Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-24T06:38:13.007Z Has data issue: false hasContentIssue false
  • English
  • Français

The Calabi–Yau equation on the Kodaira–Thurston manifold

Part of: Global differential geometry Symplectic geometry, contact geometry

Published online by Cambridge University Press:  24 September 2010

Valentino Tosatti  and
Ben Weinkove
Valentino Tosatti
Affiliation:
Mathematics Department, Columbia University, 2990 Broadway, New York, NY 10027, USA ([email protected])
Ben Weinkove
Affiliation:
Mathematics Department, University of California, San Diego, 9500 Gilman Drive, #0112, La Jolla, CA 92093, USA ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that the Calabi–Yau equation can be solved on the Kodaira–Thurston manifold for all given T2-invariant volume forms. This provides support for Donaldson's conjecture that Yau's theorem has an extension to symplectic 4-manifolds with compatible but non-integrable almost complex structures.

Keywords

Calabi–YauKodaira–Thurston manifoldalmost complexsymplectic

MSC classification

Secondary: 53C99: None of the above, but in this section 53D05: Symplectic manifolds, general
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 10 , Issue 2 , April 2011 , pp. 437 - 447
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Abbena, E., An example of an almost Kähler manifold which is not Kählerian, Boll. Un. Mat. Ital. (6) A3(3) (1984), 383–392.Google Scholar
2.Apostolov, V. and Drăghici, T., The curvature and the integrability of almost-Kähler manifolds: a survey, in Symplectic and Contact Topology: Interactions and Perspectives (Toronto, ON/Montreal, QC, 2001), Fields Institute Communications, Volume 35, pp. 25–53 (American Mathematical Society, Providence, RI, 2003).Google Scholar
3.Cordero, L. A., Fernández, M. and Gray, A., Symplectic manifolds with no Kähler structure, Topology 25(3) (1986), 375–380.CrossRefGoogle Scholar
4.Donaldson, S. K., Two-forms on four-manifolds and elliptic equations, in Inspired by S. S. Chern (World Scientific, 2006).Google Scholar
5.Drăghici, T., Li, T.-J. and Zhang, W., Symplectic forms and cohomology decomposition of almost complex 4-manifolds, Int. Math. Res. Not. 2010(1) (2010), 1–17.Google Scholar
6.Fernández, M., Gotay, M. J. and Gray, A., Compact parallelizable four-dimensional symplectic and complex manifolds, Proc. Am. Math. Soc. 103(4) (1988), 1209–1212.Google Scholar
7.Geiges, H., Symplectic structures on T 2-bundles over T 2, Duke Math. J. 67(3) (1992), 539–555.CrossRefGoogle Scholar
8.Goldberg, S. I. and Har'El, Z., Mappings of almost Hermitian manifolds, J. Diff. Geom. 14(1) (1979), 67–80.Google Scholar
9.Kodaira, K., On the structure of compact complex analytic surfaces, I, Am. J. Math. 86 (1964), 751–798.CrossRefGoogle Scholar
10.Li, T.-J., Symplectic 4-manifolds with Kodaira dimension zero, J. Diff. Geom. 74(2) (2006), 321–352Google Scholar
11.Li, T.-J., Symplectic Calabi–Yau surfaces, in Geometry and analysis, Volume I, Advanced Lectures in Mathematics, Volume 17 (International Press, 2010).Google Scholar
12.Li, T.-J. and Zhang, W., Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Commun. Analysis Geom. 17(4) (2009), 651–683.Google Scholar
13.Muškarov, O., Two remarks on Thurston's example, in Complex Analysis and Applications '85 (Varna, 1985), pp. 461–468 (Publishing House of the Bulgarian Academy of Sciences, Sofia, 1986).Google Scholar
14.Sekigawa, K., On some 4-dimensional compact Einstein almost Kähler manifolds, Math. Annalen 271(3) (1985), 333–337.CrossRefGoogle Scholar
15.Thurston, W. P., Some simple examples of symplectic manifolds, Proc. Am. Math. Soc. 55(2) (1976), 467–468.Google Scholar
16.Tosatti, V. and Weinkove, B., The Calabi–Yau equation, symplectic forms and almost complex structures, in Geometry and analysis, Volume I, pp. 475–493, Advanced Lectures in Mathematics, Volume 17 (International Press, 2010).Google Scholar
17.Tosatti, V., Weinkove, B. and Yau, S.-T., Taming symplectic forms and the Calabi–Yau equation, Proc. Lond. Math. Soc. 97(2) (2008), 401–424.Google Scholar
18.Weinkove, B., The Calabi–Yau equation on almost-Kähler four-manifolds, J. Diff. Geom. 76(2) (2007), 317–349.Google Scholar
19.Yau, S.-T., On the Ricci curvature of a compact Kähler manifold and the complex Monge– Ampére equation, I, Commun. Pure Appl. Math. 31(3) (1978), 339–411.CrossRefGoogle Scholar