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:
Let G be a semi-simple algebraic group over C, B a Borel subgroup of G and T a maximal torus in B. A beautiful unpublished result of Dale Peterson says that if G is simply laced, then every rationally smooth point of a Schubert variety X in G/B is nonsingular in X. The purpose of this paper is to generalize this result to arbitrary T -stable subvarieties of G/B, the only restriction being that G contains no G 2 factors. A key idea in Peterson's proof is to deform the tangent space Ty(X) to X at a nonsingular T -fixed point y along the orbit of y under a root subgroup in B, which is open in a T -invariant curve C (we say a T -curve) in X. In more generality, if a T -variety X is nonsingular along the open T -orbit in a T -curve C and x ∈ C T , we may consider the limit τ C (X, x) of the tangent spaces Tz(X) as z approaches x along C. We call τ C (X, x) the Peterson translate of X at x along C. Peterson showed that a Schubert variety X in G/B, where G is semi-simple, is nonsingular at x ∈ X T as long as all τ C (X, x) coincide for all such good T -curves..Our first result generalizes this theorem to any irreducible T -variety X, provided the fixed point x is attractive under much weaker hypotheses. We then prove that if X is a T -variety in G/B where G contains no G 2 factors, then every Peterson translate τ C (X, x) is contained in the linear span Θx(X) of the reduced tangent cone to X at x. (This fails when G = G 2 .) Combining these two results leads to our characterization of the smooth T -fixed points of such X. In particular, we show that if G is simply laced, then X is nonsingular at a T -fixed point x if and only if it is rationally smooth at x and x lies on at least two good T -curves Peterson's ADE result is an immediate consequence of this. In addition, we obtain a uniform description of the nonsingular T -fixed points of a Schubert variety in G/B modulo the G 2 restriction. In particular, a Schubert variety X in such a G/B is nonsingular if and only if all the reduced tangent cones of X are linear.Finally, we also obtain versions of the above results for all algebraic homogeneous spaces G/P modulo the G 2 restrictions.
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