Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Kähler hyperbolic manifolds and Chern number inequalities
HTML articles powered by AMS MathViewer

by Ping Li PDF
Trans. Amer. Math. Soc. 372 (2019), 6853-6868 Request permission


We show in this article that Kähler hyperbolic manifolds satisfy a family of optimal Chern number inequalities and that the equality cases can be attained by some compact ball quotients. These present restrictions to complex structures on negatively curved compact Kähler manifolds, thus providing evidence for the rigidity conjecture of S.-T. Yau. The main ingredients in our proof are Gromov’s results on the $L^2$-Hodge numbers, the $-1$-phenomenon of the $\chi _y$-genus and Hirzebruch’s proportionality principle. Similar methods can be applied to obtain parallel results on Kähler nonelliptic manifolds. In addition to these, we term a condition called “Kähler exactness”, which includes Kähler hyperbolic and nonelliptic manifolds and has been used by B.-L. Chen and X. Yang in their work, and we show that the canonical bundle of a Kähler exact manifold of the general type is ample. Some of its consequences and remarks are discussed as well.
  • M. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974) Astérisque, No. 32-33, Soc. Math. France, Paris, 1976, pp. 43–72. MR 0420729
  • Werner Ballmann, Lectures on Kähler manifolds, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Zürich, 2006. MR 2243012, DOI 10.4171/025
  • Jianguo Cao and Frederico Xavier, Kähler parabolicity and the Euler number of compact manifolds of non-positive sectional curvature, Math. Ann. 319 (2001), no. 3, 483–491. MR 1819879, DOI 10.1007/PL00004444
  • Shiing-shen Chern, On curvature and characteristic classes of a Riemann manifold, Abh. Math. Sem. Univ. Hamburg 20 (1955), 117–126. MR 75647, DOI 10.1007/BF02960745
  • Bing-Long Chen and Xiaokui Yang, Compact Kähler manifolds homotopic to negatively curved Riemannian manifolds, Math. Ann. 370 (2018), no. 3-4, 1477–1489. MR 3770171, DOI 10.1007/s00208-017-1521-7
  • B.-L. Chen and X. Yang, On Euler characteristic and fundamental groups of compact manifolds, arXiv:1711.03309v2 (2018).
  • Olivier Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001. MR 1841091, DOI 10.1007/978-1-4757-5406-3
  • O. Debarre, Cohomological characterizations of the complex projective space, arXiv:1512.04321 (2015).
  • Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
  • M. Gromov, Kähler hyperbolicity and $L_2$-Hodge theory, J. Differential Geom. 33 (1991), no. 1, 263–292. MR 1085144, DOI 10.4310/jdg/1214446039
  • F. Hirzebruch, Characteristic numbers of homogeneous domains, Seminars on Analytic Functions, vol. 2, Institute for Advanced Studies, Princeton, NJ, 1957, pp. 92–104.
  • Friedrich Hirzebruch, Automorphe Formen und der Satz von Riemann-Roch, Symposium internacional de topología algebraica International symposium on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 129–144 (German). MR 0103280
  • F. Hirzebruch, Topological methods in algebraic geometry, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. MR 0202713
  • F. Hirzebruch, On the Euler characteristic of manifolds with $c_1=0$. A letter to V. Gritsenko, Algebra i Analiz 11 (1999), no. 5, 126–129; English transl., St. Petersburg Math. J. 11 (2000), no. 5, 805–807. MR 1734349
  • Jürgen Jost and Kang Zuo, Vanishing theorems for $L^2$-cohomology on infinite coverings of compact Kähler manifolds and applications in algebraic geometry, Comm. Anal. Geom. 8 (2000), no. 1, 1–30. MR 1730897, DOI 10.4310/CAG.2000.v8.n1.a1
  • Shoshichi Kobayashi, Hyperbolic complex spaces, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 318, Springer-Verlag, Berlin, 1998. MR 1635983, DOI 10.1007/978-3-662-03582-5
  • Ping Li, $-1$-phenomena for the pluri $\chi _y$-genus and elliptic genus, Pacific J. Math. 273 (2015), no. 2, 331–351. MR 3317769, DOI 10.2140/pjm.2015.273.331
  • Ping Li, The Hirzebruch $\chi _y$-genus and Poincaré polynomial revisited, Commun. Contemp. Math. 19 (2017), no. 5, 1650048, 19. MR 3670789, DOI 10.1142/S0219199716500486
  • Anatoly S. Libgober and John W. Wood, Uniqueness of the complex structure on Kähler manifolds of certain homotopy types, J. Differential Geom. 32 (1990), no. 1, 139–154. MR 1064869
  • Wolfgang Lück, $L^2$-invariants: theory and applications to geometry and $K$-theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 44, Springer-Verlag, Berlin, 2002. MR 1926649, DOI 10.1007/978-3-662-04687-6
  • Xiaonan Ma and George Marinescu, Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics, vol. 254, Birkhäuser Verlag, Basel, 2007. MR 2339952, DOI 10.1007/978-3-7643-8115-8
  • M. S. Narasimhan and S. Ramanan, Generalised Prym varieties as fixed points, J. Indian Math. Soc. (N.S.) 39 (1975), 1–19 (1976). MR 424819
  • S. M. Salamon, On the cohomology of Kähler and hyper-Kähler manifolds, Topology 35 (1996), no. 1, 137–155. MR 1367278, DOI 10.1016/0040-9383(95)00006-2
  • S. M. Salamon, Cohomology of Kähler manifolds with $c_1=0$, Manifolds and geometry (Pisa, 1993) Sympos. Math., XXXVI, Cambridge Univ. Press, Cambridge, 1996, pp. 294–310. MR 1410078
  • Yum Tong Siu, The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Ann. of Math. (2) 112 (1980), no. 1, 73–111. MR 584075, DOI 10.2307/1971321
  • Shing Tung Yau, Calabi’s conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 5, 1798–1799. MR 451180, DOI 10.1073/pnas.74.5.1798
  • Shing Tung Yau, Problem section, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 669–706. MR 645762
  • Fangyang Zheng, First Pontrjagin form, rigidity and strong rigidity of nonpositively curved Kähler surface of general type, Math. Z. 220 (1995), no. 2, 159–169. MR 1355023, DOI 10.1007/BF02572607
  • Fangyang Zheng, Kodaira dimensions and hyperbolicity of nonpositively curved compact Kähler manifolds, Comment. Math. Helv. 77 (2002), no. 2, 221–234. MR 1915039, DOI 10.1007/s00014-002-8337-z
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 32Q45, 57R20, 58J20
  • Retrieve articles in all journals with MSC (2010): 32Q45, 57R20, 58J20
Additional Information
  • Ping Li
  • Affiliation: School of Mathematical Sciences, Tongji University, Shanghai 200092, People’s Republic of China
  • MR Author ID: 902503
  • Email:;
  • Received by editor(s): May 26, 2018
  • Published electronically: August 28, 2019
  • Additional Notes: The author was partially supported by the National Natural Science Foundation of China (Grant No. 11722109).
    Part of this article was completed when the author visited the Fields Institute in Toronto in May 2018. The author would like to thank the Institute for the hospitality.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 6853-6868
  • MSC (2010): Primary 32Q45, 57R20, 58J20
  • DOI:
  • MathSciNet review: 4024540