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.
Similar content being viewed by others
Notes
References
Altmann, K.: The dualizing sheaf on first-order deformations of toric surface singularities. J. Reine Angew. Math. 753, 137–158 (2019)
Ahmadinezhad, H., Ziquan, Z.: K-stability of Fano varieties via admissible flags. arXiv:2003.13788 (2020)
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)
Alper, J., Halpern-Leistner, D., Heinloth, J.: Existence of moduli spaces for algebraic stacks. arXiv:1812.01128 (2018)
Blum, H., Jonsson, M.: Thresholds, valuations, and K-stability. Adv. Math. 365, 107062 (2020)
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)
Blum, H., Liu, Y., Zhou, C.: Optimal destabilizations of K-unstable Fano varieties via stability thresholds. Geom. Topol. (to appear) arxiv:1907.05399 (2019)
Blum, H., Liu, Y., Xu, C,: Openness of K-semistability for Fano varieties. arXiv:1907.02408 (2019)
Blum, H., Xu, C.: Uniqueness of K-polystable degenerations of Fano varieties. Ann. Math. (2) 190(2), 609–656 (2019)
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)
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)
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)
Datar, V., Székelyhidi, G.: Kähler–Einstein metrics along the smooth continuity method. Geom. Funct. Anal. 26(4), 975–1010 (2016)
Dervan, R.: Uniform stability of twisted constant scalar curvature Kähler metrics. Int. Math. Res. Not. IMRN 15, 4728–4783 (2016)
Donaldson, S.K.: Scalar curvature and stability of toric varieties. J. Differ. Geom. 62(2), 289–349 (2002)
Donaldson, S.K.: Lower bounds on the Calabi functional. J. Differ. Geom. 70(3), 453–472 (2005)
Donaldson, S., Sun, S.: Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry. Acta Math. 213(1), 63–106 (2014)
Fujita, K., Odaka, Y.: On the K-stability of Fano varieties and anticanonical divisors. Tohoku Math. J. 70(4), 511–521 (2018)
Fujita, K.: A valuative criterion for uniform K-stability of \(\mathbb{Q}\)-Fano varieties. J. Reine Angew. Math. 751, 309–338 (2019)
Futaki, A., Mabuchi, T.: Bilinear forms and extremal Kähler vector fields associated with Kähler classes. Math. Ann. 301(2), 199–210, 0025-5831 (1995)
Heinloth, J.: Semistable reduction for \(G\)-bundles on curves. J. Algebraic Geom. 17(1), 167–183 (2008)
Halpern-Leistner, D.: On the structure of instability in moduli theory. arXiv:1411.0627v4 (2014)
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)
Hacon, C.D., McKernan, J., Xu, C.: ACC for log canonical thresholds. Ann. Math. (2) 180(2), 523–571 (2014)
Hall, J., Rydh, D.: Coherent Tannaka duality and algebraicity of Hom-stacks. Algebra Number Theory 13(7), 1633–1675, 1937-0652 (2019)
Halpern-Leistner, D., Preygel, A.: Mapping stacks and categorical notions of properness, Compositio Mathematica (to appear). arXiv:1402.3204 (2020)
Harder, G., Narasimhan, M.S.: On the cohomology groups of moduli spaces of vector bundles on curves. Math. Ann., 212, 215–248 (1974/75)
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
Kempf, George R.: Instability in invariant theory. Ann. Math. (2) 108(2), 299–316 (1978)
Kollár, J.: Singularities of the minimal model program, Cambridge Tracts in Mathematics, 200. Cambridge University Press, Cambridge, With a collaboration of Sándor Kovács (2013)
Kollár, J.: Families of varieties of general type. Book in preparation. https://web.math.princeton.edu/~kollar/book/modbook20170720.pdf (2017)
Kollár, J.: Families of divisors. arXiv:1910.00937 (2019)
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)
Langton, S.G.: Valuative criteria for families of vector bundles on algebraic varieties. Ann. Math. (2) 101, 88–110 (1975)
Li, C.: K-semistability is equivariant volume minimization. Duke Math. J. 166(16), 3147–3218 (2017)
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)
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)
Li, C., Wang, X., Xu, C.: Algebraicity of metric tangent cones and equivariant K-stability. J. Am. Math. Soc. 34(4), 1175–1214 (2021)
Li, C., Xu, C.: Special test configuration and K-stability of Fano varieties. Ann. Math. (2) 180(1), 197–232 (2014)
Liu, Y., Xu, C., Zhuang, Z.: Finite generation for valuations computing stability thresholds and applications to K-stability. arXiv:2102.09405 (2021)
Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, Results in Mathematics and Related Areas (2), Vol. 34. Springer, Berlin, (1994)
Odaka, Y.: On the moduli of Kähler–Einstein Fano manifolds, Proc. Kinosaki symposium (2013)
Odaka, Y.: A generalization of the Ross–Thomas slope theory. Osaka. J. Math. 50(1), 171–185, 0030–6126 (2013)
Ross, J., Szekelyhidi, G.: Twisted Kähler–Einstein metrics. Pure Appl. Math. Q. 17(3), 1025–1044 (2021)
Shatz, S.S.: The decomposition and specialization of algebraic families of vector bundles. Compos. Math. 35(2), 163–187 (1977)
Székelyhidi, G.: Optimal test-configurations for toric varieties. J. Differ. Geom. 80(3), 501–523 (2008)
Tian, G.: Kähler–Einstein metrics with positive scalar curvature. Invent. Math. 130(1), 1–37 (1997)
Tian, G., Wang, F.: On the existence of conic Kahler–Einstein metrics. arXiv:1903.12547 (2019)
Wang, X.: Height and GIT weight. Math. Res. Lett. 19(4), 909–926 (2012)
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)
Xia, M.: On sharp lower bounds for Calabi type functionals and destabilizing properties of gradient flows. arXiv:1901.07889 (2019)
Xu, C.: A minimizing valuation is quasi-monomial. Ann. Math. (2) 191(3), 1003–1030 (2020)
Xu, C.: Toward finite generation of higher rational rank valuations. Mat. Sb., Matematicheskiĭ Sbornik, 212(3), 157–174 (2021)
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)
Xu, C., Ziquan, Z.: Uniqueness of the minimizer of the normalized volume function. Cam. J. Math. 9(1), 149–176 (2021)
Zhuang, Z.: Optimal destabilizing centers and equivariant K-stability. Invent. Math. (to appear). arXiv:2004.09413 (2020)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Accepted:
Published:
DOI: https://doi.org/10.1007/s00029-021-00694-7