Algebraic geometry and analytic geometry

In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties. Let X be a projective complex algebraic variety. Because X is a complex variety, its set of complex points X(C) can be given the structure of a compact complex analytic space. This analytic space is denoted Xan. Similarly, if is a sheaf on X, then there is a corresponding sheaf on Xan. This association of an analytic object to an algebraic one is a functor. The prototypical theorem relating X and Xan says that for any two coherent sheaves and on X, the natural homomorphism: is an isomorphism. Here is the structure sheaf of the algebraic variety X and is the structure sheaf of the analytic variety Xan. In other words, the category of coherent sheaves on the algebraic variety X is equivalent to the category of analytic coherent sheaves on the analytic variety Xan, and the equivalence is given on objects by mapping to . (Note in particular that itself is coherent, a result known as the Oka coherence theorem, and also, it was proved in “Faisceaux Algebriques Coherents” () that the structure sheaf of the algebraic variety is coherent.) Another important statement is as follows: For any coherent sheaf on an algebraic variety X the homomorphisms are isomorphisms for all qs. This means that the q-th cohomology group on X is isomorphic to the cohomology group on Xan. The theorem applies much more generally than stated above (see the formal statement below). It and its proof have many consequences, such as Chow's theorem, the Lefschetz principle and Kodaira vanishing theorem.

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Concepts associés (23)

Cours associés (21)

Séances de cours associées (84)

MATH-510: Algebraic geometry II - schemes and sheaves

The aim of this course is to learn the basics of the modern scheme theoretic language of algebraic geometry.

MATH-126: Geometry for architects II

Ce cours traite des 3 sujets suivants : la perspective, la géométrie descriptive, et une initiation à la géométrie projective.

MATH-657: Deformation Theory

We will study classical and modern deformation theory of schemes and coherent sheaves. Participants should have a solid background in scheme-theory, for example being familiar with the first 3 chapter

Algebraic geometry and analytic geometry

Kodaira embedding theorem

In mathematics, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex numbers, amongst compact Kähler manifolds. In effect it says precisely which complex manifolds are defined by homogeneous polynomials. Kunihiko Kodaira's result is that for a compact Kähler manifold M, with a Hodge metric, meaning that the cohomology class in degree 2 defined by the Kähler form ω is an integral cohomology class, there is a complex-analytic embedding of M into complex projective space of some high enough dimension N.

Coherent sheaf cohomology

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the existence of sections of line bundles or of more general coherent sheaves; such sections can be viewed as generalized functions. Cohomology provides computable tools for producing sections, or explaining why they do not exist. It also provides invariants to distinguish one algebraic variety from another.

Déplacez-vous dans les principes géométriques de l'architecture gothique, en mettant l'accent sur les techniques de courbure de surface et de stéréotomie.

Explore le concept de variété défini comme la fermeture de VCA et ses applications en géométrie algébrique.

Explore les espaces tangents en géométrie algébrique, les définissant comme des sous-espaces de dimensions finies de dérivations sur des variétés.