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positive current : ウィキペディア英語版
positive current
In mathematics, more particularly in complex geometry,
algebraic geometry and complex analysis, a positive current
is a positive (''n-p'',''n-p'')-form over an ''n''-dimensional complex manifold,
taking values in distributions.
For a formal definition, consider a manifold ''M''.
Currents on ''M'' are (by definition)
differential forms with coefficients in distributions. ; integrating
over ''M'', we may consider currents as "currents of integration",
that is, functionals
:\eta \mapsto \int_M \eta\wedge \rho
on smooth forms with compact support. This way, currents
are considered as elements in the dual space to the space
\Lambda_c^
*(M) of forms with compact support.
Now, let ''M'' be a complex manifold.
The Hodge decomposition \Lambda^i(M)=\bigoplus_\Lambda^(M)
is defined on currents, in a natural way, the ''(p,q)''-currents being
functionals on \Lambda_c^(M).
A positive current is defined as a real current
of Hodge type ''(p,p)'', taking non-negative values on all positive
''(p,p)''-forms.
== Characterization of Kähler manifolds ==

Using the Hahn–Banach theorem, Harvey and Lawson proved the following criterion of existence of Kähler metrics.〔R. Harvey and H. B. Lawson, "An intrinsic characterisation of Kahler manifolds," Invent. Math 74 (1983) 169-198.〕
Theorem: Let ''M'' be a compact complex manifold. Then ''M'' does not admit a Kähler structure if and only if ''M'' admits a non-zero positive (1,1)-current \Theta which is a (1,1)-part of an exact 2-current.
Note that the de Rham differential maps 3-currents to 2-currents, hence \Theta is a differential of a 3-current; if \Theta is a current of integration of a complex curve, this means that this curve is a (1,1)-part of a boundary.
When ''M'' admits a surjective map \pi:\; M \mapsto X to a Kähler manifold with 1-dimensional fibers, this theorem leads to the following result of complex algebraic geometry.
Corollary: In this situation, ''M'' is non-Kähler if and only if the homology class of a generic fiber of \pi is a (1,1)-part of a boundary.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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