A Note on Kähler–Ricci Flow on Fano Threefolds

Abstract

In this note, we show that the solution of Kähler–Ricci flow on every Fano threefold from Family No. 2.23 in the Mori–Mukai’s list develops type II singularity. In fact, we show that no Fano threefold from Family No. 2.23 admits Kähler–Ricci soliton and the Gromov–Hausdorff limit of the Kähler–Ricci flow must be a singular $\mathbb {Q}$-Fano variety. This gives new examples of Fano manifolds of the lowest dimension on which Kähler–Ricci flow develops type II singularity.

Keywords

Kähler–Ricci soliton / K-stability / Fano threefold / Kähler–Ricci flow / Primary 53E30 / Secondary 32Q26

References

[1] Araujo, C., Castravet, A.-M., Cheltsov, I., Fujita, K., Kaloghiros, A.-S., Martinez-Garcia, J., Shramov, C., Süß, H., Viswanathan, N.: The Calabi Problem for Fano Threefolds. Lecture Notes in Mathematics, vol. 485. Cambridge University Press, Cambridge (2023)

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

[3] Blum, H., Liu, Y.C., Xu, C.Y., Zhuang, Z.Q.: The existence of the Kähler–Ricci soliton degeneration. Forum Math. Pi 11, e9 (2023)

[4] Cao HD. Deformation of Kähler metrics to Kähler–Einstein metrics on compact Kähler manifolds. Invent. Math.. 1985, 81(2): 359-372

[5] Chen XX, Sun S, Wang B. Kähler–Ricci flow, Kähler–Einstein metric, and K-stability. Geom. Topol.. 2018, 22(6): 3145-3173

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

[7] Delcroix T. Examples of K-unstable Fano manifolds. Ann. Inst. Fourier (Grenoble). 2022, 72(5): 2079-2108

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

[9] Fujita K. On $K$-stability and the volume functions of $\mathbb{Q} $-Fano varieties. Proc. Lond. Math. Soc.. 2016, 113(5): 541-582

[10] Griffiths, P., Harris, J.: Principles of Algebraic Geometry. John Wiley & Sons, New York (1994)

[11] Hamilton RS. Three-manifolds with positive Ricci curvature. J. Differ. Geom.. 1982, 17(2): 255-306

[12] Han, J.Y., Li, C.: Algebraic uniqueness of Kähler–Ricci flow limits and optimal degenerations of Fano varieties. arXiv:2009.01010 (To appear in Geometry and Topology)

[13] 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)

[14] Iskovskikh, V.A., Prokhorov, Y.G.: Albegraic Geometry. V. Fano Varieties. Encyclopaedia Math. Sci., vol. 47. Springer-Verlag, Berlin (1999)

[15] Lee, J.-H., Park, K.-D., Yoo, S.: K-stability of Gorenstein Fano group compactifications with rank two. Internat. J. Math. 33(13), 2250083 (2022)

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

[17] Li, Y., Tian, G., Zhu, X.H.: Singular limits of Kähler–Ricci flow on Fano $G$-manifolds. arXiv:1807.09167 (To appear in Amer. J. Math.)

[18] Mori, S., Mukai, S.: Classification of Fano $3$-folds with $B_{2}\ge 2$. Manuscripta Math. 36(2), 147–162 (1981)

[19] Przyjalkowski, V.V., Cheltsov, I.A., Shramov, C.A.: Fano threefolds with infinite automorphism groups. Izv. Ross. Akad. Nauk Ser. Mat. 83(4), 226–280 (2019)

[20] Reid, M.: The complete intersection of two or more quadrics. Ph.D. Thesis, Trinity College, Cambridge (1972)

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

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

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

[24] Tian G, Zhu XH. Convergence of Kähler–Ricci flow. J. Am. Math. Soc.. 2007, 20(3): 675-699

[25] Tian, G., Zhu, X.H.: Convergence of the Kähler–Ricci flow on Fano manifolds. J. Reine Angew. Math. 678, 223–245 (2013)

[26] Wang F, Zhu XH. Tian’s partial ${C}^0$-estimate implies Hamilton–Tian’s conjecture. Adv. Math.. 2021, 381 107619

[27] Xu CY. K-stability of Fano varieties: an algebro-geometric approach. EMS Surv. Math. Sci.. 2021, 8(1–2): 265-354

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Peking University