On the holomorphic length of a complex projective variety

We wish to thank R. Piergallini who called [2] to our attention.     

A KAM phenomenon for singular holomorphic vector fields

Let X be a germ of holomorphic vector field at the origin of Cⁿ and vanishing there. We assume that X is a good perturbation of a “nondegenerate” singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection of a polydisc with an analytic set of the form “resonant monomials = constants”. Such a biholomorphism conjugates the restriction of X to one of its invariant varieties to the restriction of a linear diagonal vector field to a toric variety. Moreover, we show that the set of “frequencies” defining the invariant sets is of positive measure.     

Construction of invariant curves for singular holomorphic vector fields

Camacho and Sad proved the existence of invariant analytic curves for germs of singular holomorphic foliations over a two dimensional complex analytic variety . Their proof is only of existential nature. Here we provide a simple constructive proof by giving criteria to choose a singular point at each blowing-up that follows an analytic invariant curve.

     

Laplacians on the holomorphic tangent bundle of a Kaehler manifold

Let M be a connected complex manifold endowed with a Hermitian metric g. In this paper, the complex horizontal and vertical Laplacians associated with the induced Hermitian metric 〈·, ·〉 on the holomorphic tangent bundle T ^1,0 M of M are defined, and their explicit expressions are obtained. Using the complex horizontal and vertical Laplacians associated with the Hermitian metric 〈·, ·〉 on T ^1,0 M, we obtain a vanishing theorem of holomorphic horizontal p forms which are compactly supported in T ^1,0 M under the condition that g is a Kaehler metric on M.     

Characterization of a monodromic singular point of a planar vector field

The Newton diagram and the lowest-degree quasi-homogeneous terms of an analytic planar vector field allow us to determine whether an isolated singular point of the vector field is monodromic or has a characteristic trajectory.     

On complete holomorphic vector fields

We show that, on holomorphic manifolds that have a plurisubharmonic exhaustion function and that do not carry nonconstant bounded plurisubharmonic functions (e.g. ), holomorphic vector fields that are complete in positive time are complete in complex time.

     

On singular fibres of Lagrangian fibrations over holomorphic symplectic manifolds

We classify singular fibres over general points of the discriminant locus of projective Lagrangian fibrations over 4-dimensional holomorphic symplectic manifolds. The singular fibre F is the following either one: F is isomorphic to the product of an elliptic curve and a Kodaira singular fibre up to finite unramified covering or F is a normal crossing variety consisting of several copies of a minimal elliptic ruled surface of which the dual graph is Dynkin diagram of type or . Moreover, we show all types of the above singular fibres actually occur. Received: 10 March 2000 / Revised version: 29 September 2000 / Published online: 24 September 2001