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Asymptotic Slopes of the Aubin–Yau Functional and Calculation of the Donaldson–Futaki Invariant

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Abstract

We derive an explicit formula for the asymptotic slope of the Aubin–Yau functional along a Bergman geodesic on a surface of complex dimension 2, extending the work of Phong–Sturm (On asymptotics for the Mabuchi energy functional, World Science Publishing, River Edge, 2004) on Riemann surfaces. This is equivalent to an explicit calculation of the Donaldson–Futaki invariant of a test configuration. The slope is given as a rational linear combination of period integrals of rational functions that sum to a rational number. The result gives a way to check directly whether a two-dimensional projective variety is K-stable.

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References

  1. Berman, R.: \(K\)-Polystability of Q-Fano varieties admitting Kähler-Einstein metrics. arXiv:1205.6214v2

  2. Berndtsson, B.: Convexity on the space of Kähler metrics. Ann. Fac. Sci. Toulouse Math. 22(4), 713746 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, X., Donaldson, S.K., 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  MATH  Google Scholar 

  4. Chen, X., Donaldson, S.K., 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  MATH  Google Scholar 

  5. Chen, X., Donaldson, S.K., 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  MATH  Google Scholar 

  6. Donaldson, S.K.: Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc. 50, 1–26 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  7. Donaldson, S.K.: Scalar curvature and projective embeddings I. J. Differ. Geom. 59, 479–522 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Donaldson, S.K.: Scalar curvature and stability of toric varieties. J. Differ. Geom. 59, 289–349 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Donaldson, S.K., Sun, S.: Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry. arXiv:1206.2609

  10. Ermolaeva, T.O., Tsikh, A.K.: Integration of rational functions over \({\mathbb{R}}^n\) by means of toric compactifications and multidimensional residues. Sbornik: Math. 187(9), 1301–1318 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Klevtsov, S., Zelditch, S.: Stability and integration over Bergman metrics. J. High Energy Phys. 7, 100 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Paul, S.T.: Geometric analysis of Chow-Mumford stability. Adv. Math. 182(2), 333–356 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Phong, D.H., Stein, E.M.: The Newton polyhedron and oscillatory integral operators. Acta Math. 179, 105–152 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Phong, D.H., Sturm, J.: Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions. Ann. Math. 152, 277–329 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Phong, D.H., Sturm, J.: Stability, energy functionals, and Kähler-Einstein metrics. Commun. Anal. Geom. 11, 565–597 (2003)

    Article  MATH  Google Scholar 

  16. Phong, D.H., Sturm, J.: On Asymptotics for the Mabuchi Energy Functional. Abstract and Applied Analysis, pp. 271–286. World Science Publishing, River Edge (2004)

    MATH  Google Scholar 

  17. Phong, D.H., Sturm, J.: Lectures on stability and constant scalar curvature, Handbook of Geometric Analysis, No. 3, pp. 357–436, Adv. Lect. Math. (ALM), 14. International Press, Somerville (2010)

  18. Tian, G.: \(K\)-stability and Kähler-Einstein metrics. arXiv:1211.4669

  19. Uhlenbeck, K., Yau, S.-T.: On the existence of Hermitian-Yang-Mills connections in stable vector bundles. Commun. Pure Appl. Math. 39–S, 257–293 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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  MATH  Google Scholar 

  21. Yau, S.-T.: Open problems in geometry. Proc. Symposia Pure Math. 54, 1–28 (1993)

    MathSciNet  MATH  Google Scholar 

  22. Zhang, S.W.: Heights and reductions of semistable varieties. Compos. Math. 104, 77–105 (1996)

    MATH  Google Scholar 

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Acknowledgements

I would like to thank my Ph.D. advisor D.H. Phong for his guidance and helpful insight into the problem, and also Steve Zelditch for his encouragement. Also thanks to Karsten Gimre for help in checking calculations during the preparation of this work and for his extensive assistance with the use of Mathematica. The author was partially supported by NSF Grant DMS-12-66033.

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  1. Department of Mathematics, Cornell University, Ithaca, NY, 14850, USA

    Daniel Rubin

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  1. Daniel Rubin
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Rubin, D. Asymptotic Slopes of the Aubin–Yau Functional and Calculation of the Donaldson–Futaki Invariant. J Geom Anal 28, 2812–2833 (2018). https://doi.org/10.1007/s12220-017-9935-8

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  • DOI: https://doi.org/10.1007/s12220-017-9935-8

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