Kähler manifold

A Kähler manifold is a complex manifold with a positive real closed $(1,1)$-form $\omega$. The associated Riemannian metric is $g(u,v)=\omega (u,Jv)$.

Here a Calabi-Yau threefold is a compact complex three-dimensional Kähler manifold with trivial canonical bundle and $H^1(M,\mathbb{C})=0$. The Hodge decomposition theorem for compact Kähler manifolds gives $H^1(M,{\mathcal{O}}_M)=0$, and Serre duality for compact complex manifolds then gives $H^2(M,{\mathcal{O}}_M)=0$. This supplies a positive integral Kähler class and hence a projective embedding.

The Kähler cone of a compact Kähler manifold is the open cone of real $(1,1)$ cohomology classes represented by positive closed $(1,1)$ forms. If every real degree-two cohomology class has type $(1,1)$, density of rational classes supplies a rational Kähler class. An integral multiple is the First Chern class of a positive holomorphic line bundle, and the Kodaira embedding theorem implies projectivity.

The de Rham cohomology class of a Kähler form is a Kähler class. Such classes form the Kähler cone. They have positive integral on every complex curve, and their top self-intersection is positive.

Locally, a Kähler metric is the complex Hessian $g_{i\bar{j}}={\partial }_i{\partial }_{\bar{j}}K$ of a real Kähler potential. Adding a holomorphic function and its complex conjugate to $K$ leaves the metric unchanged.

A Hodge metric is a Kähler metric whose Kähler form represents an integral cohomology class in real de Rham cohomology. Equivalently, its periods over integral two-cycles are integers. Some conventions require the class of $\omega /(2\pi )$ to be integral instead; this changes the normalization but not existence.

At any point of a Kähler manifold, a complex-linear coordinate change normalizes the Hermitian metric matrix. Closedness makes its holomorphic first derivatives symmetric in the two unbarred indices. A quadratic holomorphic coordinate change removes those derivatives, and Hermitian symmetry removes the antiholomorphic derivatives. Conversely, having this first-order normalization at every point makes the fundamental Hermitian form closed.

For a Kähler metric $g$ with complex structure $J$, the associated real two-form is $\omega (u,v)=g(Ju,v)$. It has type $(1,1)$, is positive in the sense $\omega (v,Jv)=g(v,v)>0$ for $v\ne 0$, and is closed. Conversely a real closed positive $(1,1)$ form determines a Kähler metric. A local geometric Kähler potential writes it as $i\partial \bar{\partial }\phi$.

On a nonempty compact Kähler manifold of complex dimension $n$, the Kähler form is closed and ${\int }_X{\omega }^n>0$. If ${\omega }^k=d\eta$ for $1\le k\le n$, then ${\omega }^n=d(\eta \wedge {\omega }^{n-k})$; Stokes theorem contradicts the positive integral. Thus the de Rham cohomology is nonzero in every even degree through $2n$.

For a closed real $(1,1)$ form, the Poincaré lemma gives a real one-form $\eta$ with $d\eta =\omega$. Its $(0,1)$ part is $\bar{\partial }$-closed, so the Dolbeault-Poincaré lemma gives ${\eta }^{0,1}=\bar{\partial }\psi$. Reality gives ${\eta }^{1,0}=\partial \bar{\psi }$, hence $\omega =i\partial \bar{\partial }(2\mathrm{Im}\psi )$. Positivity is a separate requirement for this form to define a metric.

A smooth real rotation-invariant potential is constant on every positive-radius circle, so its logarithmic-radius function $u$ is smooth on $\mathbb{R}$. Off the origin, $f_{z\bar{z}}=e^{-t}{u}^{\prime \prime }(t)$. Since $f$ is smooth on the entire plane, this coefficient extends smoothly through the origin. A finite Taylor expansion, rather than a convergent series assumption, gives $f(z)=f(0)+b|z{|}^2+O(|z{|}^4)$.

For a potential already smooth on all of $\mathbb{C}$, the associated form is a Kähler form exactly when the displayed two positivity conditions hold. The first gives a positive metric coefficient off the origin; the second is positivity of $f_{z\bar{z}}(0)$. The associated real metric is $2f_{z\bar{z}}(d{x}^2+d{y}^2)$. Positivity away from the origin alone permits a degenerate metric at the origin.

A geometric Kähler potential is locally a real smooth function $\phi$ with $\omega =i\partial \bar{\partial }\phi$ for the Kähler form. Its complex Hessian must be positive definite to define a Kähler metric. Conventions sometimes absorb a factor of $1/2$ into $\phi$. Potentials differing by the real part of a holomorphic function determine the same form. This topic concerns ordinary complex geometry; the existing undisambiguated Kähler potential entry concerns superfields.

For $s=|z_1{|}^2+|z_2{|}^2>0$ and $a>0$, a geometric Kähler potential with derivative ${\phi }^{\prime }(s)=\sqrt{s^2+a^2}/(2s)$ defines a Kähler metric on ${\mathbb{C}}^2\setminus \{0\}$. Its tangential and radial eigenvalues are $\sqrt{s^2+a^2}/(2s)$ and $s/(2\sqrt{s^2+a^2})$. Their product is $1/4$, giving the same Riemannian volume form as ${\omega }_0=(i/2)\sum d{z}_j\wedge d{\bar{z}}_j$, although the metric is different. This construction makes preservation of volume compatible with anisotropic stretching.

A Kähler quotient imposes a moment-map level condition and divides by the compact symmetry group. Under suitable stability assumptions it is equivalent to a quotient by the complexified group and inherits a Kähler metric away from singular orbits.

The Lefschetz operator of a Kähler manifold is exterior multiplication by its Kähler form,Its formal adjoint is denoted by $\Lambda =L^{\ast }$ and lowers the bidegree of a differential form by $(1,1)$.

This wedge operator participates in the Kähler identities; the theorem on integral $(1,1)$ classes is a different result.

For a compact Kähler manifold of complex dimension $n$, wedging with the $(n-k)$th power of its Kähler class is the displayed isomorphism for $0\le k\le n$. The Hard Lefschetz isomorphism on Dolbeault cohomology and Hodge decomposition theorem for compact Kähler manifolds give the result by summing the bidegrees. In particular, $L:H^k\to H^{k+2}$ is injective when $k<n$: if $L\alpha =0$, then $L^{n-k}\alpha =0$. Hence $b_k\le b_{k+2}$ below the middle dimension.

On a compact Kähler manifold, the Lefschetz operator preserves harmonic forms, so the indicated power acts on harmonic representatives of Dolbeault cohomology. It is injective by injectivity of powers of the Lefschetz operator; Hodge symmetry and Hodge duality make its finite-dimensional source and target have equal dimension, so it is an isomorphism.

Wedge multiplication by a closed $(1,1)$ form induces maps on Dolbeault cohomology. If ${\omega }^{\prime }-\omega =\bar{\partial }\theta$, then $({\omega }^{\prime }{)}^k-{\omega }^k=\bar{\partial }(\theta \wedge {\sum }_{j=0}^{k-1}({\omega }^{\prime }{)}^j\wedge {\omega }^{k-1-j})$. Wedging with a closed representative shows that the maps agree. For a change of Kähler metric potential, take $\theta =-i\partial f$.

For degree $k\le n$, a form is primitive if $\Lambda \alpha =0$, equivalently $L^{n-k+1}\alpha =0$. In middle degree $k=n$ this is equivalent to $L\alpha =0$. On a compact Kähler manifold, the same definitions apply to cohomology via harmonic differential forms.

On a complex two-dimensional Kähler manifold, a primitive $(1,1)$ form satisfies $\ast \alpha =-\alpha$, making it an anti-self-dual differential form. For a nonzero real primitive harmonic form on a compact surface, $\int \alpha \wedge \alpha =-\Vert \alpha {\Vert }_{L^2}^2<0$.

In complex dimension $n$, the Lefschetz operator of a Kähler manifold and adjoint Lefschetz operator satisfy $[\Lambda ,L]=(n-k)\mathrm{id}$ on degree $k$. In middle degree, this yields $\Vert L\alpha {\Vert }^2=\Vert \Lambda \alpha {\Vert }^2$.

For the Hermitian Lefschetz operator on a complex $n$-manifold, the Lefschetz commutator and the commutator derivation identity imply this formula. When expanding a product commutator, one must evaluate $[L,\Lambda ]$ on the degree after the powers of $L$ to its right have acted.

The adjoint of $L=\omega \wedge -$ lowers real degree by two and bidegree by $(1,1)$. On a compact Kähler manifold it acts on cohomology through harmonic differential forms, because it commutes with the Hodge Laplacian. It does not generally preserve all closed forms.

For the Lefschetz operator of a Kähler manifold and its adjoint, the Kähler identities include

On a Kähler manifold the Chern connection components satisfy $[\Lambda ,{\bar{\partial }}_E]=-i(D_E^{\prime }{)}^{\ast }$ and $[\Lambda ,D_E^{\prime }]=i{\bar{\partial }}_E^{\ast }$. Holomorphic normal coordinates and a Chern connection in a normal holomorphic frame reduce them to the scalar identities pointwise. Closedness of the Kähler form also gives $[\Lambda ,{\bar{\partial }}_E^{\ast }]=0$.

For a Hermitian holomorphic bundle over a Kähler manifold, let $\Theta$ act by exterior multiplication. The bundle-valued Kähler identities give $[\Lambda ,{\Delta }_{\bar{\partial }}]=-i(\Theta \wedge -{)}^{\ast }$, and taking adjoints yields $[{\Delta }_{\bar{\partial }},L]=i\Theta \wedge -$. This measures the failure of the bundle-valued Dolbeault Laplacian to commute with the Lefschetz operator.

The Lefschetz operator commutes with the bundle-valued Dolbeault Laplacian on the entire smooth form space exactly when the Chern connection is flat. The Lefschetz-Dolbeault curvature commutator gives one direction. Conversely test its curvature-wedge operator on degree-zero sections with arbitrary prescribed values at any point to force every vector-bundle curvature endomorphism to vanish.

For the Chern connection of a Hermitian holomorphic vector bundle on a Kähler manifold, the two Dolbeault Laplacians differ by $[iF\wedge ,\Lambda ]$, where $F$ is its curvature form of a connection and $\Lambda$ is the adjoint Lefschetz operator. The curvature term controls vanishing estimates.

The Dolbeault Laplacian is ${\Delta }_{\bar{\partial }}=\bar{\partial }{\bar{\partial }}^{\ast }+{\bar{\partial }}^{\ast }\bar{\partial }$. On a Kähler manifold, the Kähler identities imply

On a compact Hermitian manifold without boundary, this operator is the inverse of the Dolbeault Laplacian on the orthogonal complement of its harmonic forms and is zero on the harmonic space. Elliptic theory gives ${\Delta }_{\bar{\partial }}G=G{\Delta }_{\bar{\partial }}=I-H$, with $H$ the harmonic projection, and ensures that $G$ sends smooth forms to smooth forms. It commutes with $\bar{\partial }$ and ${\bar{\partial }}^{\ast }$; their commutation with the Laplacian and orthogonality to its kernel give this by uniqueness of the inverse. If $\eta$ is $\bar{\partial }$-exact, then $\eta =\bar{\partial }{\bar{\partial }}^{\ast }G\eta$. On a Kähler manifold, the same operator serves the holomorphic Laplacian because ${\Delta }_{\partial }={\Delta }_{\bar{\partial }}$.

On a Kähler manifold, expanding $d=\partial +\bar{\partial }$ and using the Kähler identities makes the mixed anticommutators vanish and gives ${\Delta }_{\partial }={\Delta }_{\bar{\partial }}$. Therefore

The Kähler Laplacian identity ${\Delta }_d^{\mathrm{H}}=2{\Delta }_{\bar{\partial }}$ implies that every holomorphic function on a Kähler manifold is a harmonic function. On functions, $\bar{\partial }u=0$ and ${\bar{\partial }}^{\ast }u=0$ by degree, so both terms of the Dolbeault Laplacian vanish. Compactness is unnecessary. Changing the sign convention for the scalar Laplacian does not change this conclusion.

On a Kähler manifold, the Lefschetz operator of a Kähler manifold $L=\omega \wedge -$ commutes with the Dolbeault Laplacian:Consequently, if $\alpha$ is ${\Delta }_{\bar{\partial }}$-harmonic, then every $\alpha \wedge {\omega }^k=L^k\alpha$ is also harmonic.

For a compact Hermitian manifold, elliptic theory gives the orthogonal decompositionThe harmonic space ${\mathcal{H}}_{\bar{\partial }}^{p,q}=\ker {\Delta }_{\bar{\partial }}$ is finite-dimensional and represents Dolbeault cohomology.

On a compact Hermitian manifold of complex dimension $n$, $C\alpha =\ast \bar{\alpha }$, with complex-linear Hodge star, is a conjugate-linear bijection from Dolbeault Laplacian harmonic $(p,q)$-forms to harmonic $(n-p,n-q)$-forms. The identities ${\bar{\partial }}^{\ast }=-\ast \partial \ast$ and ${\ast }^2=(-1{)}^{p+q}$ show that $\bar{\partial }\alpha ={\bar{\partial }}^{\ast }\alpha =0$ implies the same two equations for $C\alpha$. Its square is $(-1{)}^{p+q}$. Thus these harmonic spaces have equal complex dimension even without the Kähler condition. The natural map is conjugate-linear, not complex-linear.

On a compact Kähler manifold, a pure-type differential form that is $d$-closed and is either $\partial$-exact or $\bar{\partial }$-exact is $\partial \bar{\partial }$-exact. In particular, if $\eta$ is $\bar{\partial }$-exact and $\partial \eta =0$, thenfor a form $\varphi$ of bidegree one lower in each component.

On a compact Kähler manifold, the difference of cohomologous Kähler forms is a real, de Rham cohomology exact $(1,1)$-form. The ddbar lemma gives $\tilde{\omega }-\omega =\partial \bar{\partial }h$. Reality implies $\partial \bar{\partial }\bar{h}=-(\tilde{\omega }-\omega )$, so $f=(h-\bar{h})/(2i)$ is real and has the displayed potential equation. The difference of two real potentials is a pluriharmonic function, hence constant on each connected component. On a connected manifold the only ambiguity is one additive constant.

If $\alpha$ is a $d$-closed $(p,q)$-form on a compact Kähler manifold, then $\alpha$ is ddbar lemma exact if and only if it is orthogonal to every harmonic differential form of type $(p,q)$. The forward implication follows by integration by parts. For the reverse implication, Dolbeault Hodge decomposition first makes $\alpha$ $\bar{\partial }$-exact, and the ddbar lemma then makes it $\partial \bar{\partial }$-exact.

For a $d$-closed pure-type form $\alpha$ orthogonal to harmonic forms on a compact Kähler manifold, let $G_{\bar{\partial }}$ be the inverse of the Dolbeault Laplacian on the harmonic complement, zero on the harmonic space. The Kähler Laplacian identity makes it commute with $\partial$ and $\bar{\partial }$. Thus $\alpha =\bar{\partial }{\bar{\partial }}^{\ast }{G}_{\bar{\partial }}\alpha$ and $G_{\bar{\partial }}\alpha =\partial {\partial }^{\ast }{G}_{\bar{\partial }}^2\alpha$. The mixed anticommutator ${\bar{\partial }}^{\ast }\partial +\partial {\bar{\partial }}^{\ast }=0$ gives $\alpha =\partial \bar{\partial }\beta$. This is an explicit potential for the ddbar lemma, with one lower degree of each type; using the inverse of the Hodge Laplacian instead multiplies the displayed expression by four.

On a compact Kähler manifold, if a differential form $\alpha$ is $d$-closed and $d^c$-exact, then it is $d{d}^c$-exact: there is a form $\beta$ two degrees lower such that $\alpha =d{d}^c\beta$. This real-form statement is equivalent, after decomposing by type, to the ddbar lemma.

The Fubini-Study metric is the standard unitary-invariant Kähler metric on Complex projective space, induced from the unit sphere by the Hopf fibration. Its associated Kähler form is the Fubini-Study form; scaling conventions determine the area of a projective line.

The Fubini-Study form is the standard Kähler form on complex projective space. With the integral normalization, $[{\omega }_{FS}/(2\pi )]=c_1(\mathcal{O}(1))$.

The two-form is associated with the Fubini-Study metric.

The form ${\omega }_{\mathrm{int}}=\frac{i}{2\pi }\partial \bar{\partial }\log (1+\sum |z_j{|}^2)$ patches on Complex projective space. The dual tautological metric on $\mathcal{O}(1)$ identifies it with normalized Chern curvature, so $[{\omega }_{\mathrm{int}}]=c_1(\mathcal{O}(1))$. With the unnormalized convention $i\partial \bar{\partial }\log (1+\sum |z_j{|}^2)$, divide by $2\pi$ to obtain this integral class.

The scalar circle group action on ${\mathbb{C}}^{n+1}$ with the standard symplectic form has moment map $\mu =-|z{|}^2/2$ in the convention ${\iota }_{X_H}\omega =dH$. Reduction of the radius-$R$ sphere gives Complex projective space and the form ${\omega }_R=(i{R}^2/2)\partial \bar{\partial }\log (1+|w{|}^2)$. Its area on a complex projective line is $\pi R^2$. Thus $R=\sqrt{2}$ yields the Fubini-Study form normalized to projective-line area $2\pi$, while $R=1$ yields half that form. The Hopf fibration identifies the quotient.

Normalize the Fubini-Study form so that a complex projective line has symplectic area $\pi$. The displayed map takes the unit symplectic ball in ${\mathbb{C}}^n$ diffeomorphically onto the complement of the last-coordinate hyperplane. Its lift to the unit sphere has last coordinate positive real, so that coordinate contributes zero to the pulled-back standard symplectic form. The map is therefore a symplectomorphism.

Two disjoint symplectic balls of radii $r_1,r_2$ embedding into the unit four-ball must satisfy the displayed inequality. Extend their transported standard complex structures on smaller concentric balls to a compatible almost complex structure on the projective plane. A degree-one J-holomorphic curve through both centres has area $\pi$. The Monotonicity theorem for a J-holomorphic curve bounds its area in each smaller ball below by $\pi {\rho }_i^2$. Disjointness allows these bounds to be added; letting ${\rho }_i$ increase to $r_i$ proves the result.

For a compact Kähler manifold,and every class has a unique harmonic representative of each type.

This is the compact Kähler specialization of Hodge theory.

For a compact Kähler manifold of complex dimension $d$ and $k\le d$, this is the cohomology killed by the indicated power of a Kähler class. The Hodge–Riemann bilinear relations polarize its Hodge structure. On a K3 surface, primitive degree-two cohomology is the orthogonal complement of the chosen positive class.

For a nonzero primitive class $v\in H^{p,q}$ of degree $k=p+q$ on a compact Kähler manifold of complex dimension $d$, the displayed expression is positive. Together with orthogonality of complementary Hodge bidegrees, this makes primitive cohomology a polarized Hodge structure when the Kähler class is rational, after clearing denominators. Positivity on the full cohomology requires the signs prescribed by its primitive decomposition.

The Kähler Laplacian identity makes the space of complex harmonic differential forms of degree $r$ the direct sum of its $(p,q)$ parts with $p+q=r$. Conjugation identifies the $(p,q)$ and $(q,p)$ harmonic spaces. If $r$ is odd these are distinct and pair off, so the Betti number in degree $r$ is even. This is a topological obstruction to existence of a Kähler metric, exhibited by a Hopf surface with first Betti number one.

On a connected compact Kähler manifold of complex dimension two, the intersection form restricted to real $(1,1)$ cohomology has signature $(1,h^{1,1}-1)$. The Kähler class spans a positive line, and its orthogonal complement consists of primitive classes, which are negative definite. Thus a totally isotropic subspace of real $(1,1)$ cohomology has dimension at most one.