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Holomorphic vector fields with totally degenerate zeroes

Published online by Cambridge University Press:  04 December 2007

JUN-MUK HWANG
JUN-MUK HWANG
Affiliation:
Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720 Current address: Department of Mathematics, Seoul National University, Seoul 151-742, Korea. e-mail: jmhwang@math.snu.ac.kr
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Abstract

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A zero set of a holomorphic vector field is totally degenerate, if the endomorphism of the conormal sheaf induced by the vector field is identically zero. By studying a class of foliations generalizing foliations of ${\rm C}^*$-actions, we show that if a projective manifold admits a holomorphic vector field with a smooth totally degenerate zero component, then the manifold is stably birational to that component of the zero set. When the vector field has an isolated totally degenerate zero, we prove that the manifold is rational. This is a special case of Carrell's conjecture.

Keywords

holomorphic vectgor fieldsholomorphic foliationsrationality
Type
Research Article
Information
Compositio Mathematica , Volume 105 , Issue 1 , January 1997 , pp. 65 - 77
Copyright
© 1997 Kluwer Academic Publishers