Effective estimates on the very ampleness of the canonical line bundle of locally Hermitian symmetric spaces

Author:
Sai-Kee Yeung

Journal:
Trans. Amer. Math. Soc. **353** (2001), 1387-1401

MSC (2000):
Primary 14E25, 32J27, 32Q05, 32Q40

DOI:
https://doi.org/10.1090/S0002-9947-00-02777-X

Published electronically:
December 15, 2000

MathSciNet review:
1806736

Full-text PDF

Abstract | References | Similar Articles | Additional Information

We study the problem about the very ampleness of the canonical line bundle of compact locally Hermitian symmetric manifolds of non-compact type. In particular, we show that any sufficiently large unramified covering of such manifolds has very ample canonical line bundle, and give estimates on the size of the covering manifold, which is itself a locally Hermitian symmetric manifold, in terms of geometric data such as injectivity radius or degree of coverings.

**[A]**Atiyah, M. F., 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**54:8741****[B]**Borel, A., Compact Clifford-Klein forms of symmetric spaces. Topology 2 (1963), 111-122. MR**26:3823****[BGM]**Berger, M., Gauduchon, P., Mazet, E., Le Spectre d'une Varieté Riemannienne, Springer Lecture Notes in Mathematics No. 194, Springer-Verlag, N.Y., 1971. MR**43:8025****[Do1]**Donnelly, H., Asymptotic expansions for the compact quotients of properly discontinuous group actions, Illinois J. Math. 23 (1979), 485-496. MR**80h:58049****[Do2]**Donnelly, H., Elliptic operators and covers of Riemannian manifolds, Math. Zeit. 223(1996), 303-308. MR**98a:58163****[HT]**Hwang, J.-M. and To, W.-K., On Seshadri constants of canonical bundles of compact complex hyperbolic spaces, Compositio Math. 118 (1999), 203-215. MR**2000i:32034****[Lü]**Lück, W., Approximating -invariants by their finite-dimensional analogues. Geom. Funct. Anal. 4 (1994), 455-481. MR**95g:58234****[MP]**Minakshisundaram, S., Pleijel, A., Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds. Canadian J. Math. 1, (1949). 242-256. MR**11:108b****[MS]**Mostow, G. D., Siu, Y.-T., A compact Kähler surface of negative curvature not covered by the ball. Ann. of Math. (2) 112 (1980), no. 2, 321-360. MR**82f:53075****[P]**Patodi, V. K., Curvature and the eigenforms of the Laplace operator, J. Diff. Geom. 5(1971) 233-249. MR**45:1201****[Y1]**Yeung, S. K., Betti numbers on a tower of coverings, Duke Math. J. 73(1994), 201-226. MR**95e:58182****[Y2]**Yeung, S. K., Very ampleness of line bundles and canonical embedding of coverings of manifolds, to appear in Compositio. Math.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
14E25,
32J27,
32Q05,
32Q40

Retrieve articles in all journals with MSC (2000): 14E25, 32J27, 32Q05, 32Q40

Additional Information

**Sai-Kee Yeung**

Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395;
Department of Mathematics, The University of Hong Kong, Hong Kong

Email:
yeung@math.purdue.edu

DOI:
https://doi.org/10.1090/S0002-9947-00-02777-X

Keywords:
Very ampleness,
canonical embedding

Received by editor(s):
April 10, 2000

Published electronically:
December 15, 2000

Additional Notes:
The author was partially supported by grants from the National Science Foundation

Article copyright:
© Copyright 2000
American Mathematical Society