In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
and especially
differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, a Kähler manifold is a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
with three mutually compatible structures: a
complex structure, a
Riemannian structure, and a
symplectic structure
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...
. The concept was first studied by
Jan Arnoldus Schouten and
David van Dantzig in 1930, and then introduced by
Erich Kähler in 1933. The terminology has been fixed by
André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was one of the most influential mathematicians of the twentieth century. His influence is du ...
. Kähler geometry refers to the study of Kähler manifolds, their geometry and topology, as well as the study of structures and constructions that can be performed on Kähler manifolds, such as the existence of special connections like
Hermitian Yang–Mills connections, or special metrics such as
Kähler–Einstein metric
In differential geometry, a Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be Kähler–Einstein if it admits a Kähler–Einstein metric. The ...
s.
Every
smooth complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
projective variety
In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, th ...
is a Kähler manifold.
Hodge theory
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 a central part of
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, proved using Kähler metrics.
Definitions
Since Kähler manifolds are equipped with several compatible structures, they can be described from different points of view:
Symplectic viewpoint
A Kähler manifold is a
symplectic manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sy ...
equipped with an
integrable almost-complex structure which is
compatible with the
symplectic form
In mathematics, a symplectic vector space is a vector space V over a field F (for example the real numbers \mathbb) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping \omega : V \times V \to F that is
; Bilinear: ...
, meaning that the
bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
:
on the
tangent space
In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...
of
at each point is symmetric and
positive definite (and hence a Riemannian metric on
).
Complex viewpoint
A Kähler manifold is a
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
with a
Hermitian metric whose
associated 2-form is
closed. In more detail,
gives a positive definite
Hermitian form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear map, linear in each of its arguments, but a sesquilinear f ...
on the tangent space
at each point of
, and the 2-form
is defined by
:
for tangent vectors
and
(where
is the complex number
). For a Kähler manifold
, the Kähler form
is a real closed
(1,1)-form. A Kähler manifold can also be viewed as a Riemannian manifold, with the Riemannian metric
defined by
:
Equivalently, a Kähler manifold
is a
Hermitian manifold of complex dimension
such that for every point
of
, there is a
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deri ...
coordinate chart around
in which the metric agrees with the standard metric on
to order 2 near
. That is, if the chart takes
to
in
, and the metric is written in these coordinates as
, then
:
for all
,
Since the 2-form
is closed, it determines an element in
de Rham cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...
, known as the Kähler class.
Riemannian viewpoint
A Kähler manifold is a
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
of even dimension
whose
holonomy group is contained in the
unitary group
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation
* Unitarity (physics)
* ''E''-unitary inverse semi ...
. Equivalently, there is a complex structure
on the tangent space of
at each point (that is, a real
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...
from
to itself with
) such that
preserves the metric
(meaning that
) and
is preserved by
parallel transport
In differential geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on ...
.
Kähler potential
A
smooth real-valued function
on a complex manifold is called
strictly plurisubharmonic if the real closed (1,1)-form
:
is positive, that is, a Kähler form. Here
are the
Dolbeault operators. The function
is called a Kähler potential for
.
Conversely, by the complex version of the
Poincaré lemma
In mathematics, the Poincaré lemma gives a sufficient condition for a closed differential form to be exact (while an exact form is necessarily closed). Precisely, it states that every closed ''p''-form on an open ball in R''n'' is exact for ''p'' ...
, known as the
local -lemma, every Kähler metric can locally be described in this way. That is, if
is a Kähler manifold, then for every point
in
there is a neighborhood
of
and a smooth real-valued function
on
such that
. Here
is called a local Kähler potential for
. There is no comparable way of describing a general Riemannian metric in terms of a single function.
Space of Kähler potentials
Whilst it is not always possible to describe a Kähler form ''globally'' using a single Kähler potential, it is possible to describe the ''difference'' of two Kähler forms this way, provided they are in the same
de Rham cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...
class. This is a consequence of the
-lemma from
Hodge theory
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 ...
.
Namely, if
is a compact Kähler manifold, then the cohomology class
is called a Kähler class. Any other representative of this class,
say, differs from
by
for some one-form
. The
-lemma further states that this exact form
may be written as
for a smooth function
. In the local discussion above, one takes the local Kähler class
on an open subset
, and by the Poincaré lemma any Kähler form will locally be cohomologous to zero. Thus the local Kähler potential
is the same
for
locally.
In general if