In
mathematics and especially
differential geometry, 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 ...
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 Ham ...
. The concept was first studied by
Jan Arnoldus Schouten and
David van Dantzig in 1930, and then introduced by
Erich Kähler
Erich Kähler (; 16 January 1906 – 31 May 2000) was a German mathematician with wide-ranging interests in geometry and mathematical physics, who laid important mathematical groundwork for algebraic geometry and for string theory.
Education an ...
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 a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
. 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 connection In mathematics, and in particular gauge theory and complex geometry, a Hermitian Yang–Mills connection (or Hermite-Einstein connection) is a Chern connection associated to an inner product on a holomorphic vector bundle over a Kähler manifold th ...
s, or special metrics such as
Kähler–Einstein metrics.
Every
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebraic ...
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 is a Kähler manifold.
Hodge theory is a central part of
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
, 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 equipped with an
integrable almost-complex structure ''J'' which is
compatible
Compatibility may refer to:
Computing
* Backward compatibility, in which newer devices can understand data generated by older devices
* Compatibility card, an expansion card for hardware emulation of another device
* Compatibility layer, compon ...
with the
symplectic form ω, 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 linear ...
:
on the
tangent space of ''X'' at each point is symmetric and
positive definite (and hence a Riemannian metric on ''X'').
Complex viewpoint
A Kähler manifold is a
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 c ...
''X'' with a
Hermitian metric ''h'' whose
associated 2-form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
''ω'' is
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
. In more detail, ''h'' 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 in each of its arguments, but a sesquilinear form allow ...
on the tangent space ''TX'' at each point of ''X'', and the 2-form ''ω'' is defined by
:
for tangent vectors ''u'' and ''v'' (where ''i'' is the complex number
). For a Kähler manifold ''X'', the Kähler form ''ω'' is a real closed
(1,1)-form. A Kähler manifold can also be viewed as a Riemannian manifold, with the Riemannian metric ''g'' defined by
:
Equivalently, a Kähler manifold ''X'' is a
Hermitian manifold
In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on e ...
of complex dimension ''n'' such that for every point ''p'' of ''X'', 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 derivati ...
coordinate chart In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout math ...
around ''p'' in which the metric agrees with the standard metric on C
''n'' to order 2 near ''p''. That is, if the chart takes ''p'' to 0 in C
''n'', and the metric is written in these coordinates as , then
:
for all ''a'', ''b'' in
Since the 2-form ''ω'' is closed, it determines an element in
de Rham cohomology , known as the Kähler class.
Riemannian viewpoint
A Kähler manifold is a
Riemannian manifold ''X'' of even dimension 2''n'' whose
holonomy group is contained in the
unitary group U(''n''). Equivalently, there is a complex structure ''J'' on the tangent space of ''X'' at each point (that is, a real
linear map from ''TX'' to itself with ) such that ''J'' preserves the metric ''g'' (meaning that ) and ''J'' is preserved by
parallel transport.
Kähler potential
A
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebraic ...
real-valued function ρ 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 ''ρ'' is called a Kähler potential for ''ω''.
Conversely, by the complex version of the
Poincaré lemma, 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 ''p'' in ''X'' there is a neighborhood ''U'' of ''p'' and a smooth real-valued function ''ρ'' on ''U'' such that
. Here ''ρ'' is called a local Kähler potential for ''ω''. 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 class. This is a consequence of the
-lemma from
Hodge theory.
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