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On a Special Case of Carrell's Conjecture

✍ Scribed by Jerzy Konarski

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
127 KB
Volume
182
Category
Article
ISSN
0021-8693

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✦ Synopsis

In this paper we study a conjecture of J. B. Carrell on the rationality of a compact Kahler manifold admitting a holomorphic vector field with isolated zeroes. The conjecture, formulated in terms of ‫ރ‬ q -actions, says that if ‫ރ‬ q is acting on a nonsingular projective variety X with exactly one fixed point, then X is rational. We prove this is true under the additional assumption that in the tangent space at the fixed point there is only one fixed direction. To prove this result we embed X as a fibre in a family X U equipped with a suitable ‫ރ‬ U -action. Then we use a ‫ރ‬ U -equivariant projection onto the tangent space to X U at the sink. ᮊ 1996 Academic Press, Inc.

Ž

In this paper we shall study the following conjecture of J. B. Carrell see w x w x. 8 , cf. also 7 :

Conjecture 1. If a compact Kahler manifold X admits a holomorphic ¨Ž vector field with isolated zeroes, then X is rational i.e., X is bimeromor-. phic to a projective space .

As is well known, the above conjecture is equivalent to the following: Conjecture 2. If X is a nonsingular projective variety admitting an algebraic ‫ރ‬ q -action with exactly one fixed point, then X is rational. Ž w x. Conjecture 2 has been proved up to dimension 6 see 6, 3 . Here we shall prove that it is true in the following special case: THEOREM 3. Let X be a nonsingular projecti¨e ¨ariety with an action of ‫ރ‬ q . Assume that there is only one fixed point p in X and that in the tangent space T X there is only a line of fixed points with respect to the tangent p ‫ރ‬ q -action. Then X is rational.

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