Convexity and Concavity from Complex Variables to Algebraic Geometry
Giuseppe Tomassini
TL;DR
The paper develops algebraic-geometric analogues of convexity, concavity, and envelopes, introducing foundational notions such as weakly regular functions, morphism extensions, and affine envelopes. A central result proves the existence of a basis of affine neighborhoods for affine subvarieties, and the concept of an affine envelope is developed with a path toward a general affine envelope for suitable varieties. It then connects convexity notions to semi-affineness via strongly and weakly $ ext{O}$-convex conditions, and formulates an algebraic Levi problem, while concavity is explored through ${ m Env}_d(K)$-envelopes and $d$-concavity. The discussion extends to line bundles, establishing criteria for convexity/concavity along fibers, finite generation of section algebras, and modifications, thereby laying groundwork for broader applications to vector bundles and cohomology in algebraic geometry.
Abstract
The topics of Convexity and Concavity and Envelopes are central in Complex Analysis and extensively investigated. The aim of this paper is to find a possible counterpart in Algebraic Geometry. The article presents preliminary results on this topic,