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Measure Preserving Holomorphic Vector Fields, Invariant Anti-Canonical Divisors and Gibbs Stability

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Abstract

Let X be a compact complex manifold whose anti-canonical line bundle — KX is big. We show that X admits no non-trivial holomorphic vector fields if it is Gibbs stable (at any level). The proof is based on a vanishing result for measure preserving holomorphic vector fields on X of independent interest. As an application it shown that, in general, if — KX is big, there are no holomorphic vector fields on X that are tangent to a non-singular irreducible anti-canonical divisor S on X. More generally, the result holds for varieties with log terminal singularities and log pairs. Relations to a result of Berndtsson about generalized Hamiltonians and coercivity of the quantized Ding functional are also pointed out.

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References

  1. D. Auroux, Mirror symmetry and T-duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol. GGT, 1 (2007), 51–91.

    MathSciNet  MATH  Google Scholar 

  2. V. V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom., 3 (1994), 493–535.

    MathSciNet  MATH  Google Scholar 

  3. R. J. Berman, Analytic torsion, vortices and positive Ricci curvature, arXiv:1006.2988 (2010).

  4. R. J. Berman, A thermodynamical formalism for Monge—Ampère equations, Moser-Trudinger inequalities and Kähler—Einstein metrics, Adv. Math., 248 (2013), 1254–1297.

    Article  MathSciNet  Google Scholar 

  5. R. J. Berman, Käahler—Einstein metrics, canonical random point processes and birational geometry, in: Algebraic Geometry (Salt Lake City, 2015), Proc. Sympos. Pure Math., vol. 97.1, Amer. Math. Soc. (Providence, RI, 2018), pp. 29–74.

    MATH  Google Scholar 

  6. R. J. Berman: Emergent complex geometry, arXiv:2109.00307 (2021).

  7. R. J. Berman, The probabilistic vs the quantization approach to Kähler—Einstein geometry, arXiv:2109.06575 (2021).

  8. R. J. Berman: Kähler-Einstein metrics and Archimedean zeta functions, arXiv.org/2112.04791 (2021).

  9. R. J. Berman, S. Boucksom, V. Guedj and A. Zeriahi, A variational approach to complex Monge—Ampère equations, Publ. Math. Inst. Hautes Études Sci., 117 (2012), 179–245.

    Article  Google Scholar 

  10. R. J. Berman, P. Eyssidieu, S. Boucksom, V. Guedj and A. Zeriahi, Kähler—Einstein metrics and the Kähler—Ricci flow on log Fano varieties, Reine Angew. Math., 751 (2019), 27–89.

    Article  MathSciNet  Google Scholar 

  11. B. Berndtsson, A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry, Invent. Math., 200 (2015), 149–200.

    Article  MathSciNet  Google Scholar 

  12. B. Berndtsson: The openness conjecture and complex Brunn-Minkowski inequalities, in: Complex Geometry and Dynamics, Abel Symp., vol. 10, Springer (Cham, 2015), pp. 29–44.

    Chapter  Google Scholar 

  13. P. Cascini and Y. Gongyo, On the anti-canonical ring and varieties of Fano type, Saitama Math. J., 30 (2013), 27–38.

    MathSciNet  MATH  Google Scholar 

  14. X. X. Chen, S. Donaldson and S. Sun, Kähler—Einstein metrics on Fano manifolds, I, II, III, J. Amer. Math. Soc., 28 (2015), 183–197, 199–234, 235–278.

    Article  MathSciNet  Google Scholar 

  15. J.-P. Demailly, Appendix to I. Cheltsov and C. Shramov’s article “Log canonical thresholds of smooth Fano three-folds”: On Tian’s invariant and log canonical thresholds, Uspekhi Mat. Nauk, 63 (2008), 73–180.

    Google Scholar 

  16. S. K. Donaldson, Some numerical results in complex differential geometry, Pure Appl. Math. Q., 5 (2009), 571–618.

    Article  MathSciNet  Google Scholar 

  17. S. K. Donaldson, Kähler metrics with cone singularities along a divisor, in: Essays in Mathematics and its Applications, Springer (Heidelberg, 2012), pp. 49–79.

    Chapter  Google Scholar 

  18. K. Fujita and Y. Odaka, On the K-stability of Fano varieties and anticanonical divisors, Tohoku Math. J., 70 (2018), 511–521.

    Article  MathSciNet  Google Scholar 

  19. Q. Guan and X. Zhou, A proof of Demailly’s strong openness conjecture, Ann. of Math. (2), 182 (2015), 605–616.

    Article  MathSciNet  Google Scholar 

  20. S. Kobayashi, Transformation Groups in Differential Geometry, Springer (2012).

  21. J. Kollar, Singularities of pairs, in: Algebraic Geometry (Santa Cruz, 1995), Proc. Sympos. Pure Math., vol. 62, Part 1, Amer. Math. Soc. (Providence, RI, 1997), 221–287.

    Google Scholar 

  22. R. K. Lazarsfeld, Positivity in Algebraic Geometry, I-II, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer (2004).

  23. C. Li, G-uniform stability and Käahler-Einstein metrics on Fano varieties, Invent. Math., 227 (2022), 661–744.

    Article  MathSciNet  Google Scholar 

  24. Y. Liu, C. Xu and Z. Zhuang, Finite generation for valuations computing stability thresholds and applications to K-stability, arXiv:2102.09405 (2021).

  25. X. Ma and G. Marinescu, Holomorphic Morse Inequalities and Bergman Kernels, Progress in Math., Birkhäauser (2007).

  26. Y. Odaka, The GIT stability of polarized varieties via discrepancy, Ann. of Math. (2), 177 (2013), 645–661.

    Article  MathSciNet  Google Scholar 

  27. R. Ohta and S. Okawa, On ruled surfaces with big anti-canonical divisor and numerically trivial divisors on weak log Fano surfaces, Manuscripta Math., 166 (2021), 1–17.

    Article  MathSciNet  Google Scholar 

  28. Y. A. Rubinstein, G. Tian and K. Zhang, Basis divisors and balanced metrics, J. Reine Angew. Math. (to appear), arXiv:2008.08829 (2020).

  29. C. Schnell and D. Chen, Surfaces with big anti-canonical class, https://www.math.stonybrook.edu/~cschnell/pdf/papers/anticanonical.pdf.

  30. J. Song and X. Wang, The greatest Ricci lower bound, conical Einstein metrics and Chern number inequality, Geom. Topol., 20 (2016) 49–102.

    Article  MathSciNet  Google Scholar 

  31. M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc., 324 (1991), 793–821.

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Thanks to Rolf Andreasson and Bo Berndtsson for discussions and helpful comments and the referee for comments that helped to improve the exposition.

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Authors and Affiliations

  1. Chalmers University of Technology, Kaptensgatan 28D, 414 59, Göteborg, Sweden

    R. J. Berman

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  1. R. J. Berman
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Correspondence to R. J. Berman.

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Dedicated to Laszlo Lempert on the occasion of his 70th anniversary

This work was supported by grants from the Knut and Alice Wallenberg foundation, the Göran Gustafsson foundation and the Swedish Research Council.

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Berman, R.J. Measure Preserving Holomorphic Vector Fields, Invariant Anti-Canonical Divisors and Gibbs Stability. Anal Math 48, 347–375 (2022). https://doi.org/10.1007/s10476-022-0154-6

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  • DOI: https://doi.org/10.1007/s10476-022-0154-6

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