Advertisement

Holomorphic Invariants of Bounded Domains

  • 125 Accesses

Abstract

In this survey paper, we give a review of some recent developments on holomorphic invariants of bounded domains, which include squeezing functions, Fridman’s invariants, p-Bergman kernels, and spaces of \(L^p\) integrable holomorphic functions.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.

References

  1. 1.

    Alexander, H.: Extremal holomorphic imbeddings between the ball and polydisc. Proc. Am. Math. Soc. 68(2), 200–202 (1978)

  2. 2.

    Avelin, B., Hed, L., Persson, H.: A note on the hyperconvexity of pseudoconvex domains beyond Lipschitz regularity (English summary). Potential Anal. 43(3), 531–545 (2015)

  3. 3.

    Balakumar, G.P., Mahajan, P., Verma, K.: Bounds for invariant distances on pseudoconvex Levi corank one domains and applications. Ann. Fac. Sci. Toulouse Math. (6) 24(2), 281–388 (2015)

  4. 4.

    Berndtsson, B.: Complex Brunn–Minkowski theory and positivity of vector bundles (e-preprint). arXiv:1807.05844

  5. 5.

    Berndtsson, B.: Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains. Ann. Inst. Fourier (Grenoble) 56(6), 1633–1662 (2006)

  6. 6.

    Berndtsson, B.: Curvature of vector bundles associated to holomorphic fibrations. Ann. Math. (2) 169(2), 531–560 (2009)

  7. 7.

    Berndtsson, B., Păun, M.: Bergman kernels and the pseudoeffectivity of relative canonical bundles. Duke Math. J. 145(2), 341–378 (2008)

  8. 8.

    Berndtsson, B., Păun, M.: Bergman kernels and subadjunction. arXiv:1002.4145

  9. 9.

    Błocki, Z.: Suita conjecture and the Ohsawa–Takegoshi extension theorem. Invent. Math. 193(1), 149–158 (2013)

  10. 10.

    Chen, B.-Y.: The Bergman metric on Teichmüller space. Int. J. Math. 15(10), 1085–1091 (2004)

  11. 11.

    Chen, B.-Y., Zhang, J.: On graphs of holomorphic motions (private communication)

  12. 12.

    Chi, C.-Y.: Pseudonorms and theorems of Torelli type. J. Differ. Geom. 104(2), 239–273 (2016)

  13. 13.

    Chi, C.-Y., Yau, S.-T.: A geometric approach to problems in birational geometry. Proc. Natl Acad. Sci. USA 105(48), 18696–18701 (2008)

  14. 14.

    Demailly, J.P.: Mesures de Monge-Ampére et mesures pluriharmoniques. Math. Z. 194, 519–564 (1987)

  15. 15.

    Deng, F., Zhang, X.: Fridman’s invariant, squeezing functions, and exhausting domains (e-preprint). arXiv:1810.12724

  16. 16.

    Deng, F., Guan, Q., Zhang, L.: Some properties of squeezing functions on bounded domains. Pac. J. Math. 257, 319–341 (2012)

  17. 17.

    Deng, F., Guan, Q., Zhang, L.: Properties of squeezing functions and global transformations of bounded domains. Trans. Am. Math. Soc. 368, 2679–2696 (2016)

  18. 18.

    Deng, F., Fornaess, J.E., Wold, E.F.: Exposing boundary points of strongly pseudoconvex subvarieties in complex spaces. Proc. Am. Math. Soc. 146(6), 2473–2487 (2018)

  19. 19.

    Deng, F., Wang, Z., Zhang, L., Zhou, X.: New characterization of plurisubharmonic functions and positivity of direct image sheaves (e-preprint). arXiv:1809.10371

  20. 20.

    Deng, F., Wang, Z., Zhang, L., Zhou, X.: Linear invariants of complex manifolds and their plurisubharmonic variations (e-preprint). arXiv:1901.08920

  21. 21.

    Diederich, K., Fornaess, J.E.: Comparison of the Bergman and the Kobayashi metric. Math. Ann. 254, 257–262 (1980)

  22. 22.

    Diederich, K., Fornaess, J.E.: Boundary behavior of the Bergman metric. Asian J. Math. 22(2), 291–298 (2018)

  23. 23.

    Diederich, K., Fornaess, J.E., Wold, E.F.: Exposing points on the boundary of a strictly pseudoconvex or a locally convexifiable domain of finite 1-type. J. Geom. Anal. 24(4), 2124–2134 (2014)

  24. 24.

    Diederich, K., Fornaess, J.E., Wold, E.F.: A characterization of the ball in \({\mathbb{C}}^n\). Int. J. Math. 27(9), 1650078 (2016)

  25. 25.

    Fornaess, J.E., Rong, F.: Estimate of the squeezing function for a class of bounded domains. Math. Ann. 371(3–4), 1087–1094 (2018)

  26. 26.

    Fornaess, J.E., Shcherbina, N.: A domain with non-plurisubharmonic squeezing function. J. Geom. Anal. 28(1), 13–21 (2018)

  27. 27.

    Fornaess, J.E., Wold, E.F.: An estimate for the squeezing function and estimates of invariant metrics. In: Complex Analysis and Geometry. Springer Proceedings in Mathematics and Statistics, vol. 144, pp. 135–147. Springer, Tokyo (2015)

  28. 28.

    Fornaess, J.E., Wold, E.F.: A non-strictly pseudoconvex domain for which the squeezing function tends to 1 towards the boundary. Pac. J. Math. 297(1), 79–86 (2018)

  29. 29.

    Frankel, S,: Applications of affine geometry to geometric function theory in Several Complex Variables. Proc. Symp. Pure Math. 52(Part 2), 183–208 (1991)

  30. 30.

    Fridman, B.L.: Biholomorphic invariants of a hyperbolic manifold and some application. Trans. Am. Math. Soc. 276, 685–698 (1983)

  31. 31.

    Guan, Q., Zhou, X.: A solution of an \(L^2\) extension problem with an optimal estimate and applications. Ann. Math. (2) 181(3), 1139–1208 (2015)

  32. 32.

    Hacon, C., Popa, M., Schnell, C.: Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Paun. In: Local and Global Methods in Algebraic Geometry. Contemporary Mathematics, vol. 712, , pp. 143–195. American Mathematical Society, Providence (2018)

  33. 33.

    Hosono, G., Inayama, T.: A converse of Hörmander’s \(L^2\)-estimate and new positivity notions for vector bundles (e-preprint). arXiv:1901.02223

  34. 34.

    Joo, S., Kim, K.-T.: On boundary points at which the squeezing function tends to one. J. Geom. Anal. 28(3), 2456–2465 (2018)

  35. 35.

    Kim, K.-T., Zhang, L.: On the uniform squeezing property of bounded convex domains in \(\mathbb{C}^n\). Pac. J. Math. 282(2), 341–358 (2016)

  36. 36.

    Lakic, N.: An isometry theorem for quadratic differentials on Riemann surfaces of finite genus. Trans. Am. Math. Soc. 349(7), 2951–2967 (1997)

  37. 37.

    Liu, K., Sun, X., Yau, S.-T.: Canonical metrics on the moduli space of Riemann surfaces I. J. Differ. Geom. 68(3), 571–637 (2004)

  38. 38.

    Liu, K., Sun, X., Yau, S.-T.: Canonical metrics on the moduli space of Riemann Surfaces II. J. Differ. Geom. 69, 163–216 (2005)

  39. 39.

    Mahajan, P., Verma, K.: A comparison of two biholomorphic invariants (e-preprint). arXiv:1807.09451v2

  40. 40.

    Mahajan, P., Verma, K.: Some aspects of the Kobayashi and Carathéodory metrics on pseudoconvex domains. J. Geom. Anal. 22(2), 491–560 (2012)

  41. 41.

    Markovic, V.: Biholomorphic maps between Teichmüller spaces. Duke Math. J. 120(2), 405–431 (2003)

  42. 42.

    Nikolov, N., Andreev, L.: Boundary behavior of the squeezing functions of \({\mathbb{C}}\)-convex domains and plane domains. Int. J. Math. 28(5), 1750031 (2017)

  43. 43.

    Nikolov, N., Trybula, M.: Estimates for the squeezing function near strictly pseudoconvex boundary points with applications (e-preprint). arXiv:1808.07892

  44. 44.

    Nikolov, N., Verma, K.: On the squeezing function and Fridman invariants (e-preprint). arXiv:181010739

  45. 45.

    Ning, J., Zhang, H., Zhou, X.: On p-Bergman kernel for bounded domains in \({\mathbb{C}}^n\). Commun. Anal. Geom. 24(4), 887–900 (2016)

  46. 46.

    Păun, M., Takayama, S.: Positivity of twisted relative pluricanonical bundles and their direct images. J. Algebr. Geom. 27(2), 211–272 (2018)

  47. 47.

    Rudin, W.: \(L^{p}\)-isometries and equimeasurability. Indiana Univ. Math. J. 25(3), 215–228 (1976)

  48. 48.

    Takayama, S.: Singularities of Narasimhan-Simha type metrics on direct images of relative pluricanonical bundles. Ann. Inst. Fourier (Grenoble) 66(2), 753–783 (2016)

  49. 49.

    Wold, E.F.: Asymptotics of invariant metrics in the normal direction and a new characterisation of the unit disk. Math. Z. 288(3–4), 875–887 (2018)

  50. 50.

    Yau, S.-T.: On the pseudonorm project of birational classification of algebraic varieties. In: Geometry and Analysis on Manifolds. Progress in Mathematics, vol. 308, pp. 327–339. Birkhäuser/Springer, Cham (2015)

  51. 51.

    Yeung, S.-K.: Quasi-isometry of metrics on Teichmuller spaces. Int. Math. Res. Not. 2005, 239–255 (2005)

  52. 52.

    Yeung, S.-K.: Geometry of domains with the uniform squeezing property. Adv. Math. 221(2), 547–569 (2009)

  53. 53.

    Zhang, L.: Intrinsic derivative, curvature estimates and squeezingfunction. Sci. China Math. 60(6), 1149–1162 (2017)

  54. 54.

    Zhou, X., Zhu, L.: An optimal \(L^2\) extension theorem on weakly pseudoconvex Kähler manifolds. J. Differ. Geom. 110(1), 135–186 (2018)

  55. 55.

    Zhou, X., Zhu, L.: Siu’s lemma, optimal \(L^2\) extension, and applications to pluricanonical sheaves. Math. Ann. https://doi.org/10.1007/s00208-018-1783-8 (2018)

  56. 56.

    Zimmer, A.: A gap theorem for the complex geometry of convex domains. Trans. Am. Math. Soc. 370(10), 7489–7509 (2018)

  57. 57.

    Zimmer, A.: Characterizing strong pseudoconvexity, obstructions to biholomorphisms, and Lyapunov exponents. Math. Ann. https://doi.org/10.1007/s00208-018-1715-7 (2018)

Download references

Acknowledgements

The authors thank the referee for drawing the references [40] and [3] to our attention. The first author is partially supported by the University of Chinese Academy of Sciences and by NSFC Grants. The second author was partially supported by the Fundamental Research Funds for the Central Universities and by the NSFC Grant NSFC-11701031. The third and fourth authors are partially supported by NSFC Grants.

Author information

Correspondence to Liyou Zhang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Deng, F., Wang, Z., Zhang, L. et al. Holomorphic Invariants of Bounded Domains. J Geom Anal (2019). https://doi.org/10.1007/s12220-019-00229-9

Download citation

Keywords

  • Squeezing function
  • Fridman invariant
  • p-Bergman kernel
  • Plurisubharmonic variation

Mathematics Subject Classification

  • 32F45
  • 32H02