In
mathematics, a Moishezon manifold is a compact
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a ...
such that the
field of
meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. ...
s on each component has
transcendence degree equal the
complex dimension In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety. These are spaces in which the local neighborhoods of points (or of non-singular points in the case of a variety) are modeled on ...
of the component:
:
Complex
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex number
...
have this property, but the converse is not true:
Hironaka's example In geometry, Hironaka's example is a non-Kähler complex manifold that is a deformation of Kähler manifolds found by . Hironaka's example can be used to show that several other plausible statements holding for smooth varieties of dimension at mo ...
gives a smooth 3-dimensional Moishezon manifold that is not an algebraic variety or
scheme. showed that a Moishezon manifold is a
projective algebraic variety if and only if it admits a
Kähler metric. showed that any Moishezon manifold carries an
algebraic space structure; more precisely, the category of Moishezon spaces (similar to Moishezon manifolds, but are allowed to have singularities) is equivalent with the category of algebraic spaces that are proper over .
References
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Algebraic geometry
Analytic geometry
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