature and projective embeddings. I. J. Differ. Geom. 59(3), 479–522 (2001)

20

Donaldson, S. : Kähler–Einstein metrics and algebraic structures on limit spaces. Surv. Differ. Geom. Adv. Geom. Math. Phys. 21, 85–94 (2016)

21

Donaldson, S., Sun, S. : Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry. Acta Math. 213(1), 63–106 (2014)

22

Donaldson, S., Sun, S. : Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, II. J. Differ. Geom. 107(2), 327–371 (2017)

23

Ein, L., Lazarsfeld, R., Smith, K. : Uniform approximation of Abhyankar valuation ideals in smooth function fields. Am. J. Math. 125(2), 409–440 (2003)

24

Fujita, K. : Optimal bounds for the volumes of Kähler–Einstein Fano manifolds. Am. J. Math. 140(2), 391–414 (2018)

25
Fulton, W. : Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 2nd edn. Springer, Berlin (1998)
26

Futaki, A. : An obstruction to the existence of Einstein Kähler metrics. Invent. Math. 43, 437–443 (1983)

27

Gigena, S. : Integral invariants of convex cones. J. Differ. Geom. 13, 191–222 (1978)

28

Hacon, C., McKernan, J., Xu, C. : ACC for log canonical thresholds. Ann. Math. 180(2), 523–571 (2014)

29

Hein, H. -J., Sun, S. : Calabi–Yau manifolds with isolated conical singularities. Publ. Math. IHES 126, 73–130 (2017)

30

Ilten, N., Süss, H. : K-stability for Fano manifolds with torus action of complexity 1. Duke Math. J. 166(1), 177–204 (2017)

31

Ishii, S. : Extremal functions and prime blow-ups. Commun. Algebra 32(3), 819–827 (2004)

32

Jonsson, M., Mustaţă, M. : Valuations and asymptotic invariants for sequences of ideals. Ann. Inst. Fourier (Grenoble) 62(6), 2145–2209 (2012)

33

Kaveh, K., Khovanskii, A. : Newton–Okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Ann. Math. 176(2), 925–978 (2012)

34

Kaveh, K., Khovanskii, A. : Convex bodies and multiplicities of ideals. Proc. Steklov Inst. Math. 286(1), 268–284 (2014)

35

Kollár, J., Mori, S. : Birational Geometry of Algebraic Varieties. Cambridge Tracts in Math, vol. 134. Cambridge University Press, Cambridge (1998)

36

Kollár, J. : Singularities of the Minimal Model Program. Cambridge Tracts in Math, vol. 200. Cambridge University Press, Cambridge (2013)

37

Lazarsfeld, R., Mustaţă, M. : Convex bodies associated to linear series. Ann. Sci. Éc. Norm. Supér. 42(5), 783–835 (2009)

38

Li, C. : Minimizing normalized volumes of valuations. Math. Z. 289(1–2), 491–513 (2018)

39

Li, C. : Yau–Tian–Donaldson correspondence for K-semistable Fano manifolds. J. Reine Angew. Math. 733, 55–85 (2017)

40

Li, C. : K-semistability is equivariant volume minimization. Duke Math. J. 166(16), 3147–3218 (2017)

41

Liu, Y. : The volume of singular Kähler–Einstein Fano varieties. Compos. Math. 154(6), 1131–1158 (2018)

42
Li, C., Liu, Y. : Kähler-Einstein metrics and volume minimization. Adv. Math. arXiv: 1602.05094
43

Liendo, A., Süss, H. : Normal singularities with torus actions. Tohoku Math. J. Second Ser. 65(1), 105–130 (2013)

44
Li, C., Wang, X., Xu, C. : On proper moduli space of smoothable Kähler–Einstein Fano varieties. arXiv: 1411.0761v3
45
Li, C., Wang, X., Xu, C. : Algebraicity of the metric tangent cones and equivariant K-stability. arXiv: 1805.03393
46

Li, C., Xu, C. : Special test configurations and K-stability of Fano varieties. Ann. Math. (2) 180(1), 197–232 (2014)

47
Li, C., Xu, C. : Stability of valuations and Kollár components. arXiv: 1604.05398
48
Liu, Y., Xu, C. : K-stability of cubic threefolds. arXiv: 1706.01933
49

Nicaise, J., Mustaţă, M. : Weight functions on non-Archimedean analytic spaces and the Kontsevich–Soibelman skeleton. Algebraic Geom. 2(3), 365–404 (2015)

50

Martelli, D., Sparks, J., Yau, S. -T. : The geometric dual of a-maximisation for toric Sasaki–Einstein manifolds. Commun. Math. Phys. 268, 39–65 (2006)

51

Martelli, D., Sparks, J., Yau, S. -T. : Sasaki–Einstein manifolds and volume minimisation. Commun. Math. Phys. 280, 611–673 (2008)

52

Mustaţă, M. : On multiplicities of graded sequences of ideals. J. Algebra 256, 229–249 (2002)

53

Nicaise, J., Xu, C. : The essential skeleton of a degeneration of algebraic varieties. Am. J. Math. 138(6), 1645–1667 (2016)

54

Okounkov, A. : Brunn–Minkowski inequality for multiplicities. Invent. Math. 125(3), 405–411 (1996)

55

Odaka, Y., Xu, C. : Log-canonical models of singular pairs and its applications. Math. Res. Lett. 19(2), 325–334 (2012)

56

Petersen, L., Süss, H. : Torus invariant divisors. Israel J. Math. 182, 481–504 (2011)

57
Piltant, O. : Graded algebras associated with a valuation (preprint Ecole Polytechnique)
58

Ross, J., Thomas, R. : Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics. J. Differ. Geom. 88(1), 109–159 (2011)

59
Spotti, C., Sun, S. : Explicit Gromov–Hausdorff compactifications of moduli spaces of Kähler–Einstein Fano manifolds. arXiv: 1705.00377
60

Spotti, C., Sun, S., Yao, C. : Existence and deformations of Kähler–Einstein metrics on smoothable Q-Fano varieties. Duke Math. J. 165(16), 3043–3083 (2016)

61

Teissier, B. : Valuations, deformations, and toric geometry. Valuat. Theory Appl. 2, 361–459 (2003)

62

Teissier, B. : Overweight deformations of affine toric varieties and local uniformization. Valuat. Theory Interact. 2014, 474–565 (2014)

63

Tevelev, J. : On a question of Teissier. Collect. Math. 65(1), 61–66 (2014)

64

Tian, G. : On Calabi's conjecture for complex surfaces with positive first Chern class. Invent. Math. 101, 101–172 (1990)

65

Tian, G. : Kähler–Einstein metrics with positive scalar curvature. Invent. Math. 137, 1–37 (1997)

66
Tian, G. : Existence of Einstein Metrics on Fano Manifolds, In: Dai X., Rong X. (eds. ) Metric and Differential Geometry Progress in Mathematics, vol. 297, pp. 119–159. Birkhäuser, Basel (2012)
67

Tian, G. : Partial C0-estimate for Kähler–Einstein metrics. Commun. Math. Stat. 1(2), 105–113 (2013)

68

Tian, G. : K-stability and Kähler–Einstein metrics. Commun. Pure Appl. Math. 68(7), 1085–1156 (2015)

69

Xu, C. : Finiteness of algebraic fundamental groups. Compos. Math. 150(3), 409–414 (2014)

70
Zariski, O., Samuel, P. : Commutative Algebra II. In: Graduate Texts in Mathematics, vol. 29, Springer, New York (1975)
Peking Mathematical Journal
Pages 1-79
Cite this article:
Li, C., Xu, C. Stability of Valuations: Higher Rational Rank. Peking Math J 1, 1-79 (2018). https://doi.org/10.1007/s42543-018-0001-7

241

Views

28

Crossref

Altmetrics

Received: 23 September 2017
Revised: 26 January 2018
Accepted: 22 July 2018
Published: 24 October 2018
© Peking University 2018
Return