Complex manifolds with spanning sets of holomorphic vector fields

Question

I want to understand compact complex manifolds $(M^{2n}, J)$ with the following property: there exists a collection $\{X_i\}_{i=1}^L$ of holomorphic vector fields (sections of $(T^{1,0}_{\mathbb C} M)$) such that for all $p \in M$, $\{X_i(p)\}_{i=1}^L$ spans $(T^{1,0}_{\mathbb C} M)_p$. For instance, it is satisfied by complex tori, and while it has no nonvanishing vector fields, this property is satisfied by $\mathbb {CP}^n$. I have various questions:

1) How general a phenomenon is this, in the sense of, are there large classes of manifolds with this property?

2) It seems to be implied by the condition known as being strongly dominable: for all $p \in M$ there exists a holomorphic map $f : \mathbb C^n \to M$ such that $f(0) = p$ at $f$ is a local biholomorphism at $0$. In particular, if $M$ is strongly dominable then we can cover $M$ by finitely many open sets which are mapped to biholomorphically by the corresponding maps $f$ above. Then the pushforwards of the standard coordinate vector fields on each copy of $\mathbb C^n$ provide the collection $\{X_i\}_{i=1}^L$. How large is this class of manifolds? Is there a difference between this condition and the one described above?

3) Is there more rigidity of such manifolds if we also impose that $c_1 \leq 0$?

Answer 1

On any compact complex manifold, the set of all global holomorphic vector fields is a finite dimensional Lie algebra. They span the tangent space at every point just when each component of the manifold is a homogeneous for the action of its biholomorphism group.

Some points in the proof: By compactness, all holomorphic vector fields on any compact complex manifold are complete vector fields, so they generate 1-parameter complex subgroups of the biholomorphism group. You can find a complete proof that the biholomorphism group of a compact complex manifold is a finite dimensional complex Lie group acting holomorphically, with Lie algebra the set of all holomorphic vector fields, in Theorem 1.1 (and its proof), p. 77, Kobayashi, Transformation Groups in Differential Geometry, Springer, 1995. Kobayashi also proves that this group is finite if the manifold is Kobayashi hyperbolic or of general type. It is pretty clear that the Lie algebra of global holomorphic vector fields is finite dimensional, as it is the set of holomorphic sections of a coherent sheaf on a compact manifold. But in case the global holomorphic vector fields span every tangent space, all orbits of the biholomorphism group are open sets, clearly disjoint, so if the manifold is connected there is only one open set, i.e. the biholomorphism group action is transitive.

You can get a lot of information about homogeneous complex manifolds in the book by Dmitri Akhiezer, Lie group actions in complex analysis, Aspects of Mathematics, vol. E27, Friedr. Vieweg, Braunschweig and Wiesbaden, 1995, vii + 201 pp., ISBN 3-528-06420-X. In particular, Akhiezer proves that every compact complex homogeneous manifold is a bundle of parallelizable compact complex manifolds over a flag manifold. Clearly $c_1$ is represented by the cycle which is the zero locus of the wedge product of $n$ vector fields, $n$ equalling the dimension of the manifold $M$.

In particular, if the manifold $M$ is Kaehler with $c_1\le 0$, it is Calabi-Yau, and therefore a complex torus. But from the bundle structure, we get that the base can't be positive dimensional flag manifold, since we get positive Chern class there.

Every compact complex parallelizable manifold is $G/\Gamma$ for $\Gamma \subset G$ a cocompact lattice in a complex Lie group.

Complex homogeneous manifolds are strongly dominable, since the Lie group of biholomorphisms has an exponential map: pick a point $m_0 \in M$ and take each holomorphic vector field $v$ to the point $e^v m_0$. By completeness of the flow, this holomorphic map is defined on the Lie algebra of global vector fields. If we restrict to any subspace of those vector fields which spans $T_{m_0} M$, we get strong domination.

Kummer surfaces may well be strongly dominable without being homogeneous, because the relevant vector fields are not globally defined. Strong domination does not imply the existence of global nonzero vector fields. Kummer surfaces are not homogeneous, and indeed have finite symmetry groups, as do all K3 surfaces.