Flexibility Properties in Complex Analysis and Affine Algebraic Geometry

Abstract

Abstract. In the last decades affine algebraic varieties and Stein manifolds with big (infinite-dimensional) automorphism groups have been intensively studied. Several notions expressing that the automorphisms group is big have been proposed. All of them imply that the manifold in question is an Oka-Forstnerič manifold. This important notion has also recently merged from the intensive studies around the homotopy principle in Complex Analysis. This homotopy principle, which goes back to the 1930's, has had an enormous impact on the development of the area of Several Complex Variables and the number of its applications is constantly growing. In this overview article we present 3 classes of properties: 1. density property, 2. flexibility 3. Oka-Forstnerič. For each class we give the relevant definitions, its most significant features and explain the known implications between all these properties. Many difficult mathematical problems could be solved by applying the developed theory, we indicate some of the most spectacular ones.

Interpolation by conformal minimal surfaces and directed holomorphic curves

“…This representation formula has greatly influenced the study of minimal surfaces in R n by providing powerful tools coming from Complex Analysis in one and several variables. In particular, Runge and Mergelyan theorems for open Riemann surfaces (see Bishop [18] and also [41,37]) and, more recently, the modern Oka Theory (we refer to the monograph by Forstnerič [27] and to the surveys by Lárusson [36], Forstnerič and Lárusson [29], Forstnerič [26], and Kutzschebauch [35]) have been exploited in order to develop a uniform approximation theory for conformal minimal surfaces in the Euclidean spaces which is analogous to the one of holomorphic functions in one complex variable and has found plenty of applications; see [12,15,7,16,20,11,10,28] and the references therein. In this paper we extend some of the methods invented for developing this approximation theory in order to provide also interpolation on closed discrete subsets of the underlying complex structure.…”

“…In this subsection we recall the notion of Oka manifold and state some of the properties of such manifolds which will be exploited in our argumentation. A comprehensive treatment of Oka theory can be found in [27]; for a briefer introduction to the topic we refer to [36,29,26,35]. Definition 2.5.…”

Surjective Holomorphic Maps onto Oka Manifolds

“…A major source of examples of algebraically elliptic manifolds are the algebraically flexible manifods; see e.g. [22,Definition 12]. An algebraic manifold X is said to be algebraically flexible of it admits finitely many algebraic vector fields V 1 , .…”

The algebraic density property for affine toric varieties

“…This has remarkable applications for geometric questions in complex analysis, we refer the reader to survey articles ✩ The first and second authors were partially supported by Schweizerische Nationalfonds Grant 200021-140235/1 and the third author was partially supported by FONDECYT Project 11121151 and by Dirección de Investigación, Talca University. [21,16,18] and the recent book [10]. For smooth affine algebraic varieties, the algebraic density property (ADP) was also introduced by Varolin.…”