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On properness of K-moduli spaces and optimal degenerations of Fano varieties

Abstract

We establish an algebraic approach to prove the properness of moduli spaces of K-polystable Fano varieties and reduce the problem to a conjecture on destabilizations of K-unstable Fano varieties. Specifically, we prove that if the stability threshold of every K-unstable Fano variety is computed by a divisorial valuation, then such K-moduli spaces are proper. The argument relies on studying certain optimal destabilizing test configurations and constructing a \(\Theta \)-stratification on the moduli stack of Fano varieties.

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Notes

  1. 1.

    This definition differs from that in [4, 22] by a sign convention to conform to the convention in the K-stability literature that non-negativity of the Futaki invariant corresponds to semistability.

  2. 2.

    The current arXiv version of [22] states a weaker version of Theorem 8.3 that applies to real valued numerical invariants, and also includes the condition of uniqueness of HN filtrations. The theorem we discuss here appears in an update of [22] that is not yet available on the arXiv.

References

  1. 1.

    Altmann, K.: The dualizing sheaf on first-order deformations of toric surface singularities. J. Reine Angew. Math. 753, 137–158 (2019)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Ahmadinezhad, H., Ziquan, Z.: K-stability of Fano varieties via admissible flags. arXiv:2003.13788 (2020)

  3. 3.

    Alper, J., Blum, H., Halpern-Leistner, D., Xu, C.: Reductivity of the automorphism group of K-polystable Fano varieties. Invent. Math. 222(3), 995–1032 (2020)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Alper, J., Halpern-Leistner, D., Heinloth, J.: Existence of moduli spaces for algebraic stacks. arXiv:1812.01128 (2018)

  5. 5.

    Blum, H., Jonsson, M.: Thresholds, valuations, and K-stability. Adv. Math. 365, 107062 (2020)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Blum, H., Liu, Y.: Openness of uniform K-stability in families of \({\mathbb{Q}}\)-Fano varieties, Ann. Sci. Éc. Norm. Supér. (to appear) arXiv:1808.09070 (2018)

  7. 7.

    Blum, H., Liu, Y., Zhou, C.: Optimal destabilizations of K-unstable Fano varieties via stability thresholds. Geom. Topol. (to appear) arxiv:1907.05399 (2019)

  8. 8.

    Blum, H., Liu, Y., Xu, C,: Openness of K-semistability for Fano varieties. arXiv:1907.02408 (2019)

  9. 9.

    Blum, H., Xu, C.: Uniqueness of K-polystable degenerations of Fano varieties. Ann. Math. (2) 190(2), 609–656 (2019)

    MathSciNet  Article  Google Scholar 

  10. 10.

    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)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Chen, W.: Boundedness of varieties of Fano type with alpha-invariants and volumes bounded below. Publ. Res. Inst. Math. Sci. 56(3), 539–559 (2020)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Codogni, G., Patakfalvi, Z.: Positivity of the CM line bundle for families of K-stable klt Fano varieties. Invent. Math. 223(3), 811–894 (2021)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Datar, V., Székelyhidi, G.: Kähler–Einstein metrics along the smooth continuity method. Geom. Funct. Anal. 26(4), 975–1010 (2016)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Dervan, R.: Uniform stability of twisted constant scalar curvature Kähler metrics. Int. Math. Res. Not. IMRN 15, 4728–4783 (2016)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Donaldson, S.K.: Scalar curvature and stability of toric varieties. J. Differ. Geom. 62(2), 289–349 (2002)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Donaldson, S.K.: Lower bounds on the Calabi functional. J. Differ. Geom. 70(3), 453–472 (2005)

    MathSciNet  Article  Google Scholar 

  17. 17.

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

    MathSciNet  Article  Google Scholar 

  18. 18.

    Fujita, K., Odaka, Y.: On the K-stability of Fano varieties and anticanonical divisors. Tohoku Math. J. 70(4), 511–521 (2018)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Fujita, K.: A valuative criterion for uniform K-stability of \(\mathbb{Q}\)-Fano varieties. J. Reine Angew. Math. 751, 309–338 (2019)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Futaki, A., Mabuchi, T.: Bilinear forms and extremal Kähler vector fields associated with Kähler classes. Math. Ann. 301(2), 199–210, 0025-5831 (1995)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Heinloth, J.: Semistable reduction for \(G\)-bundles on curves. J. Algebraic Geom. 17(1), 167–183 (2008)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Halpern-Leistner, D.: On the structure of instability in moduli theory. arXiv:1411.0627v4 (2014)

  23. 23.

    Hisamoto, T.: On the limit of spectral measures associated to a test configuration of a polarized Kähler manifold. J. Reine Angew. Math. 713, 129–148 (2016)

    MathSciNet  MATH  Google Scholar 

  24. 24.

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

    MathSciNet  Article  Google Scholar 

  25. 25.

    Hall, J., Rydh, D.: Coherent Tannaka duality and algebraicity of Hom-stacks. Algebra Number Theory 13(7), 1633–1675, 1937-0652 (2019)

  26. 26.

    Halpern-Leistner, D., Preygel, A.: Mapping stacks and categorical notions of properness, Compositio Mathematica (to appear). arXiv:1402.3204 (2020)

  27. 27.

    Harder, G., Narasimhan, M.S.: On the cohomology groups of moduli spaces of vector bundles on curves. Math. Ann., 212, 215–248 (1974/75)

  28. 28.

    Jiang, C.: Boundedness of \(\mathbb{Q}\)-Fano varieties with degrees and alpha-invariants bounded from below, Ann. Sci. Éc. Norm. Supér. (4), Annales Scientifiques de l’École Normale Supérieure. Quatrième Série, 53, 2020, 5, 1235–1248

  29. 29.

    Kempf, George R.: Instability in invariant theory. Ann. Math. (2) 108(2), 299–316 (1978)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Kollár, J.: Singularities of the minimal model program, Cambridge Tracts in Mathematics, 200. Cambridge University Press, Cambridge, With a collaboration of Sándor Kovács (2013)

  31. 31.

    Kollár, J.: Families of varieties of general type. Book in preparation. https://web.math.princeton.edu/~kollar/book/modbook20170720.pdf (2017)

  32. 32.

    Kollár, J.: Families of divisors. arXiv:1910.00937 (2019)

  33. 33.

    Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties. In: Cambridge Tracts in Mathematics, Vol. 134. Cambridge University Press, Cambridge, With the collaboration of C. H. Clemens and A. Corti (1998)

  34. 34.

    Langton, S.G.: Valuative criteria for families of vector bundles on algebraic varieties. Ann. Math. (2) 101, 88–110 (1975)

    MathSciNet  Article  Google Scholar 

  35. 35.

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

    MathSciNet  Article  Google Scholar 

  36. 36.

    Li, C., Liu, Y., Xu, C.: A guided tour to normalized volume, Geometric analysis. In: Honor of Gang Tian’s 60th birthday, Progress in Mathematics, Vol. 333, pp. 167–219. Birkhäuser/Springer, Cham (2020)

  37. 37.

    Li, C.W., Wang, X., Xu, C.: On the proper moduli spaces of smoothable Kähler–Einstein Fano varieties. Duke Math. J. 168(8), 1387–1459 (2019)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Li, C., Wang, X., Xu, C.: Algebraicity of metric tangent cones and equivariant K-stability. J. Am. Math. Soc. 34(4), 1175–1214 (2021)

    Article  Google Scholar 

  39. 39.

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

    MathSciNet  Article  Google Scholar 

  40. 40.

    Liu, Y., Xu, C., Zhuang, Z.: Finite generation for valuations computing stability thresholds and applications to K-stability. arXiv:2102.09405 (2021)

  41. 41.

    Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, Results in Mathematics and Related Areas (2), Vol. 34. Springer, Berlin, (1994)

  42. 42.

    Odaka, Y.: On the moduli of Kähler–Einstein Fano manifolds, Proc. Kinosaki symposium (2013)

  43. 43.

    Odaka, Y.: A generalization of the Ross–Thomas slope theory. Osaka. J. Math. 50(1), 171–185, 0030–6126 (2013)

  44. 44.

    Ross, J., Szekelyhidi, G.: Twisted Kähler–Einstein metrics. Pure Appl. Math. Q. 17(3), 1025–1044 (2021)

    Article  Google Scholar 

  45. 45.

    Shatz, S.S.: The decomposition and specialization of algebraic families of vector bundles. Compos. Math. 35(2), 163–187 (1977)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Székelyhidi, G.: Optimal test-configurations for toric varieties. J. Differ. Geom. 80(3), 501–523 (2008)

    MathSciNet  Article  Google Scholar 

  47. 47.

    Tian, G.: Kähler–Einstein metrics with positive scalar curvature. Invent. Math. 130(1), 1–37 (1997)

    MathSciNet  Article  Google Scholar 

  48. 48.

    Tian, G., Wang, F.: On the existence of conic Kahler–Einstein metrics. arXiv:1903.12547 (2019)

  49. 49.

    Wang, X.: Height and GIT weight. Math. Res. Lett. 19(4), 909–926 (2012)

    MathSciNet  Article  Google Scholar 

  50. 50.

    Wang, X.-J., Zhu, X.: Kähler–Ricci solitons on toric manifolds with positive first Chern class. Adv. Math. 188(1), 87–103, 0001-8708 (2004)

  51. 51.

    Xia, M.: On sharp lower bounds for Calabi type functionals and destabilizing properties of gradient flows. arXiv:1901.07889 (2019)

  52. 52.

    Xu, C.: A minimizing valuation is quasi-monomial. Ann. Math. (2) 191(3), 1003–1030 (2020)

    MathSciNet  Article  Google Scholar 

  53. 53.

    Xu, C.: Toward finite generation of higher rational rank valuations. Mat. Sb., Matematicheskiĭ Sbornik, 212(3), 157–174 (2021)

  54. 54.

    Xu, C., Zhuang, Z.: On positivity of the CM line bundle on K-moduli spaces. Ann. of Math. (2) Second Ser. 192(3), 1005–1068 (2020)

    MathSciNet  Article  Google Scholar 

  55. 55.

    Xu, C., Ziquan, Z.: Uniqueness of the minimizer of the normalized volume function. Cam. J. Math. 9(1), 149–176 (2021)

    Article  Google Scholar 

  56. 56.

    Zhuang, Z.: Optimal destabilizing centers and equivariant K-stability. Invent. Math. (to appear). arXiv:2004.09413 (2020)

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Acknowledgements

The authors thank Jarod Alper and Jochen Heinloth for helpful conversations. HB was partially supported by NSF Grant DMS-1803102. DHL was partially supported by a Simons Foundation Collaboration grant and NSF CAREER Grant DMS-1945478. YL was partially supported by NSF Grant DMS-2001317. CX was partially supported by NSF Grant DMS-1901849.

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Correspondence to Harold Blum.

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Blum, H., Halpern-Leistner, D., Liu, Y. et al. On properness of K-moduli spaces and optimal degenerations of Fano varieties. Sel. Math. New Ser. 27, 73 (2021). https://doi.org/10.1007/s00029-021-00694-7

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Keywords

  • Fano varieties
  • K-stability
  • Moduli

Mathematics Subject Classification

  • 14D20
  • 14J45