Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-25T07:10:33.842Z Has data issue: false hasContentIssue false
  • English
  • Français

The Chern–Ricci flow on complex surfaces

Part of: Global differential geometry Differential operators in several variables

Published online by Cambridge University Press:  02 December 2013

Valentino Tosatti  and
Ben Weinkove
Valentino Tosatti
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA email [email protected]
Ben Weinkove
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Chern–Ricci flow is an evolution equation of Hermitian metrics by their Chern–Ricci form, first introduced by Gill. Building on our previous work, we investigate this flow on complex surfaces. We establish new estimates in the case of finite time non-collapsing, analogous to some known results for the Kähler–Ricci flow. This provides evidence that the Chern–Ricci flow carries out blow-downs of exceptional curves on non-minimal surfaces. We also describe explicit solutions to the Chern–Ricci flow for various non-Kähler surfaces. On Hopf surfaces and Inoue surfaces these solutions, appropriately normalized, collapse to a circle in the sense of Gromov–Hausdorff. For non-Kähler properly elliptic surfaces, our explicit solutions collapse to a Riemann surface. Finally, we define a Mabuchi energy functional for complex surfaces with vanishing first Bott–Chern class and show that it decreases along the Chern–Ricci flow.

Keywords

Chern–Ricci flowcompact complex surfaceHermitian metric

MSC classification

Secondary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 53C55: Hermitian and Kählerian manifolds 32W20: Complex Monge-Ampère operators
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 12 , December 2013 , pp. 2101 - 2138
Copyright
© The Author(s) 2013 

References

Barth, W. P., Hulek, K., Peters, C. A. M. and Van de Ven, A., Compact complex surfaces (Springer, Berlin, 2004).Google Scholar
Belgun, F. A., On the metric structure of non-Kähler complex surfaces, Math. Ann. 317 (2000), 1–40.Google Scholar
Błocki, Z., On the uniform estimate in the Calabi–Yau theorem, II, Sci. China Math. 54 (2011), 1375–1377.CrossRefGoogle Scholar
Bogomolov, F. A., Surfaces of class ${\mathrm{V II} }_{0} $ and affine geometry, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 710–761.Google Scholar
Borzellino, J. E., Riemannian geometry of orbifolds, PhD thesis, UCLA (1992).Google Scholar
Brieskorn, E. and Van de Ven, A, Some complex structures on products of homotopy spheres, Topology 7 (1968), 389–393.CrossRefGoogle Scholar
Brînzănescu, V., Néron–Severi group for nonalgebraic elliptic surfaces. II. Non-Kählerian case, Manuscripta Math. 84 (1994), 415–420.CrossRefGoogle Scholar
Brunella, M., A characterization of Inoue surfaces, Comment. Math. Helv., to appear, arXiv:1011.2035.Google Scholar
Brunella, M., Locally conformally Kähler metrics on Kato surfaces, Nagoya Math. J. 202 (2011), 77–81.Google Scholar
Buchdahl, N., On compact Kähler surfaces, Ann. Inst. Fourier 49 (1999), 287–302.Google Scholar
Buchdahl, N., A Nakai–Moishezon criterion for non-Kähler surfaces, Ann. Inst. Fourier (Grenoble) 50 (2000), 1533–1538.Google Scholar
Calabi, E. and Eckmann, B., A class of compact, complex manifolds which are not algebraic, Ann. of Math. (2) 58 (1953), 494–500.CrossRefGoogle Scholar
Cherrier, P., Équations de Monge–Ampère sur les variétés Hermitiennes compactes, Bull. Sci. Math (2) 111 (1987), 343–385.Google Scholar
Dinew, S. and Kołodziej, S., Pluripotential estimates on compact Hermitian manifolds, in Advances in geometric analysis, Advanced Lectures in Mathematics, vol. 21 (International Press, Somerville, MA, 2012), 69–86.Google Scholar
Federer, H., Geometric measure theory (Springer, New York, 1969).Google Scholar
Fong, F. T.-H. and Zhang, Z., The collapsing rate of the Kähler–Ricci flow with regular infinite time singularity, J. Reine Angew. Math., arXiv:1202.3199.Google Scholar
Fujiki, A., A theorem on bimeromorphic maps of Kähler manifolds and its applications, Publ. Res. Inst. Math. Sci. Kyoto Univ. 17 (1981), 735–754.CrossRefGoogle Scholar
Fujiki, A. and Pontecorvo, M., Anti-self-dual biHermitian structures on Inoue surfaces, J. Differential Geom. 85 (2010), 15–71.Google Scholar
Fukaya, K., Theory of convergence for Riemannian orbifolds, Japan. J. Math. (N.S.) 12 (1986), 121–160.CrossRefGoogle Scholar
Gauduchon, P. and Ornea, L., Locally conformally Kähler metrics on Hopf surfaces, Ann. Inst. Fourier (Grenoble) 48 (1998), 1107–1127.CrossRefGoogle Scholar
Gill, M., Convergence of the parabolic complex Monge–Ampère equation on compact Hermitian manifolds, Comm. Anal. Geom. 19 (2011), 277–303.Google Scholar
Godbillon, C., Feuilletages. Études géométriques (Birkhäuser, Basel, 1991).Google Scholar
Griffiths, P. and Harris, J., Principles of algebraic geometry, Pure and Applied Mathematics (Wiley-Interscience, New York, 1978).Google Scholar
Gross, M., Tosatti, V. and Zhang, Y., Collapsing of abelian fibred Calabi–Yau manifolds, Duke Math. J. 162 (2013), 517–551.CrossRefGoogle Scholar
Guan, B. and Li, Q., Complex Monge–Ampère equations and totally real submanifolds, Adv. Math. 225 (2010), 1185–1223.CrossRefGoogle Scholar
Harvey, R. and Lawson, H. B., An intrinsic characterization of Kähler manifolds, Invent. Math. 74 (1983), 169–198.CrossRefGoogle Scholar
Kodaira, K., On the structure of compact complex analytic surfaces, II, Amer. J. Math. 88 (1966), 682–721.CrossRefGoogle Scholar
Kołodziej, S., The Monge–Ampère equation on compact Kähler manifolds, Indiana Univ. Math. J. 52 (2003), 667–686.Google Scholar
Inoue, M., On surfaces of Class $VI{I}_{0} $, Invent. Math. 24 (1974), 269–310.Google Scholar
Inoue, M., Kobayashi, S. and Ochiai, T., Holomorphic affine connections on compact complex surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 247–264.Google Scholar
Lamari, A., Le cône kählérien d’une surface, J. Math. Pures Appl. (9) 78 (1999), 249–263.CrossRefGoogle Scholar
LeBrun, C., Anti-self-dual Hermitian metrics on blown-up Hopf surfaces, Math. Ann. 289 (1991), 383–392.Google Scholar
Li, J., Yau, S.-T. and Zheng, F., On projectively flat Hermitian manifolds, Comm. Anal. Geom. 2 (1994), 103–109.CrossRefGoogle Scholar
Liu, K. and Yang, X., Geometry of Hermitian manifolds, Internat. J. Math. 23 (2012), 1250055.Google Scholar
Lott, J., On the long-time behavior of type-III Ricci flow solutions, Math. Ann. 339 (2007), 627–666.Google Scholar
Lott, J., Dimensional reduction and the long-time behavior of Ricci flow, Comment. Math. Helv. 85 (2010), 485–534.Google Scholar
Lott, J. and Sesum, N., Ricci flow on three-dimensional manifolds with symmetry, Comment. Math. Helv., to appear, arXiv:1102.4384.Google Scholar
Maehara, K., On elliptic surfaces whose first Betti numbers are odd, in Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) (Kinokuniya Book Store, Tokyo, 1978), 565–574.Google Scholar
Montgomery, R., A tour of subRiemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91 (American Mathematical Society, Providence, RI, 2002).Google Scholar
Phong, D. H. and Sturm, J., On stability and the convergence of the Kähler–Ricci flow, J. Differential Geom. 72 (2006), 149–168.Google Scholar
Phong, D. H. and Sturm, J., The Dirichlet problem for degenerate complex Monge-Ampere equations, Comm. Anal. Geom. 18 (2010), 145–170.Google Scholar
Rong, X., Convergence and collapsing theorems in Riemannian geometry, in Handbook of geometric analysis, No. 2, Advanced Lectures in Mathematics (ALM), vol. 13 (Int. Press, Somerville, MA, 2010), 193–299.Google Scholar
Schweitzer, M., Autour de la cohomologie de Bott–Chern, Preprint (2007), arXiv:0709.3528.Google Scholar
Song, J. and Tian, G., The Kähler–Ricci flow on surfaces of positive Kodaira dimension, Invent. Math. 170 (2007), 609–653.CrossRefGoogle Scholar
Song, J. and Tian, G., The Kähler–Ricci flow through singularities, Preprint (2009), arXiv:0909.4898.Google Scholar
Song, J. and Weinkove, B., Contracting exceptional divisors by the Kähler–Ricci flow, II, Proc. Lond. Math. Soc. (3), to appear, arXiv:1102.1759.Google Scholar
Song, J. and Weinkove, B., Contracting exceptional divisors by the Kähler–Ricci flow, Duke Math. J. 162 (2013), 367–415.Google Scholar
Song, J. and Weinkove, B., An introduction to the Kähler–Ricci flow, in An Introduction to the Kähler–Ricci flow, Lecture Notes in Mathematics, vol. 2086 (Springer, 2013), 89–188.CrossRefGoogle Scholar
Streets, J. and Tian, G., A parabolic flow of pluriclosed metrics, Int. Math. Res. Not. IMRN 2010 (2010), 3101–3133.Google Scholar
Streets, J. and Tian, G., Hermitian curvature flow, J. Eur. Math. Soc. (JEMS) 13 (2011), 601–634.Google Scholar
Streets, J. and Tian, G., Regularity results for pluriclosed flow, Geom. Topol. 17 (2013), 2389–2429.Google Scholar
Teleman, A., Projectively flat surfaces and Bogomolov’s theorem on class $VI{I}_{0} $-surfaces, Int. J. Math. 5 (1994), 253–264.Google Scholar
Teleman, A., Donaldson theory on non-Kählerian surfaces and class VII surfaces with ${b}_{2} = 1$, Invent. Math. 162 (2005), 493–521.CrossRefGoogle Scholar
Teleman, A., The pseudo-effective cone of a non-Kählerian surface and applications, Math. Ann. 335 (2006), 965–989.Google Scholar
Tian, G., Canonical metrics in Kähler geometry, in Notes taken by Meike Akveld, Lectures in Mathematics ETH Zürich (Birkhäuser, Basel, 2000).Google Scholar
Tian, G. and Zhang, Z., On the Kähler–Ricci flow on projective manifolds of general type, Chinese Ann. Math. Ser. B 27 (2006), 179–192.Google Scholar
Tits, J., Espaces homogènes complexes compacts, Comment. Math. Helv. 37 (1962/1963), 111–120.CrossRefGoogle Scholar
Tosatti, V. and Weinkove, B., The complex Monge–Ampère equation on compact Hermitian manifolds, J. Amer. Math. Soc. 23 (2010), 1187–1195.Google Scholar
Tosatti, V. and Weinkove, B., On the evolution of a Hermitian metric by its Chern–Ricci form, Preprint (2012), arXiv:1201.0312.Google Scholar
Tricerri, F., Some examples of locally conformal Kähler manifolds, Rend. Semin. Mat. Univ. Politec. Torino 40 (1982), 81–92.Google Scholar
Tsuji, H., Existence and degeneration of Kähler–Einstein metrics on minimal algebraic varieties of general type, Math. Ann. 281 (1988), 123–133.Google Scholar
Vaisman, I., Non-Kähler metrics on geometric complex surfaces, Rend. Semin. Mat. Univ. Politec. Torino 45 (1987), 117–123.Google Scholar
Wall, C. T. C., Geometric structures on compact complex analytic surfaces, Topology 25 (1986), 119–153.CrossRefGoogle Scholar