**Moishezon manifold**
**Definition**
A Moishezon manifold is a complex manifold whose field of meromorphic functions has transcendence degree equal to its complex dimension. Equivalently, it is a compact complex manifold that is bimeromorphic to a projective algebraic variety.
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# Moishezon Manifold
A **Moishezon manifold** is a fundamental concept in complex geometry, representing a class of complex manifolds that are closely related to projective algebraic varieties. Introduced by Boris Moishezon in the 1960s, these manifolds serve as a bridge between the analytic and algebraic categories, providing insight into the structure of complex manifolds that are „almost” algebraic. The notion of Moishezon manifolds plays a significant role in the classification theory of complex manifolds and in the study of algebraic approximations of complex spaces.
## Overview
In complex geometry, a central problem is to understand the relationship between complex manifolds and algebraic varieties. While every projective algebraic variety naturally carries the structure of a complex manifold, the converse is not true: not all complex manifolds arise from algebraic varieties. Moishezon manifolds are those complex manifolds that, although not necessarily algebraic themselves, share many algebraic properties and can be transformed into projective varieties through bimeromorphic modifications.
Formally, a compact complex manifold (X) of complex dimension (n) is called a Moishezon manifold if the transcendence degree of its field of meromorphic functions over (mathbb{C}) is equal to (n). This condition implies that (X) admits „enough” meromorphic functions to separate points and approximate algebraic behavior.
## Historical Context
The concept was introduced by Boris Moishezon in his work on the classification of complex surfaces and higher-dimensional complex manifolds. Moishezon’s motivation was to characterize those complex manifolds that are „close” to being projective algebraic varieties, and to understand the extent to which complex analytic methods can be used to study algebraic geometry.
Moishezon manifolds emerged as a natural generalization of projective varieties, and their study has led to important developments in complex geometry, including the theory of modifications, the use of meromorphic functions, and the classification of complex surfaces.
## Definition and Characterizations
### Transcendence Degree of Meromorphic Functions
Let (X) be a compact complex manifold of complex dimension (n). The field of meromorphic functions on (X), denoted (mathcal{M}(X)), consists of all meromorphic functions defined on (X). The transcendence degree of (mathcal{M}(X)) over (mathbb{C}) is the maximal number of algebraically independent meromorphic functions on (X).
– **Definition:** (X) is a Moishezon manifold if
[
text{tr.deg}_{mathbb{C}} mathcal{M}(X) = n.
]
This means there exist (n) meromorphic functions on (X) that are algebraically independent over (mathbb{C}).
### Bimeromorphic Equivalence to Projective Varieties
An equivalent characterization is that a Moishezon manifold is bimeromorphic to a projective algebraic variety. A bimeromorphic map is a holomorphic map that is an isomorphism outside subsets of codimension at least two.
– **Theorem:** A compact complex manifold (X) is Moishezon if and only if there exists a projective algebraic variety (Y) and a bimeromorphic map (f: X dashrightarrow Y).
This equivalence highlights the algebraic nature of Moishezon manifolds and their role as analytic models of algebraic varieties.
### Relation to Algebraic Dimension
The algebraic dimension (a(X)) of a compact complex manifold (X) is defined as the transcendence degree of (mathcal{M}(X)) over (mathbb{C}). Thus, Moishezon manifolds are precisely those with maximal algebraic dimension:
[
a(X) = dim_{mathbb{C}} X.
]
## Properties of Moishezon Manifolds
### Algebraic Approximation
Moishezon manifolds can be approximated by projective varieties in the sense that they admit modifications (blow-ups and blow-downs) that transform them into projective varieties. This property is crucial in the classification theory of complex manifolds and in the study of algebraic approximations.
### Stability under Modifications
The class of Moishezon manifolds is stable under bimeromorphic modifications. If (X) is Moishezon and (Y) is obtained from (X) by a finite sequence of blow-ups or blow-downs, then (Y) is also Moishezon.
### Relation to Kähler Manifolds
While all projective varieties are Kähler manifolds, Moishezon manifolds need not be Kähler. However, a celebrated result by Moishezon states that a Moishezon manifold admits a Kähler metric if and only if it is projective algebraic. This result provides a criterion to distinguish projective varieties among Moishezon manifolds.
### Examples
– **Projective Varieties:** Every projective algebraic variety is a Moishezon manifold.
– **Non-Kähler Moishezon Manifolds:** There exist Moishezon manifolds that are not Kähler, constructed via complex analytic methods involving modifications of non-Kähler manifolds.
## Construction and Examples
### Blow-ups and Modifications
One of the main tools in constructing Moishezon manifolds is the process of blowing up subvarieties. Starting from a projective variety, blowing up along suitable centers produces new Moishezon manifolds. Conversely, certain complex manifolds can be modified by blow-ups to become projective.
### Surfaces
In complex dimension two, Moishezon manifolds coincide with algebraic surfaces up to bimeromorphic equivalence. The classification of complex surfaces shows that every Moishezon surface is bimeromorphic to a projective surface.
### Higher Dimensions
In dimensions three and higher, the situation is more complicated. Moishezon manifolds form a strictly larger class than projective varieties, and their classification remains an active area of research.
## Relation to Other Classes of Complex Manifolds
### Kähler Manifolds
As noted, Moishezon manifolds are not necessarily Kähler. The existence of a Kähler metric on a Moishezon manifold characterizes projective varieties within this class.
### Fujiki Class (mathcal{C})
The Fujiki class (mathcal{C}) consists of compact complex manifolds bimeromorphic to Kähler manifolds. Moishezon manifolds are contained in this class, but the inclusion is strict.
### Algebraic Dimension and Kodaira Dimension
Moishezon manifolds have maximal algebraic dimension, but their Kodaira dimension can vary. The Kodaira dimension measures the growth of pluricanonical sections and is an important invariant in classification theory.
## Applications and Significance
### Classification Theory
Moishezon manifolds provide a framework for understanding the classification of complex manifolds, especially in dimensions two and three. They serve as a testing ground for conjectures relating algebraic and analytic geometry.
### Algebraic Approximation Problems
The study of Moishezon manifolds is closely related to the problem of approximating complex manifolds by algebraic varieties. This has implications in deformation theory and moduli problems.
### Complex Surface Theory
In the theory of complex surfaces, Moishezon manifolds correspond to algebraic surfaces up to bimeromorphic equivalence, making them central to the Enriques–Kodaira classification.
## Technical Tools and Methods
### Meromorphic Function Fields
The field of meromorphic functions on a complex manifold is a key object in defining and studying Moishezon manifolds. Techniques from algebraic geometry, such as transcendence degree and algebraic independence, are applied in the analytic setting.
### Bimeromorphic Geometry
Bimeromorphic maps and modifications are essential tools in the study of Moishezon manifolds. They allow the comparison of complex manifolds up to birational equivalence, facilitating the transfer of algebraic properties.
### Hodge Theory and Kähler Metrics
Although Moishezon manifolds need not be Kähler, Hodge theory and the existence of Kähler metrics play a role in distinguishing projective varieties within the class of Moishezon manifolds.
## Open Problems and Research Directions
### Characterization of Moishezon Manifolds
While the definition of Moishezon manifolds is well-established, characterizing them in terms of geometric or topological properties remains an area of active research.
### Extension to Singular Spaces
Generalizing the notion of Moishezon manifolds to complex spaces with singularities is an ongoing topic, with implications for the minimal model program and complex analytic geometry.
### Relation to Other Geometric Structures
Exploring the connections between Moishezon manifolds and other geometric structures, such as symplectic or complex contact structures, is a developing field.
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## Summary
Moishezon manifolds occupy a central position in complex geometry as complex manifolds that are analytically close to projective algebraic varieties. Defined by the maximal transcendence degree of their meromorphic function fields, they provide a rich interplay between analytic and algebraic methods. Their study has deepened the understanding of complex manifolds, birational geometry, and the classification of complex spaces.
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**Meta Description:**
A Moishezon manifold is a compact complex manifold whose field of meromorphic functions has maximal transcendence degree, making it bimeromorphic to a projective algebraic variety. This article explores their definition, properties, and significance in complex geometry.