Positive current - Alchetron, The Free Social Encyclopedia

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

η ↦ ∫ M η ∧ ρ

on smooth forms with compact support. This way, currents are considered as elements in the dual space to the space Λ c ∗ ( M ) of forms with compact support.

Now, let M be a complex manifold. The Hodge decomposition Λ i ( M ) = ⨁ p + q = i Λ p , q ( M ) is defined on currents, in a natural way, the (p,q)-currents being functionals on Λ c p , q ( 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.

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 Θ which is a (1,1)-part of an exact 2-current.

Note that the de Rham differential maps 3-currents to 2-currents, hence Θ is a differential of a 3-current; if Θ 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 π : M ↦ 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 π is a (1,1)-part of a boundary.

References

Positive current Wikipedia

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