In his study on the structure of the complex Lie algebra of holomorphic vector fields on a compact Kähler manifold, Lichnerowicz ([3] Theorem 2, see also [1] and [4]) shows that if the first Chern class of the manifold is positive semi-definite, then to each harmonic (O.l)-form (i.e. anti-holomorphic 1-form) η, there exists a holomorphic vector field X such that the (O.1)-form i(X)k is d″-cohomologous to η, where k is the Kähler form. The purpose of this note is to indicate that this result is a consequence of an existence theorem for solutions of a certain self-adjoint elliptic partial differential equation.
Holomorphic vector fields and the first Chern class of a Hodge manifold