Pluripotential Kähler–Ricci flows

Guedj, Vincent¹ ; Lu, Chinh H² ; Zeriahi, Ahmed¹

¹ Institut de Mathématiques de Toulouse, Université de Toulouse, CNRS, Toulouse, France
² Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, Orsay, France

Geometry & topology, Tome 24 (2020) no. 3, pp. 1225-1296.

Résumé

We develop a parabolic pluripotential theory on compact Kähler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge–Ampère equations. We provide a parabolic analogue of the celebrated Bedford–Taylor theory and apply it to the study of the Kähler–Ricci flow on varieties with log terminal singularities.

Détail

DOI : 10.2140/gt.2020.24.1225

Classification : 53C44, 32W20, 58J35
Keywords: parabolic Monge–Ampère equation, pluripotential solution, Perron envelope, Kähler–Ricci flow

Bibliographie

[1] E Bedford, B A Taylor, The Dirichlet problem for a complex Monge–Ampère equation, Invent. Math. 37 (1976) 1 | DOI

[2] E Bedford, B A Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982) 1 | DOI

[3] R J Berman, S Boucksom, P Eyssidieux, V Guedj, A Zeriahi, Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties, J. Reine Angew. Math. 751 (2019) 27 | DOI

[4] R J Berman, S Boucksom, V Guedj, A Zeriahi, A variational approach to complex Monge–Ampère equations, Publ. Math. Inst. Hautes Études Sci. 117 (2013) 179 | DOI

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

[6] U Cegrell, S Kołodziej, A Zeriahi, Maximal subextensions of plurisubharmonic functions, Ann. Fac. Sci. Toulouse Math. 20 (2011) 101 | DOI

[7] T C Collins, G Székelyhidi, The twisted Kähler–Ricci flow, J. Reine Angew. Math. 716 (2016) 179 | DOI

[8] T C Collins, V Tosatti, Kähler currents and null loci, Invent. Math. 202 (2015) 1167 | DOI

[9] E Di Nezza, C H Lu, Uniqueness and short time regularity of the weak Kähler–Ricci flow, Adv. Math. 305 (2017) 953 | DOI

[10] S Dinew, An inequality for mixed Monge–Ampère measures, Math. Z. 262 (2009) 1 | DOI

[11] P Eyssidieux, V Guedj, A Zeriahi, A priori L∞–estimates for degenerate complex Monge–Ampère equations, Int. Math. Res. Not. 2008 (2008) | DOI

[12] P Eyssidieux, V Guedj, A Zeriahi, Singular Kähler–Einstein metrics, J. Amer. Math. Soc. 22 (2009) 607 | DOI

[13] P Eyssidieux, V Guedj, A Zeriahi, Viscosity solutions to degenerate complex Monge–Ampère equations, Comm. Pure Appl. Math. 64 (2011) 1059 | DOI

[14] P Eyssidieux, V Guedj, A Zeriahi, Weak solutions to degenerate complex Monge–Ampère flows, II, Adv. Math. 293 (2016) 37 | DOI

[15] P Eyssidieux, V Guedj, A Zeriahi, Convergence of weak Kähler–Ricci flows on minimal models of positive Kodaira dimension, Comm. Math. Phys. 357 (2018) 1179 | DOI

[16] V Guedj, C H Lu, A Zeriahi, Stability of solutions to complex Monge–Ampère flows, Ann. Inst. Fourier (Grenoble) 68 (2018) 2819 | DOI

[17] V Guedj, C H Lu, A Zeriahi, Pluripotential solutions versus viscosity solutions to complex Monge–Ampère flows, preprint (2019)

[18] V Guedj, C H Lu, A Zeriahi, Weak subsolutions to complex Monge–Ampère equations, J. Math. Soc. Japan 71 (2019) 727 | DOI

[19] V Guedj, H C Lu, A Zeriahi, The pluripotential Cauchy–Dirichlet problem for complex Monge–Ampère flows, preprint (2018)

[20] V Guedj, A Zeriahi, Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal. 15 (2005) 607 | DOI

[21] V Guedj, A Zeriahi, Stability of solutions to complex Monge–Ampère equations in big cohomology classes, Math. Res. Lett. 19 (2012) 1025 | DOI

[22] V Guedj, A Zeriahi, Degenerate complex Monge–Ampère equations, 26, Eur. Math. Soc. (2017) | DOI

[23] V Guedj, A Zeriahi, Regularizing properties of the twisted Kähler–Ricci flow, J. Reine Angew. Math. 729 (2017) 275 | DOI

[24] R S Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982) 255 | DOI

[25] J Kollár, S Mori, Birational geometry of algebraic varieties, 134, Cambridge Univ. Press (1998) | DOI

[26] S Kołodziej, Some sufficient conditions for solvability of the Dirichlet problem for the complex Monge–Ampère operator, Ann. Polon. Math. 65 (1996) 11 | DOI

[27] S Kołodziej, The complex Monge–Ampère equation, Acta Math. 180 (1998) 69 | DOI

[28] S Kołodziej, The Monge–Ampère equation on compact Kähler manifolds, Indiana Univ. Math. J. 52 (2003) 667 | DOI

[29] D H Phong, J Song, J Sturm, B Weinkove, The Kähler–Ricci flow with positive bisectional curvature, Invent. Math. 173 (2008) 651 | DOI

[30] D H Phong, J Song, J Sturm, B Weinkove, The Kähler–Ricci flow and the ∂ operator on vector fields, J. Differential Geom. 81 (2009) 631 | DOI

[31] J Song, G Székelyhidi, B Weinkove, The Kähler–Ricci flow on projective bundles, Int. Math. Res. Not. 2013 (2013) 243 | DOI

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

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

[34] 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

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

[36] T D Tô, Regularizing properties of complex Monge–Ampère flows, J. Funct. Anal. 272 (2017) 2058 | DOI

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

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

[39] H Tsuji, Existence and degeneration of Kähler–Einstein metrics on minimal algebraic varieties of general type, Math. Ann. 281 (1988) 123 | DOI

[40] S T Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I, Comm. Pure Appl. Math. 31 (1978) 339 | DOI