Let X be any algebraic variety over \({\mathbb{C}}\). Then there is in a natural way associated to X a complex analytic space \(X_{an}\). Also for every coherent algebraic sheaf F on X there is an associated coherent analytic sheaf \(F_{an}\) on \(X_{an}\), and one has comparison maps \(\alpha_ i: H^ i(X,F)\to H^ i(X_{an},F_{an})\) on cohomology. If X is complete, all \(\alpha_ i\) are isomorphisms. We study the case of a non- complete variety, generalizing results of Hartshorne. Our main result is the following theorem: Let X be an n-dimensional projective manifold over \({\mathbb{C}}\), \(Y\subset X\) a local complete intersection in X of codimension q. Assume that the normal bundle \(N_{Y/X}\) of Y in X is globally generated and k-ample. Then for every coherent algebraic sheaf F on \(X\setminus Y\) the comparison maps \(\alpha_ i: H^ i(X\setminus Y,F)\to H^ i((X\setminus Y)_{an},F_{an})\) are bijective (surjective) for \(i>q+k\) \((i=q+k)\). Furthermore all cohomology groups are finitedimensional for \(i\geq q+k.\) This theorem applies in particular to rational homogeneous projective spaces \(X=G/P\). The special case \(X={\mathbb{P}}^ n\) is due to Hartshorne.