On a Special Case of Carrellʹs Conjecture Original Research Article

In this paper we study a conjecture of J. B. Carrell on the rationality of a compact Kähler manifold admitting a holomorphic vector field with isolated zeroes. The conjecture, formulated in terms of image+-actions, says that if image+is acting on a nonsingular projective varietyXwith exactly one fixed point, thenXis 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 embedXas a fibre in a familyX* equipped with a suitable image*-action. Then we use a image*-equivariant projection onto the tangent space toX* at the sink.