Holomorphic Vector Fields on Algebraic Manifolds

Abstract

Then M does not admit any non-trivial holomorphic p-form for p > m. We recall that a Hodge (or projective algebraic) manifold is a compact complex manifold which admits a holomorphic imbedding into complex projective space pN for some N. A holomorphic vector field X is one which can be represented locally as X = EXi3/azi, where each Xi is a holomorphic function. The set zero (X) = {x C M: X= O} is an analytic subspace of M, and by dim zero (X) we mean the maximum dimension of its components. Our hypothesis on the dimension includes the assumption that zero (X) is non-empty. Finally, a non-trivial form is one which does not vanish identically. As a special case of our theorem, we obtain: