Descrizione del progetto
Varietà di Oka in fase di studio
La teoria di Oka rappresenta il campo dell’analisi complessa alle prese con problemi globali sulle varietà di Stein che ammettono soluzioni analitiche in assenza di ostacoli topologici. Le varietà di Oka sono una nuova classe di varietà complesse la cui caratteristica essenziale è che consentono grandi quantità di mappature olomorfiche da varietà affini complesse. Le mappature olomorfiche sono importanti, in quanto si verificano naturalmente nei problemi fisici. Il progetto HPDR, finanziato dall’UE, intende studiare ulteriormente le proprietà delle varietà di Oka e la loro applicazione a un’ampia varietà di problemi di geometria complessa.
Obiettivo
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.
Campo scientifico
Programma(i)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Argomento(i)
Meccanismo di finanziamento
HORIZON-AG - HORIZON Action Grant Budget-BasedIstituzione ospitante
1000 Ljubljana
Slovenia