Description du projet
Les variétés d’Oka à l’étude
La théorie d’Oka est le domaine de l’analyse complexe traitant des problèmes globaux sur les variétés de Stein qui admettent des solutions analytiques en l’absence d’obstructions topologiques. Les variétés d’Oka sont une nouvelle classe de variétés complexes dont la caractéristique essentielle tient au fait qu’elles permettent une abondance de cartographies holomorphes à partir de variétés complexes affines. Les cartographies holomorphes sont importantes car elles apparaissent naturellement dans les problèmes physiques. Le projet HPDR, financé par l’UE, a pour objectif d’approfondir les propriétés des variétés d’Oka et leur application à une grande variété de problèmes en géométrie complexe.
Objectif
The aim is to develop an emerging field of complex analysis and geometry focused on holomorphic partial differential relations (HPDR). Such a relation of order r is given by a subset of the manifold of r-jets of holomorphic maps between a pair of complex manifolds, and the main question is when does a formal solution lead to an honest analytic solution. This complex analogue of Gromov’s h-principle is highly important but poorly understood. The project will focus on the following problems.
(A) Oka theory concerns the existence and approximation of holomorphic maps from Stein manifolds to complex manifolds, corresponding to HPDRs of order zero. The central notion of Oka theory is Oka manifold; this is a complex manifold such that the h-principle holds for maps from any Stein manifold into it. Recently developed techniques give a promise of major new developments on Oka manifolds and their applications to a variety of problems in complex geometry.
(B) Open first order HPDRs. Oka-theoretic methods will be applied in problems concerning holomorphic immersions and locally biholomorphic maps.
(C) First order HPDRs defined by analytic varieties in the jet bundle. Application of Oka-theoretic methods in holomorphic directed systems, with emphasis on complex contact manifolds and holomorphic Legendrian curves.
(D) Applications of Oka theory to minimal surfaces. Development of hyperbolicity theory for minimal surfaces. The Calabi-Yau problem for minimal surfaces in general Riemannian manifolds. Study of superminimal surfaces in self-dual Einstein four-manifolds via the Penrose-Bryant correspondence.
These closely interrelated topics embrace major open problems in three fields, with diverse applications.
Champ scientifique
Programme(s)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Thème(s)
Régime de financement
HORIZON-AG - HORIZON Action Grant Budget-BasedInstitution d’accueil
1000 Ljubljana
Slovénie