Independence of singularity type for numerically effective Kähler–Ricci flows

Wondo, Hosea¹ ; Zhang, Zhou¹

¹ School of Mathematics and Statistics, The University of Sydney, Sydney, NSW, Australia

Geometry & topology, Tome 29 (2025) no. 1, pp. 479-494.

Résumé

We show that the singularity type of solutions to the Kähler–Ricci flow on a numerically effective manifold does not depend on the initial metric. More precisely, if there exists a type-III solution to the Kähler–Ricci flow, then any other solution starting from a different initial metric will also be type III. This generalizes a previous result by Yashan Zhang for the semiample case, and confirms a conjecture by Valentino Tosatti.

Détail

DOI : 10.2140/gt.2025.29.479

Keywords: Kähler–Ricci flow, Kähler geometry, minimal model program, curvature estimates

Bibliographie

[1] H D Cao, Deformation of Kähler metrics to Kähler–Einstein metrics on compact Kähler manifolds, Invent. Math. 81 (1985) 359 | DOI

[2] F T H Fong, On the collapsing rate of the Kähler–Ricci flow with finite-time singularity, J. Geom. Anal. 25 (2015) 1098 | DOI

[3] F T H Fong, Y Zhang, Local curvature estimates of long-time solutions to the Kähler–Ricci flow, Adv. Math. 375 (2020) 107416 | DOI

[4] B Guo, J Song, B Weinkove, Geometric convergence of the Kähler–Ricci flow on complex surfaces of general type, Int. Math. Res. Not. 2016 (2016) 5652 | DOI

[5] R S Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255

[6] W Jian, Convergence of scalar curvature of Kähler–Ricci flow on manifolds of positive Kodaira dimension, Adv. Math. 371 (2020) 107253 | DOI

[7] Y C Lu, On holomorphic mappings of complex manifolds, PhD thesis, University of California, Berkeley (1967)

[8] K Matsuki, Introduction to the Mori program, Springer (2002) | DOI

[9] O Munteanu, G Székelyhidi, On convergence of the Kähler–Ricci flow, Comm. Anal. Geom. 19 (2011) 887 | DOI

[10] D H Phong, J Sturm, On stability and the convergence of the Kähler–Ricci flow, J. Differential Geom. 72 (2006) 149

[11] D H Phong, N Sesum, J Sturm, Multiplier ideal sheaves and the Kähler–Ricci flow, Comm. Anal. Geom. 15 (2007) 613 | DOI

[12] J Song, G Tian, The Kähler–Ricci flow on surfaces of positive Kodaira dimension, Invent. Math. 170 (2007) 609 | DOI

[13] J Song, G Tian, Canonical measures and Kähler–Ricci flow, J. Amer. Math. Soc. 25 (2012) 303 | DOI

[14] J Song, G Tian, Bounding scalar curvature for global solutions of the Kähler–Ricci flow, Amer. J. Math. 138 (2016) 683 | DOI

[15] J Song, G Tian, The Kähler–Ricci flow through singularities, Invent. Math. 207 (2017) 519 | DOI

[16] J Song, B Weinkove, Contracting exceptional divisors by the Kähler–Ricci flow, Duke Math. J. 162 (2013) 367 | DOI

[17] J Song, B Weinkove, An introduction to the Kähler–Ricci flow, from: "An introduction to the Kähler–Ricci flow" (editors S Boucksom, P Eyssidieux, V Guedj), Lecture Notes in Math. 2086, Springer (2013) 89 | DOI

[18] J Song, B Weinkove, Contracting exceptional divisors by the Kähler–Ricci flow, II, Proc. Lond. Math. Soc. 108 (2014) 1529 | DOI

[19] J Song, Y Yuan, Metric flips with Calabi ansatz, Geom. Funct. Anal. 22 (2012) 240 | DOI

[20] G Tian, Existence of Einstein metrics on Fano manifolds, from: "Metric and differential geometry" (editors X Dai, X Rong), Progr. Math. 297, Birkhäuser (2012) 119 | DOI

[21] G Tian, Some progresses on Kähler–Ricci flow, Boll. Unione Mat. Ital. 12 (2019) 251 | DOI

[22] G Tian, Z Zhang, On the Kähler–Ricci flow on projective manifolds of general type, Chinese Ann. Math. Ser. B 27 (2006) 179 | DOI

[23] G Tian, Z Zhang, Convergence of Kähler–Ricci flow on lower-dimensional algebraic manifolds of general type, Int. Math. Res. Not. 2016 (2016) 6493 | DOI

[24] V Tosatti, KAWA lecture notes on the Kähler–Ricci flow, Ann. Fac. Sci. Toulouse Math. 27 (2018) 285 | DOI

[25] V Tosatti, Y Zhang, Infinite-time singularities of the Kähler–Ricci flow, Geom. Topol. 19 (2015) 2925 | DOI

[26] V Tosatti, B Weinkove, X Yang, The Kähler–Ricci flow, Ricci-flat metrics and collapsing limits, Amer. J. Math. 140 (2018) 653 | DOI

[27] H Wu, Z Zhang, The modified Kähler–Ricci flow, II, preprint (2022)

[28] S T Yau, A general Schwarz lemma for Kähler manifolds, Amer. J. Math. 100 (1978) 197 | DOI

[29] Y Yuan, On the convergence of a modified Kähler–Ricci flow, Math. Z. 268 (2011) 281 | DOI

[30] Z Zhang, A modified Kähler–Ricci flow, Math. Ann. 345 (2009) 559 | DOI

[31] Z Zhang, Scalar curvature bound for Kähler–Ricci flows over minimal manifolds of general type, Int. Math. Res. Not. 2009 (2009) 3901 | DOI

[32] Z Zhang, Scalar curvature behavior for finite-time singularity of Kähler–Ricci flow, Michigan Math. J. 59 (2010) 419 | DOI

[33] Y Zhang, Collapsing limits of the Kähler–Ricci flow and the continuity method, Math. Ann. 374 (2019) 331 | DOI

[34] Y Zhang, Infinite-time singularity type of the Kähler–Ricci flow, J. Geom. Anal. 30 (2020) 2092 | DOI