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A [[Complex manifold|complex manifold]] which admits a [[Kähler metric|Kähler metric]]. Sometimes it is called a manifold of Kähler type, with the term  "Kähler manifold"  being reserved for manifolds actually equipped with a Kähler metric [[#References|[1]]]. Any submanifold of a Kähler manifold is a Kähler manifold. In particular, all projective complex algebraic varieties without singular points are Kähler manifolds, and, moreover, their Kähler metric is induced by the [[Fubini–Study metric|Fubini–Study metric]] on the complex projective space. Similarly, every submanifold in the affine space  $  \mathbf C  ^ {n} $(
in particular, every [[Stein manifold|Stein manifold]]) is a Kähler manifold. Other examples of Kähler manifolds are obtained if one considers the quotient space  $  M/ \Gamma $
of a Kähler manifold  $  M $
by a discrete group  $  \Gamma $
of analytic automorphisms preserving the Kähler metric. In particular, every [[Complex torus|complex torus]] is a Kähler manifold. Any one-dimensional complex manifold is Kählerian.

The theory of harmonic forms on a compact Kähler manifold  $  M $
yields the following properties of the de Rham and Dolbeault cohomology groups on  $  M $ (see [[#References|[1]]], [[#References|[2]]], and also [[#References|[5]]], where these properties were first established for projective algebraic varieties):

$$ 
H  ^ {r} ( M, \mathbf C )  \cong \ 
\bigoplus _ {p + q = r }
H ^ {p, q } ( M),
$$

$$ 
H ^ {p, q } ( M)  \cong  H ^ {q, p } ( M),
$$

$$ 
 \mathop{\rm dim}  H ^ {2r + 1 } ( M, \mathbf C ) \  \textrm{ is even } ,
$$

$$ 
H  ^ {2r} ( M, \mathbf C )  \neq  0 \  \textrm{ if }  r
= 1, \dots, \frac{1}{2} \mathop{\rm dim _ {\mathbf C }}   M.
$$

Holomorphic forms on a compact Kähler manifold are closed. In particular,

$$ 
H  ^ {1} ( M, \mathbf C )  \cong \ 
A  ^ {1} \oplus \overline{A}\; {}  ^ {1} ,
$$

where  $  A  ^ {1} $
is the space of all holomorphic  $  1 $-forms on  $  M $. 
If  $   \mathop{\rm dim} _ {\mathbf C }   M = 1 $, 
the number

$$ 
 \mathop{\rm dim}  A  ^ {1}  = \ 
{
\frac{1}{2}
 }  \mathop{\rm dim}  H  ^ {1} ( M, \mathbf C )
$$

is the genus of  $  M $
as a compact Riemann surface (cf. also [[Genus of a surface|Genus of a surface]]; [[Riemann surface|Riemann surface]]). The above properties provide the basis for the construction of examples of non-Kähler compact manifolds, the simplest of which is the Hopf surface, which is diffeomorphic to  $  S  ^ {1} \times S  ^ {3} $.

A Kähler manifold  $  M $
is called a [[Hodge variety|Hodge variety]] if its Kähler metric is a Hodge metric. Any projective algebraic variety without singular points is a Hodge variety relative to the metric induced by the Fubini–Study metric. Conversely, any compact complex manifold  $  M $
equipped with a Kählerian Hodge metric  $  \eta $
can be biholomorphically imbedded in a complex projective space in such a way that the metric induced on  $  M $
by the Fubini–Study metric may be expressed as  $  k \eta $
for some natural number  $  k $ (Kodaira's projective imbedding theorem [[#References|[1]]], [[#References|[3]]]). Thus, a compact complex manifold  $  M $
is isomorphic to a projective algebraic variety if and only if it is a Hodge variety. Another form of this criterion: A compact complex manifold  $  M $
is a projective algebraic variety if and only if it admits a [[Negative vector bundle|negative vector bundle]]. Kodaira's theorem can be generalized to complex spaces (see [[#References|[4]]], [[#References|[6]]]). Compact Kähler manifolds which are not Hodge varieties may be found among the two-dimensional complex tori. For example, this is the case for the torus  $  \mathbf C  ^ {2} / \Gamma $, 
where  $  \Gamma $ is the lattice spanned by the vectors  $  ( 1, 0) $, 
$  ( 0, 1) $, 
$  ( \sqrt - 2 , \sqrt - 3 ) $, 
$  ( \sqrt - 5 , \sqrt - 7 ) $(
see [[#References|[1]]], [[#References|[3]]]). Another necessary and sufficient condition for an  $  n $-dimensional compact Kähler manifold  $  M $
to be projective is that there exist  $  n $
algebraically independent meromorphic functions on  $  M $[[#References|[8]]].

Any non-compact complete Kähler manifold with positive [[Sectional curvature|sectional curvature]] is a [[Stein manifold|Stein manifold]]. The same is true of any simply-connected complete Kähler manifold of non-positive sectional curvature [[#References|[7]]].

====References====
<table><tr><td valign="top">[1]</td> <td valign="top">  R.O. Wells jr.,   "Differential analysis on complex manifolds" , Springer  (1980)</td></tr><tr><td valign="top">[2]</td> <td valign="top">  A. Weil,   "Introduction à l'Aeetude des variétés kahlériennes" , Hermann  (1958)</td></tr><tr><td valign="top">[3]</td> <td valign="top">  S.S. Chern,   "Complex manifolds without potential theory" , Springer  (1979)</td></tr><tr><td valign="top">[4]</td> <td valign="top">  R.C. Gunning,   H. Rossi,   "Analytic functions of several complex variables" , Prentice-Hall  (1965)</td></tr><tr><td valign="top">[5]</td> <td valign="top">  W.V.D. Hodge,   "The theory and application of harmonic integrals" , Cambridge Univ. Press  (1952)</td></tr><tr><td valign="top">[6]</td> <td valign="top">  H. Grauert,   "Ueber Modifikationen und exzeptionelle analytische Mengen"  ''Math. Ann.'' , '''146'''  (1962)  pp. 331–368</td></tr><tr><td valign="top">[7]</td> <td valign="top">  R.E. Greene,   H. Wu,   "A theorem in complex geometric function theory" , ''Value-distribution Theory'' , '''A''' , M. Dekker  (1974)  pp. 145–167</td></tr><tr><td valign="top">[8]</td> <td valign="top">  B.G. Moishezon,   "On $n$-dimensional compact complex manifolds possessing $n$ algebraically independent meromorphic functions I - III"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''30'''  (1966)  pp. 133–174; 345–386; 621–656  (In Russian)</td></tr></table>

====Comments====
A complex compact surface is Kähler if and only if its first [[Betti number|Betti number]] is even.

Let  $  M $
be complex compact manifold of (complex) dimension  $  m $. 
Let  $  g $
be a Kähler metric on  $  M $
and  $  \omega $
its associated  $  ( 1, 1) $-form. (If  $  g = \sum g _ {\alpha \overline \beta \; }   dz  ^  \alpha  \otimes d \overline{z}\; {}  ^  \beta  $
in local coordinates, then  $  \omega = \sum g _ {\alpha \overline \beta \; }   dz  ^  \alpha  \wedge d \overline{z}\; {}  ^  \beta  $; 
giving  $  \omega $
amounts to the same thing as giving  $  g $.) 
Denote by  $   \mathop{\rm Ric} _ {g} $
the Ricci curvature tensor of  $  ( M , g) $
and let  $  \gamma _  \omega  $
be the associated  $  ( 1, 1) $
form to  $   \mathop{\rm Ric} _ {g} $. 
Then, [[#References|[a1]]], the cohomology class of  $  \gamma _  \omega  $
is  $  2 \pi $
times the first Chern class of  $  M $: 
$  [ \gamma _  \omega  ] = 2 \pi c _ {1} ( M) $.

Now let  $  \widetilde \gamma   $
be a given  $  ( 1, 1) $
form on  $  ( M, \omega ) $
such that  $  [ \gamma _  \omega  ] = 2 \pi c _ {1} ( M) $. 
Then the Calabi conjecture [[#References|[a2]]], [[#References|[a3]]] states that there is precisely one Kähler form  $  \widetilde \omega   $
in the same class as  $  \omega $
such that  $  \widetilde \gamma   = \gamma _ {\widetilde \omega   }  $. 
E. Calabi proved uniqueness.

A related conjecture is: Let  $  M $
be a complex compact manifold which admits a negative first Chern form. Then there exists a unique Kähler form  $  \omega $
such that  $  \gamma _  \omega  = - \omega $.

Such a metric is called an Einstein–Kähler metric. (A Riemannian metric  $  g $
is called an Einstein metric if  $   \mathop{\rm Ric} _ {g} = kg $
for some constant  $  k $.)

The Calabi conjecture (existence) was proved by S.T. Yau [[#References|[a4]]], [[#References|[a5]]] and the second conjecture was proved by both T. Aubin and Yau, [[#References|[a5]]], [[#References|[a6]]].

A Kähler manifold is called a Calabi–Yau manifold if its first Chern class vanishes (so that there exists a Ricci flat on  $  M $).

These manifolds are of great current interest in theoretical physics as prime candidates for the  "missing six dimensions"  for a  $  10 $-dimensional super-string theory; i.e. the carrier space of this  $  10 $-dimensional space would be  $  M  ^ {4} \times K $, 
where  $  M  ^ {4} $
is four-dimensional Minkowski space and  $  K $
is a six-dimensional ( "small" ) compact Calabi–Yau manifold.

The Calabi–Yau manifolds with non-vanishing [[Euler characteristic|Euler number]] provide the only known  "compactifications" $  M  ^ {4} \times K $
consistent with the super-string equations of motion. A selection of references dealing with these matters is [[#References|[a8]]]–[[#References|[a12]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  S.S. Chern,   "Characteristic classes of Hermitian manifolds"  ''Ann. of Math.'' , '''47'''  (1946)  pp. 85–121</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  E. Calabi,   "The space of Kähler metrics" , ''Proc. Internat. Congress Mathematicians (Amsterdam, 1954)'' , '''2''' , Noordhoff  (1954)  pp. 206–207</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  E. Calabi,   "On Kähler manifolds with vanishing canonical class"  R.H. Fox (ed.)  D.C. Spencer (ed.)  A.W. Tucker (ed.) , ''Algebraic geometry and topology (Symp. in honor of S. Lefschetz)'' , Princeton Univ. Press  (1955)  pp. 78–89</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  S.-T. Yau,   "On Calabi's conjecture and some new results in algebraic geometry"  ''Proc. Nat. Acad. Sci. USA'' , '''74'''  (1977)  pp. 1798–1799</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  S.-T. Yau,   "On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation I"  ''Comm. Pure Appl. Math.'' , '''31'''  (1978)  pp. 339–411</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  T. Aubin,   "Equations du type de Monge–Ampère sur les variétés kähleriennes compactes"  ''C.R. Acad. Sci. Paris'' , '''283'''  (1976)  pp. 119–121</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  J.-P. Bourguignon,   "Premières formes de Chern des variétés kähleriennes compactes" , ''Sem. Bourbaki 1977/1978'' , '''Exp. 507''' , Springer  (1979)  pp. 1–21</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  T. Huebsch,   "Calabi–Yau manifolds—motivations and constructions"  ''Comm. Math. Phys.'' , '''108'''  (1987)  pp. 291–318</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  M.B. Green,   J.H. Schwarz,   E. Witten,   "Superstring theory" , '''2. Loop amplitudes, anomalies &amp; phenomenology''' , Cambridge Univ. Press  (1987)  pp. 438ff</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  P. Candelas,   G. Horowitz,   A. Strominger,   E. Witten,   "Vacuum configurations for superstrings"  ''Nuclear Physics'' , '''B258'''  (1958)  pp. 46–74</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  B. de Wit,   D.J. Smit,   N.D. Hari-Dass,   "Residual supersymmetry of compactified $d = 10$ supergravity"  ''Nuclear Physics'' , '''B283'''  (1987)  pp. 165–191</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  D. Gepner,   "Exactly solvable string compactifications on manifolds of $\operatorname{SU} ( N )$ holonomy"  ''Physics Letters'' , '''B199'''  (1987)  pp. 380–388</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  "Première classe de Chern et courbure de Ricci: preuve de la conjecture de Calabi" , ''Sem. Palaiseau 1978'' , Soc. Math. France  (1978)</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  Y.-T. Siu,   "Lectures on Hermitean–Einstein metrics for stable bundles and Kähler–Einstein metrics" , Birkhäuser  (1987)</td></tr></table>