Laminations by Complex Manifolds with singularities

Project facts

Project promoter
Complutense University of Madrid
Project Number:
ES07-0023
Target groups
Researchers or scientists
Status:
Completed
Initial project cost:
€15,500
Final project cost:
€15,000
From EEA Grants:
€ 15,000
The project is carried out in:
Spain

Description

The project goal is to obtain a better understanding of several situations appearing in the study of complex dynamical systems. It is expected to obtain a partial answer on the rigidity of foliations of the complex projective plane and results on embedding laminations by complex manifolds. The project will intent to address these challenges developing new mathematical techniques based on Several Complex Variables Analysis and Foliations Theory. The results will enhance the research on Complex Dynamical Systems. The Complex Variables group of the University of Oslo, donor partner, has done several advances in the understanding of the complements of laminations in projective spaces, as well as, obtained very deep results on embedding of Riemann surfaces. The host institution group is made up by worldwide specialists on this theory which has enhanced the development of new results in foliation theory. In Spain there is currently no research group specialized in Several Complex Variables or Pluripotential Theory.

Summary of project results

The concept of lamination is a generalization of the concept of orbits of a vector field. These objects deserve proper attention as solutions of holomorphic vector fields. In fact, many interesting results on qualitative theory of real vector fields were achieved via the extended problem on complex variables. The goal was to obtain a better understanding of several situations appearing in the study of complex dynamical systems. It was expected to obtain a partíal answer on the rigidity of foliations of the complex projective plane and results on embedding laminations by complex manifolds. Short time after the arrival of the beneficiary to University of Oslo, Prof. Wold proposed the problem of solving dbar equation on foliations by complex manifolds. The result of partners are the following: they obtain that, given k, a finite regularity, there exists a number N(k) such that if v is a LN(k) valuated form, there exists a transversely Ck section u of LN(k) with d''u=v. They have been able to use this result in order to prove an embedding theorem into a complex projective plane of a big enough dimension. Several new questions arise from this result which they will face in the future. The first one would be to find an example of a foliated space with a positive bundle with for which a power of the bundle were necessary in order to obtain a continuous solution. The beneficiary devoted his last month of the stay in Oslo to the search of such a space. Another direction partners want to follow is to solve the equation when considering vector bundles in general, or for singular hermitian metrics in line bundles. The beneficiary also studied the harmonic flow induced by a harmonic current in a lamination by Riemann surfaces embedded in a compact complex homogeneous manifold as well as the generic rigidity of foliations in the complex projective plane. However, few remarkable results were obtained in these topics.

Summary of bilateral results

The Complex Variables group of the University of Oslo, has done several advances in the understanding of the complements of laminations in projective spaces, as well as, obtained very deep results on embedding of Riemann surfaces. The host institution group is made up by worldwide specialists on this theory which has enhanced the development of new results in foliation theory. In Spain there is currently no research group specialized in Several Complex Variables or Pluripotential Theory. This fellowship was be an opportunity to introduce these techniques in mathematical research in Spain. Several branches appear from the result obtained during this stay. These were be the core of a second stay supported by a NILS Grant which beneficiary started start on the 15th of October 2014. Moreover, partners were are preparing a proposal in order to organize a SpanishNorwegian Workshop in Several Complex Variables. Different researchers from both countries have shown their interest in that project. In addition, the beneficiary and partner contacted Julius Ross at University of Cambridge. He is a specialist in Kähler geometry and has recently shown interest in foliation theory. They started a plan of collaboration in order to develop a Quantization Theory in CRmanifolds.