Abstract. In this paper we define, in two equivalent ways, the Schwarzian derivative of a map between complex manifolds equipped with complex projective connections. Also, a new, coordinate-free definition of complex projective connections is given. We show how the Schwarzian derivative is related to the projective structure of the manifolds, to projective linear transformations, and to complex geodesics.

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Univalence of holomorphic mappings is studied via two differential operators naturally associated with the mapping. The first operator is invariant under composition on the left with a projective linear mapping and the second operator is invariant under composition with holomorphic Euclidean transformations. The methods used are analogous to methods used by Osgood and Stowe in the case of conformal mappings.

Let M be a Kähler manifold supporting a positive line bundle L and Ω M a domain with C 2 , pseudoconvex boundary. If F is a coherent analytic sheaf on M , then for each x ∈ Ω, the stalk (F ⊗ L ⊗k ) x is generated by global sections over Ω, for k sufficiently large (Theorem 1). Also, for a suitable cover U of Ω, the weighted cohomology groups H p r,r (U , F ⊗ L ⊗k ) vanish for r and k sufficiently large (Theorem 2).