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Compact Kähler manifolds with nonpositive bisectional curvature

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Abstract

Let (M n, g) be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold N k with c 1 <  0. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive bisectional curvature, the Kodaira dimension is equal to the maximal rank of the Ricci tensor. We also prove a global splitting result under the assumption of certain immersed complex submanifolds.

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References

  1. T. Aubin. Équations du type Monge-Ampère sur les variétés kählériennes compactes. Bulletin des Sciences Mathmatiques (2)102(1978), 63–95.

  2. Böhm C., Wilking B.: Nonnegative curved manifolds with finite fundamental groups admit metrics with positive Ricci curvature. Geometric and Functional Analysis 17, 665–681 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cao H.-D.: Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds.. Inventiones Mathematicae 81, 359–372 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chow B., Lu P.: The maximum principle for systems of parabolic equations subject to an avoidance set. Pacific Journal of Mathematics 214, 201–222 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. D. Ferus. On the completeness of nullity foliations. The Michigan Mathematical Journal 18 (1971) 61–64.

  6. M. Gill. Collapsing of Products Along the Kähler-Ricci Flow. Transactions of the American Mathematical Society

  7. D. Guler and F. Zheng. Nonpositively curved compact Riemannian manifold with degenerate Ricci tensor. Transactions of the American Mathematical Society (3)363 (2011), 1265–1285.

  8. R. Hamilton. Three-manifolds with positive Ricci curvature. The Journal of Differential Geometry 17 (1982), 255–306.

  9. R. Hamilton. Four-manifolds with positive curvature operator. The Journal of Differential Geometry 24 (1986), 153–179

  10. R. Hamilton. The formation of singularities in the Ricci flow. Surveys in Differential Geometry International Press 2 (1995), 7–136.

  11. T. Ivey. Ricci solitons on compact three-manifolds. Differential Geometry and its Applications 3 (1993), 301–307.

  12. N. M. Mok. The uniformization theorem for compact Kähler manifolds with nonnegative holomorphic bisectional curvature. The Journal of Differential Geometry 27 (1988), 179–214.

  13. S. Mori. Projective manifolds with ample tangent bundle. Annals of Mathematics 110 (1979), 593–606.

  14. J. Song and B. Weinkove. Lecture notes on the Kähler-Ricci flow. arxiv:1212.3653.

  15. Y. T. Siu and S. T. Yau. Compact Kähler manifolds with positive bisectional curvature. Inventiones Mathematicae 59 (1980), 189–204.

  16. H. Wu and F. Zheng. Kähler manifolds with slightly positive bisectional curvature. In: Explorations in complex and Riemannian geometry, Contemp. Math., 332. Amer. Math. Soc., Providence, RI, (2003), pp. 305–325.

  17. H. Wu and F. Zheng. Compact Kähler manifolds with nonpositive bisectional curvature. The Journal of Differential Geometry 61 (2002), 263–287.

  18. J. Song, G. Tian. The Kähler-Ricci flow on minimal surfaces of positive Kodaria dimension. Inventiones Mathematicae (3)170 (2007), 609–653.

  19. J. Song, G. Tian. Canonical measures and Kähler-Ricci flow. Journal of the American Mathematical Society 25 (2012), 303–353.

  20. G. Tian, Z. Zhang. On the Kähler-Ricci flow on projective manifolds of general type. Chinese Annals of Mathematics, Series B (2)27 (2006), 179–192.

  21. H. Tsuji. Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type. Mathematische Annalen (1)281 (1988), 123–133.

  22. S. T. Yau. On the Curvature of Compact Hermitian Manifolds. Inventiones Mathematicae 25 (1974), 213-239.

  23. S. T. Yau. A general Schwarz lemma for Kähler manifolds. The American Journal of Mathematics (1)100 (1978), 197-203.

  24. S.-T. Yau. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I. Communications on Pure and Applied Mathematics (3)31 (1978), 339-411.

  25. K. Ueno. Classification theory of algebraic varieties and compact complex spaces. Lecture notes in Mathematics, Vol. 439. (1975).

  26. C. J. Yu. A note on Wu-Zheng’s splitting conjecture. Proceedings of the American Mathematical Society (5)141 (2013), 1791–1793.

  27. F. Zheng. Kodaira dimensions and hyperbolicity for nonpositively curved Kähler manifolds. Commentarii Mathematici Helvetici (2)77 (2002), 221–234.

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  1. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Gang Liu

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  1. Gang Liu
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Correspondence to Gang Liu.

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Liu, G. Compact Kähler manifolds with nonpositive bisectional curvature. Geom. Funct. Anal. 24, 1591–1607 (2014). https://doi.org/10.1007/s00039-014-0290-7

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  • DOI: https://doi.org/10.1007/s00039-014-0290-7

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