Generalized Optimal Degenerations of Fano Varieties

Abstract

We prove a generalization of the algebraic version of Tian conjecture. Precisely, for any smooth strictly increasing function

g:\mathbb{R}\to {\mathbb{R}}_{>0}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$g:\mathbb {R}\rightarrow \mathbb {R}_{>0}$$\end{document}

with

\text{log}\circ g

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{log}\circ g$$\end{document}

convex, we define the

{\mathbf{H}}^g

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textbf{H}^g$$\end{document}

-invariant on a Fano variety X generalizing the

\mathbf{H}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textbf{H}$$\end{document}

-invariant introduced by Tian–Zhang–Zhang–Zhu and show that

{\mathbf{H}}^g

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textbf{H}^g$$\end{document}

admits a unique minimizer. Such a minimizer will induce the g-optimal degeneration of the Fano variety X, whose limit space admits a

g^{\prime }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$g'$$\end{document}

-soliton. We present an example of Fano threefold which has the same g-optimal degenerations for any g.

Keywords

Fano varieties / Generalized optimal degenerations / Generalized H-invariants /

g^{\prime }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$g'$$\end{document}

-weighted K-stability')">

g^{\prime }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$g'$$\end{document}

-weighted K-stability /

g^{\prime }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$g'$$\end{document}

-soliton metrics')">

g^{\prime }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$g'$$\end{document}

-soliton metrics / 14J45 / 32Q20 / 14D20

References

[1] Abban, H., Zhuang, Z.Q.: K-stability of Fano varieties via admissible flags. Forum Math. Pi 10(43), Paper No. e15, 43 pp. (2022)

[2] Bamler, R.: Convergence of Ricci flows with bounded scalar curvature. Ann. Math. 188(3), 753–831 (2018)

[3] Birkar, C.: Anti-pluricanonical systems on Fano varieties. Ann. Math. 190(2), 345–463 (2019)

[4] Birkar C, Cascini P, Hacon CD, McKernan J. Existence of minimal models for varieties of log general type. J. Amer. Math. Soc., 2010, 23(2): 405-468

[5] Blum, H., Jonsson, M.: Thresholds, valuations, and K-stability. Adv. Math. 365, Paper No. 107062, 57 pp. (2020)

[6] Blum H, Liu YC, Xu CY. Openness of K-semistability for Fano varieties. Duke Math. J., 2022, 171(13): 2753-2797

[7] Blum, H., Liu, Y.C., Xu, C.Y., Zhuang, Z.Q.: The existence of the Kähler–Ricci soliton degeneration. Forum Math. Pi 11, Paper No. e9, 28 pp. (2023)

[8] Boucksom, S., Chen, H.Y.: Okounkov bodies of filtered linear series. Compos. Math. 147(4), 1205–1229 (2011)

[9] 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)

[10] Chen XX, Wang B. Space of Ricci flows (II)—Part B: Weak compactness of the flows. J. Different. Geom., 2020, 116(1): 1-123

[11] Chen, Z.Q.: Stable degeneration of families of klt singularities with constant local volume. Forum Math. Sigma 13, Paper No. e158, 43 pp. (2025)

[12] Delcroix, T.: Weight sensitivity in K-stability of Fano varieties. arXiv:2411.07864 (2024)

[13] Dervan, R., Székelyhidi, G.: The Kähler–Ricci flow and optimal degenerations. J. Different. Geom. 116(1), 187–203 (2020)

[14] Hacon, C.D., McKernan, J., Xu, C.Y.: On the birational automorphisms of varieties of general type. Ann. Math. 177(3), 1077–1111 (2013)

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

[16] Han, J.Y., Li, C.: On the Yau–Tian–Donaldson conjecture for generalized Kähler–Ricci soliton equations. Comm. Pure Appl. Math. 76(9), 1793–1867 (2023)

[17] Han, J.Y., Li, C.: Algebraic uniqueness of Kähler–Ricci flow limits and optimal degenerations of Fano varieties. Geom. Topol. 28(2), 539–592 (2024)

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

[19] Lazarsfeld, R., Mustaţă, M.: Convex bodies associated to linear series. Ann. Sci. Éc. Norm. Supér. (4) 42(5), 783–835 (2009)

[20] Li C. G\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$G$$\end{document}-uniform stability and Kähler-Einstein metrics on Fano varieties. Invent. Math., 2022, 227(2): 661-744

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

[22] Li C, Xu CY. Stability of valuations: Higher rational rank. Peking Math. J., 2018, 1(1): 1-79

[23] Liu YC, Xu CY, Zhuang ZQ. Finite generation for valuations computing stability thresholds and applications to K-stability. Ann. Math., 2022, 196(2): 507-566

[24] Miao MH, Tian G. A note on Kähler–Ricci flow on Fano threefolds. Peking Math. J., 2025, 8(1): 191-199

[25] Miao, M.H., Wang, L.S.: Optimal degenerations of K-unstable Fano threefolds. arXiv:2401.13999 (2024)

[26] Miao, M.H., Wang, L.S.: Kähler–Ricci solitons on Fano threefolds with non-trivial moduli. Math. Ann., 392(3), 4483–4523 (2025)

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

[28] Tian, G., Zhang, S.J., Zhang, Z.L., Zhu, X.H.: Perelman’s entropy and Kähler–Ricci flow on a Fano manifold. Trans. Amer. Math. Soc. 365(12), 6669–6695 (2013)

[29] Tian G, Zhang ZL. Regularity of Kähler–Ricci flows on Fano manifolds. Acta Math., 2016, 216(1): 127-176

[30] Wang, F., Zhu, X.H.: Tian’s partial C0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^0$$\end{document}-estimate implies Hamilton–Tian’s conjecture. Adv. Math. 381, Paper No. 107619, 29 pp. (2021)

[31] Wang, L.S.: A valuative criterion of K-polystability. arXiv:2406.06176 (2024) (To appear in Trans. Amer. Math. Soc.)

[32] Wang, L.S.: K-polystability and reduced uniform K-stability of log Fano cone singularities. arXiv:2505.10828 (2025)

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

[34] Xu, C.Y.: K-stability of Fano Varieties. New Math. Monogr., vol. 50. Cambridge University Press, Cambridge (2025)

[35] Xu, C.Y., Zhuang, Z.Q.: Stable degenerations of singularities. J. Amer. Math. Soc. 38(3), 585–626 (2025)

Funding

Key Technologies Research and Development Program(#2023YFA1010600)

RIGHTS & PERMISSIONS

Peking University

Just Accepted

This article has successfully passed peer review and final editorial review, and will soon enter typesetting, proofreading and other publishing processes. The currently displayed version is the accepted final manuscript. The officially published version will be updated with format, DOI and citation information upon launch. We recommend that you pay attention to subsequent journal notifications and preferentially cite the officially published version. Thank you for your support and cooperation.