Kähler manifold

A complex manifold which admits a 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 [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 on the complex projective space. Similarly, every submanifold in the affine space $ \mathbf C ^ {n} $ (in particular, every 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 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 [1], [2], and also [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; 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 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 [1], [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. Kodaira's theorem can be generalized to complex spaces (see [4], [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 [1], [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 $ [8].

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

Comments

A complex compact surface is Kähler if and only if its first 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, [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 [a2], [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 [a4], [a5] and the second conjecture was proved by both T. Aubin and Yau, [a5], [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 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 [a8]–[a12].

References

[1]	R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)
[2]	A. Weil, "Introduction à l'étude des variétés kahlériennes" , Hermann (1958)
[3]	S.S. Chern, "Complex manifolds without potential theory" , Springer (1979)
[4]	R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965)
[5]	W.V.D. Hodge, "The theory and application of harmonic integrals" , Cambridge Univ. Press (1952)
[6]	H. Grauert, "Ueber Modifikationen und exzeptionelle analytische Mengen" Math. Ann. , 146 (1962) pp. 331–368 Zbl 0173.33004
[7]	R.E. Greene, H. Wu, "A theorem in complex geometric function theory" , Value-distribution Theory , A , M. Dekker (1974) pp. 145–167
[8]	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)
[a1]	S.S. Chern, "Characteristic classes of Hermitian manifolds" Ann. of Math. , 47 (1946) pp. 85–121
[a2]	E. Calabi, "The space of Kähler metrics" , Proc. Internat. Congress Mathematicians (Amsterdam, 1954) , 2 , Noordhoff (1954) pp. 206–207
[a3]	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
[a4]	S.-T. Yau, "On Calabi's conjecture and some new results in algebraic geometry" Proc. Nat. Acad. Sci. USA , 74 (1977) pp. 1798–1799
[a5]	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
[a6]	T. Aubin, "Équations du type de Monge–Ampère sur les variétés kähleriennes compactes" C.R. Acad. Sci. Paris , 283 (1976) pp. 119–121
[a7]	J.-P. Bourguignon, "Premières formes de Chern des variétés kähleriennes compactes" , Sém. Bourbaki 1977/1978 , Exp. 507 , Springer (1979) pp. 1–21
[a8]	T. Huebsch, "Calabi–Yau manifolds—motivations and constructions" Comm. Math. Phys. , 108 (1987) pp. 291–318
[a9]	M.B. Green, J.H. Schwarz, E. Witten, "Superstring theory" , 2. Loop amplitudes, anomalies & phenomenology , Cambridge Univ. Press (1987) pp. 438ff
[a10]	P. Candelas, G. Horowitz, A. Strominger, E. Witten, "Vacuum configurations for superstrings" Nuclear Physics , B258 (1958) pp. 46–74
[a11]	B. de Wit, D.J. Smit, N.D. Hari-Dass, "Residual supersymmetry of compactified $d = 10$ supergravity" Nuclear Physics , B283 (1987) pp. 165–191
[a12]	D. Gepner, "Exactly solvable string compactifications on manifolds of $\operatorname{SU} ( N )$ holonomy" Physics Letters , B199 (1987) pp. 380–388
[a13]	"Première classe de Chern et courbure de Ricci: preuve de la conjecture de Calabi" , Sem. Palaiseau 1978 , Soc. Math. France (1978)
[a14]	Y.-T. Siu, "Lectures on Hermitean–Einstein metrics for stable bundles and Kähler–Einstein metrics" , Birkhäuser (1987)