Holomorphic vector fields and Kaehler manifolds

Abstract

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:

Fixed Points of the Actions on Compact Kähler Manifolds

“…As was shown in the proof of Theorem 1 in [6] the second spectral sequence degenerates. Hence for every integer m we have the inequality…”

“…3 below so that it holds true also for complete nonsingular algebraic varieties. Since F is compact Kahler, the left hand side has the natural (pure) (?- In particular h p ' q (X) = 0 for \p -g|>dim F 9 which is a special case of a theorem of Carrell and Lieberman [6] . Note that by Remark 1.…”

“…6] Let V = a(V) = V-V -IJV be the holomorphic vector field associated with a Killing vector field V on a compact Kahler manifold X as above. Following Carrell and Lieberman [6] we consider the Koszul complex 0 ->$!-» associated with V, where the differential is given by the contraction iy by V. Set K L X = Q X 1 in order to make the differential of degree +1.…”

“…Note that by Remark 1. 2 we can dispense with the arguments of [6] for the proof of Corollary 1. 7 which will be used in the proof of the theorem.…”

The generalized geometry, equivariant ∂̄∂-lemma, and torus actions

“…Historically, holomorphic vector fields on Kähler manifolds have been studied by many mathematicians. Among many other things, a famous result of Carrell and Lieberman [CL73] asserts if on a compact Kähler manifold M there exists a holomorphic vector field which has only isolated zeroes, then H p,q ∂ (M) = 0 unless p = q. Recently, assuming the holomorphic vector field is generated by a torus action, Carrell, Kaveh and Puppe [CKP04] gave a new proof of this result.…”