In mathematics, more particularly in
complex geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
,
algebraic geometry and
complex analysis, a positive current
is a
positive
Positive is a property of positivity and may refer to:
Mathematics and science
* Positive formula, a logical formula not containing negation
* Positive number, a number that is greater than 0
* Plus sign, the sign "+" used to indicate a posi ...
(''n-p'',''n-p'')-form over an ''n''-dimensional
complex manifold,
taking values in distributions.
For a formal definition, consider a manifold ''M''.
Current
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
s 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
:
on smooth forms with compact support. This way, currents
are considered as elements in the dual space to the space
of forms with compact support.
Now, let ''M'' be a complex manifold.
The
Hodge decomposition
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coho ...
is defined on currents, in a natural way, the ''(p,q)''-currents being
functionals on
.
A positive current is defined as a real
current
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
of Hodge type ''(p,p)'', taking non-negative values on all
positive
Positive is a property of positivity and may refer to:
Mathematics and science
* Positive formula, a logical formula not containing negation
* Positive number, a number that is greater than 0
* Plus sign, the sign "+" used to indicate a posi ...
''(p,p)''-forms.
Characterization of
Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ...
s
Using the
Hahn–Banach theorem
The Hahn–Banach theorem is a central tool in functional analysis.
It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
, 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 Kähler may refer to:
;People
*Alexander Kähler (born 1960), German television journalist
*Birgit Kähler (born 1970), German high jumper
*Erich Kähler (1906–2000), German mathematician
*Heinz Kähler (1905–1974), German art historian and arc ...
if and only if ''M'' admits a non-zero positive (1,1)-current
which is a (1,1)-part of an exact 2-current.
Note that the
de Rham differential maps 3-currents to 2-currents, hence
is a differential of a 3-current; if
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
to a
Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ...
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
Homology may refer to:
Sciences
Biology
*Homology (biology), any characteristic of biological organisms that is derived from a common ancestor
*Sequence homology, biological homology between DNA, RNA, or protein sequences
*Homologous chromo ...
of a generic fiber of
is a (1,1)-part of a boundary.
Notes
References
*
P. Griffiths and
J. Harris (1978), ''Principles of Algebraic Geometry'', Wiley. {{isbn, 0-471-32792-1
*
J.-P. Demailly,
$L^2$ vanishing theorems for positive line bundles and adjunction theory, Lecture Notes of a CIME course on "Transcendental Methods of Algebraic Geometry" (Cetraro, Italy, July 1994)'
Complex manifolds
Several complex variables