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On deformations of Fano manifolds

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Abstract

In this paper, we provide new necessary and sufficient conditions for the existence of Kähler–Einstein metrics on small deformations of a Fano Kähler–Einstein manifold. We also show that the Weil–Petersson metric can be approximated by the Ricci curvatures of the canonical \(L^2\) metrics on the direct image bundles. In addition, we describe the plurisubharmonicity of the energy functional of harmonic maps on the Kuranishi space of the deformation of compact Kähler–Einstein manifolds of general type.

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References

  1. Berman, R.J.: K-polystability of \({{\mathbb{Q}}}\)-Fano varieties admitting Kähler–Einstein metrics. Invent. Math. 203(3), 973–1025 (2016)

    Article  MathSciNet  Google Scholar 

  2. Brönnle, A.: Deformation constructions of extremal metrics. Ph.D. thesis, Imperial College (2011)

  3. Calabi, E.: Extremal Kähler metrics II. Differential Geometry and Complex Analysis, pp. 95–114. Springer, Berlin (1985)

  4. Cao, H.-D., Sun, X., Yau, S.-T., Zhang, Y.: Weil–Petersson metrics on deformation spaces. J. Iran. Math. Soc. 1(1), 117–128 (2020)

  5. Catanese, F.: Theory of Moduli (Montecatini Terme, 1985), Volume 1337 Moduli of algebraic surfaces. Lecture Notes in Mathematic, pp. 1–83. Springer, Berlin (1988)

  6. Chen, X., Donaldson, S., Sun, S.: Kähler–Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities. J. Am. Math. Soc. 28(1), 183–197 (2015)

    Article  Google Scholar 

  7. Chen, X., Donaldson, S., Sun, S.: Kähler–Einstein metrics on Fano manifolds. II: Limits with cone angle less than \(2\pi \). J. Am. Math. Soc. 28(1), 199–234 (2015)

    Article  Google Scholar 

  8. Chen, X., Donaldson, S., Sun, S.: Kähler–Einstein metrics on Fano manifolds III: Limits as cone angle approaches \(2\pi \) and completion of the main proof. J. Am. Math. Soc. 28(1), 235–278 (2015)

    Article  Google Scholar 

  9. Donaldson, S.K., Kronheimer, P.B.: The Geometry of Four-Manifolds. Oxford Mathematical Monographs. Oxford University Press, New York (1990)

    MATH  Google Scholar 

  10. Fujiki, A., Schumacher, G.: The moduli space of extremal compact Kähler manifolds and generalized Weil–Petersson metrics. Publ. Res. Inst. Math. Sci. 26(1), 101–183 (1990)

    Article  MathSciNet  Google Scholar 

  11. Knudsen, F.F., Mumford, D.: The Projectivity of the Moduli Space of Stable Curves. I. Preliminaries on “det”and “Div”. Math. Scand. 39(1), 19–55 (1976)

    Article  MathSciNet  Google Scholar 

  12. Futaki, A.: An obstruction to the existence of Einstein Kähler metrics. Invent. Math. 73(3), 437–443 (1983)

    Article  MathSciNet  Google Scholar 

  13. Kodaira, K.: Complex Manifolds and Deformation of Complex Structures, Volume 283, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, New York (1986). Translated from the Japanese by Kazuo Akao, With an appendix by Daisuke Fujiwara

    Book  Google Scholar 

  14. Kodaira, K., Spencer, D.C.: On the variation of almost-complex structure. Algebraic Geometry and Topology. A Symposium in Honor of S. Lefschetz, pp. 139–150. Princeton University Press, Princeton (1957)

    Chapter  Google Scholar 

  15. Koiso, N.: Einstein metrics and complex structures. Invent. Math. 73(1), 71–106 (1983)

    Article  MathSciNet  Google Scholar 

  16. Kuranishi, M.: On the locally complete families of complex analytic structures. Ann. Math. 2(75), 536–577 (1962)

    Article  MathSciNet  Google Scholar 

  17. Kuranishi, M.: New proof for the existence of locally complete families of complex structures. In: Proceedings of Conference on Complex Analysis (Minneapolis, 1964). Springer, Berlin, pp. 142–154 (1965)

  18. LeBrun, C., Simanca, S.R.: Extremal Kähler metrics and complex deformation theory. Geom. Funct. Anal. 4(3), 298–336 (1994)

    Article  MathSciNet  Google Scholar 

  19. Lu, Z.: On the lower order terms of the asymptotic expansion of Tian–Yau–Zelditch. Am. J. Math. 122(2), 235–273 (2000)

    Article  MathSciNet  Google Scholar 

  20. Matsushima, Y.: Sur les espaces homogènes kählériens d’un groupe de Lie réductif. Nagoya Math. J. 11, 53–60 (1957)

    Article  MathSciNet  Google Scholar 

  21. Phong, D.H., Ross, J., Sturm, J.: Deligne pairings and the Knudsen–Mumford expansion. J. Differ. Geom. 78(3), 475–496 (2008)

    Article  MathSciNet  Google Scholar 

  22. Phong, D.H., Song, J., Sturm, J., Weinkove, B.: The Kähler–Ricci flow with positive bisectional curvature. Invent. Math. 173(3), 651–665 (2008)

    Article  MathSciNet  Google Scholar 

  23. Phong, D.H., Song, J., Sturm, J., Weinkove, B.: The Kähler–Ricci flow and the \(\overline{\partial }\) operator on vector fields. J. Differ. Geom. 81(3), 631–647 (2009)

    Article  Google Scholar 

  24. Phong, D.H., Sturm, J.: On stability and the convergence of the Kähler–Ricci flow. J. Differ. Geom. 72(1), 149–168 (2006)

    Article  Google Scholar 

  25. Rollin, Y., Simanca, S.R., Tipler, C.: Deformation of extremal metrics, complex manifolds and the relative Futaki invariant. Math. Z. 273(1–2), 547–568 (2013)

    Article  MathSciNet  Google Scholar 

  26. Sampson, J.H.: Applications of harmonic maps to Kähler geometry. Complex Differential Geometry and Nonlinear Differential Equations (Brunswick, Maine, 1984). American Mathematical Society, Providence (1986)

    Google Scholar 

  27. Schumacher, G.: The curvature of the Petersson–Weil metric on the moduli space of Kähler–Einstein manifolds. Complex Analysis and Geometry, University, Series Mathematics, pp. 339–354. Plenum, New York (1993)

    Chapter  Google Scholar 

  28. Sun, X.: Deformation of canonical metrics I. Asian J. Math. 16(1), 141–155 (2012)

    Article  MathSciNet  Google Scholar 

  29. Sun, X., Yau, S.-T.: Deformation of Kähler–Einstein metrics. Surveys in Geometric Analysis and Relativity, Volume 20 of Advanced Lectures in Mathematics (ALM), pp. 467–489. Internatinnal Press, Somerville (2011)

    Google Scholar 

  30. Székelyhidi, G.: The Kähler–Ricci flow and \(K\)-polystability. Am. J. Math. 132(4), 1077–1090 (2010)

    Article  Google Scholar 

  31. Tian, G.: On a set of polarized Kähler metrics on algebraic manifolds. J. Differ. Geom. 32(1), 99–130 (1990)

    Article  Google Scholar 

  32. Tian, G.: Kähler–Einstein metrics with positive scalar curvature. Invent. Math. 130(1), 1–37 (1997)

    Article  MathSciNet  Google Scholar 

  33. Todorov, A.N.: The Weil–Petersson geometry of the moduli space of \({\rm SU}(n\ge 3)\) (Calabi-Yau) manifolds. I. Commun. Math. Phys. 126(2), 325–346 (1989)

    Article  Google Scholar 

  34. Toledo, D.: Hermitian curvature and plurisubharmonicity of energy on Teichmüller space. Geom. Funct. Anal. 22(4), 1015–1032 (2012)

    Article  MathSciNet  Google Scholar 

  35. Wavrik, J.J.: Obstructions to the existence of a space of moduli. Global Analysis (Papers in Honor of K Kodaira), pp. 403–414. University Tokyo Press, Tokyo (1969)

    Google Scholar 

  36. Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I. Commun. Pure Appl. Math. 31(3), 339–411 (1978)

    Article  Google Scholar 

  37. Yau, S.-T.: Nonlinear analysis in geometry. Monographies de L’Enseignement Mathématique [Monographs of L’Enseignement Mathématique], 33. Série des Conférences de l’Union Mathématique Internationale [Lecture Series of the International Mathematics Union], 8. L’Enseignement Mathématique, Geneva, 1986. 54 pp

  38. Yau, S.-T.: Open problems in geometry. Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, CA, 1990). American Mathematical Society, Providence (1993)

    Google Scholar 

  39. Zelditch, S.: Szegö kernels and a theorem of Tian. Int. Math. Res. Not. 6, 317–331 (1998)

    Article  Google Scholar 

  40. Zhang, S.: Heights and reductions of semi-stable varieties. Compos. Math. 104, 77–105 (1996)

    MathSciNet  MATH  Google Scholar 

  41. Zhang, Y.: Ph.D. thesis, Lehigh University (2014). https://preserve.lehigh.edu/etd/1689

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Acknowledgements

We would like to thank T. Collins and J. Keller for their helpful comments on an earlier version of the paper. We would also like to thank the anonymous referee for providing helpful comments. The second named author would also like to thank S. K. Donaldson, D. H. Phong, R. Schoen and X. Wang for helpful discussions.

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Authors and Affiliations

  1. Department of Mathematics, Lehigh University, Bethlehem, PA, 18015, USA

    Huai-Dong Cao & Xiaofeng Sun

  2. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Shing-Tung Yau

  3. Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

    Yingying Zhang

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  1. Huai-Dong Cao
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  2. Xiaofeng Sun
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  3. Shing-Tung Yau
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  4. Yingying Zhang
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Correspondence to Huai-Dong Cao.

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Communicated by Ngaiming Mok.

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Research of H.-D. Cao and X. Sun was supported in part by Simons Foundation Grants.

Research of Y. Zhang was supported by Tsinghua University Initiative Scientific Research Program.

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Cao, HD., Sun, X., Yau, ST. et al. On deformations of Fano manifolds. Math. Ann. 383, 809–836 (2022). https://doi.org/10.1007/s00208-021-02226-2

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