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In general, the existence of a holomorphic vector field with zeros on a compact complex manifold imposes restrictions on the topology and numerical characters of the manifold. For example, Howard has shown that a Hodge manifold admitting a holomorphic vector field X with zeros has no nontrivial holomorphic p-forms if p > dim zero(X) I-4, 5]. In this note we use a degeneracy criterion of Deligne to show that in fact much stronger restrictions hold. To prove Theorem 1 we employ the spectral sequences of hypercohomology of the Koszul complex associated to X. This gives an effective method for analysing the cohomology of M when M is compact Kaehler. We are able to generalize to the compact Kaehler case a result of Matsushima [5; Theorems 9.7 and 9.8] for vector fields on Hodge manifolds whose proof, in the Hodge case, depends on the use of Blanchard's Theorem on projective imbeddings [5; Theorem 9.4].Let/~1 denote the space of all homomorphic vector fields Z on M which annihilate all holomorphic one forms ~t on M in the sense that iz ct = 0 for all ct, where iz denotes the contraction operator associated to Z. We shall prove:

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Let ~ and 0 be smooth differential forms with compact support, of dimensions 2n and 2 n -1, respectively, defined on an open set W in C", and let q~ be any holomorphic function defined on W. We prove in this paper that the limits,~o I(ol a I~ =~ exist, where [q[ denotes the absolute value of q~, and relate them to the topology of the variety tp = 0 and its complement in W.The limits exist even when W is an open set in a paracompact and reduced complex space X of pure dimension n, although in this case the domains W(lq~l > 6) and W(lq~l = 6) are only semianalytic, in general, and integration on them means integration on their regular points, in the sense explained in [2] and [7]. No assumptions are needed about the singular sets of X or Y.We prove the existence of these limits in the last two sections of the paper (Section 6 and 7), supposing first that W is a manifold and q~ = 0 has only normal crossings, and using then resolution of singularities to handle the general case (el. Theorem 7

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