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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A bound for the dimension of the automorphism group of a homogeneous compact complex manifold


Author: Dennis M. Snow
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 2051-2055
MSC (2000): Primary 32M10; Secondary 32M05
Published electronically: December 23, 2003
MathSciNet review: 2053977
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Abstract: Let $X$ be a homogeneous compact complex manifold, and let $\operatorname{Aut}(X)$ be the complex Lie group of holomorphic automorphisms of $X$. Examples show that $\dim \operatorname{Aut} (X)$ can grow exponentially in $n = \dim X$. In this note it is shown that

\begin{displaymath}\dim \operatorname{Aut}(X) \le n^2-1+\binom{2n-1}{n-1} \end{displaymath}

when $n \ge 3$. Thus, $\dim \operatorname{Aut} (X)$ is at most exponential in $n$. The proof relies on an upper bound for the dimension of the space of sections of the anticanonical bundle, $K_Y^* = \det T_Y$, of a homogeneous projective rational manifold $Y$ of dimension $m$: $\dim H^0(Y,K_Y^*) \le \binom{2m+1}{m}$.


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Additional Information

Dennis M. Snow
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: snow.1@nd.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07295-2
PII: S 0002-9939(03)07295-2
Received by editor(s): November 10, 2002
Received by editor(s) in revised form: March 20, 2003
Published electronically: December 23, 2003
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2003 American Mathematical Society