A bound for the dimension of the automorphism group of a homogeneous compact complex manifold
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- by Dennis M. Snow PDF
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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 \[ \dim \operatorname {Aut}(X) \le n^2-1+\binom {2n-1}{n-1} \] 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 $m$: $\dim H^0(Y,K_Y^*) \le \binom {2m+1}{m}$.References
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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
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 2051-2055
- MSC (2000): Primary 32M10; Secondary 32M05
- DOI: https://doi.org/10.1090/S0002-9939-03-07295-2
- MathSciNet review: 2053977