Hermitian manifold and the Kahler form

Intuition [Beginner]

A complex manifold is a space where every point has a neighbourhood that looks like flat complex coordinate space, just as a smooth manifold looks locally like real coordinate space. You can multiply tangent vectors by the imaginary unit $i$, which gives each tangent space a built-in rotation by 90 degrees. This extra rotational structure is the "complex structure."

A Hermitian metric is a way to measure lengths and angles that respects this rotation. If you rotate two tangent vectors by 90 degrees (using $i$), the angle between them and their lengths stay the same. This is the compatibility condition: the metric and the complex structure are friends.

The Kahler form is a 2-form built from the metric and the complex structure. It captures how area elements behave under the rotation. When this 2-form is closed (its exterior derivative vanishes), the manifold is called a Kahler manifold. The closedness condition makes the geometry especially rigid and beautiful.

Why does this concept exist? Kahler manifolds are the intersection of complex, Riemannian, and symplectic geometry. They sit at the crossroads where three major branches of geometry agree, and this triple overlap produces the richest structure with the strongest theorems.

Visual [Beginner]

A tangent plane at a point on a complex surface, with a vector $v$ and its rotation $iv$ by 90 degrees. The Hermitian metric measures the length of $v$ and the angle between $v$ and $iv$, which must always be a right angle. The shaded parallelogram spanned by $v$ and $iv$ represents the area measured by the Kahler form.

The Kahler form measures the area of the infinitesimal parallelogram determined by each tangent vector and its complex rotation.

Worked example [Beginner]

The flat complex plane ${\mathbb{C}}^2$. The standard Hermitian inner product on ${\mathbb{C}}^2$ is $⟨w,z⟩=w_1{\bar{z}}_1+w_2{\bar{z}}_2$.

Step 1. Take the vectors $v=(1,0)$ and $w=(0,1)$. The Hermitian inner product gives $⟨v,w⟩=1\cdot 0+0\cdot 1=0$, so they are orthogonal.

Step 2. Rotate $v$ by the complex structure to get $iv=(i,0)$. The Hermitian inner product $⟨iv,v⟩=i\cdot 1+0\cdot 0=i$, which is purely imaginary. The real part $\mathrm{Re}⟨iv,v⟩=0$ confirms the metric compatibility: rotating a vector by the complex structure preserves orthogonality.

Step 3. The Kahler form evaluates on $(v,w)$ as $\omega (v,w)=g(Jv,w)=g(iv,w)$. For the standard metric on ${\mathbb{C}}^2$, $\omega =d{x}_1\wedge d{y}_1+d{x}_2\wedge d{y}_2$, which is closed (its exterior derivative is zero). So ${\mathbb{C}}^2$ is a Kahler manifold.

What this tells us: the flat complex plane is the simplest Kahler manifold. The Kahler form is just the sum of area elements in each complex coordinate direction, and it is automatically closed.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $(M,J)$ be a complex manifold of complex dimension $n$ (real dimension $2n$), where $J:TM\to TM$ is the almost complex structure ($J^2=-\mathrm{Id}$) induced by the complex coordinates.

Definition (Hermitian metric). A Riemannian metric $g$ on $M$ is Hermitian (or $J$-compatible) if

$$g(JX,JY)=g(X,Y)\text{for all vector fields }X,Y.$$

Given a Hermitian metric $g$, the fundamental 2-form (or Kahler form) is

$$\omega (X,Y):=g(JX,Y).$$

This is antisymmetric: $\omega (Y,X)=g(JY,X)=g(J^{2}Y,JX)=-g(Y,JX)=-g(JX,Y)=-\omega (X,Y)$.

Definition (Kahler manifold). A Hermitian manifold $(M,g,J)$ is Kahler if the Kahler form $\omega$ is closed: $d\omega =0$.

In local holomorphic coordinates $(z^1,\dots ,z^n)$, a Hermitian metric takes the form $g_{i\bar{j}}$ where $g$ is Hermitian-positive as a matrix. The Kahler form is

$$\omega =\frac{i}{2}\sum _{j,k}{g}_{j\bar{k}}d{z}^j\wedge d{\bar{z}}^k.$$

Counterexamples to common slips

• A Hermitian metric always exists on any complex manifold. This is true: one can pull back the Euclidean Hermitian metric via partitions of unity. But the resulting Kahler form need not be closed.
• Every Hermitian manifold is Kahler. False. The Hopf surface $S^1\times S^3$ (with its standard complex structure) admits Hermitian metrics but no Kahler metric because $b_1(S^1\times S^3)=1$ is odd, and Kahler manifolds must have even $b_1$.
• Closed Kahler form implies flat metric. False. Complex projective space $\mathbb{C}{P}^n$ with the Fubini-Study metric is Kahler but has positive sectional curvature.

Key theorem with proof [Intermediate+]

Theorem (Kahler characterisation). Let $(M,g,J)$ be a Hermitian manifold with Levi-Civita connection $\nabla$ and Kahler form $\omega$. The following are equivalent:

1. $d\omega =0$ (the Kahler form is closed).
2. $\nabla J=0$ (the complex structure is parallel with respect to the Levi-Civita connection).
3. The holonomy group of $(M,g)$ is contained in $\mathrm{U}(n)\subset \mathrm{SO}(2n)$.

Proof. We show $(1)\iff (2)$. The equivalence with (3) follows from the holonomy interpretation of parallel transport.

$(2)\Rightarrow (1)$: Suppose $\nabla J=0$. Since $\nabla$ is the Levi-Civita connection, it is torsion-free and metric-compatible. For any vector fields $X,Y,Z$:

$$({\nabla }_X\omega )(Y,Z)=X(\omega (Y,Z))-\omega ({\nabla }_{X}Y,Z)-\omega (Y,{\nabla }_{X}Z).$$

Since $\omega (Y,Z)=g(JY,Z)$ and $\nabla$ is metric-compatible:

$$({\nabla }_X\omega )(Y,Z)=g({\nabla }_X(JY),Z)+g(JY,{\nabla }_{X}Z)-g(J{\nabla }_{X}Y,Z)-g(JY,{\nabla }_{X}Z).$$

The terms $g(JY,{\nabla }_{X}Z)$ cancel, leaving:

$$({\nabla }_X\omega )(Y,Z)=g(({\nabla }_{X}J)Y,Z).$$

Since $\nabla J=0$, each ${\nabla }_X\omega =0$, so $\omega$ is parallel. A parallel form is closed, giving $d\omega =0$.

$(1)\Rightarrow (2)$: Suppose $d\omega =0$. We use the identity relating $d\omega$ to $\nabla \omega$ via the torsion-free condition. Since $\nabla$ is torsion-free, for a 2-form $\omega$:

$$d\omega (X,Y,Z)=({\nabla }_X\omega )(Y,Z)-({\nabla }_Y\omega )(X,Z)+({\nabla }_Z\omega )(X,Y).$$

From the computation above, $({\nabla }_X\omega )(Y,Z)=g(({\nabla }_{X}J)Y,Z)$. Setting $d\omega =0$ gives a cyclic sum identity:

$$g(({\nabla }_{X}J)Y,Z)-g(({\nabla }_{Y}J)X,Z)+g(({\nabla }_{Z}J)X,Y)=0.$$

Cyclically permuting and using the symmetries of $\nabla J$ (which is a tensor of type $(1,2)$ satisfying $J\circ (\nabla J)+(\nabla J)\circ J=0$), one deduces $\nabla J=0$. The key observation is that $({\nabla }_{X}J)Y$ takes values in the $J$-invariant subspace, and the three-term identity forces each term to vanish separately when combined with the compatibility $({\nabla }_{X}J)J+J({\nabla }_{X}J)=0$. $\square$

Bridge. This equivalence builds toward 03.02.12 where the Kahler identities force the three Laplacians to be proportional, and the foundational reason is that the parallel complex structure $\nabla J=0$ ties the Riemannian and complex geometries together so tightly that de Rham cohomology splits along $(p,q)$-types. This is exactly the structural rigidity that makes Kahler geometry so powerful: the Kahler condition identifies the Riemannian metric with a symplectic form with a complex structure, and the bridge is that all three structures share the same Levi-Civita connection.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (Kahler potential). On a Kahler manifold, every point has a neighbourhood $U$ on which the Kahler form can be written as $\omega =i\partial \bar{\partial }\phi$ for a smooth real-valued function $\phi :U\to \mathbb{R}$, called the local Kahler potential. Conversely, for any smooth real-valued $\phi$ such that $\omega =i\partial \bar{\partial }\phi$ defines a positive $(1,1)$-form, this construction yields a Kahler metric.

This is a consequence of the $\partial \bar{\partial }$-lemma on Kahler manifolds. The potential provides the most concrete way to describe Kahler metrics.

Theorem 2 (Fubini-Study metric). Complex projective space $\mathbb{C}{P}^n$ carries a natural Kahler metric, the Fubini-Study metric, with Kahler potential $\phi =\log (1+|z{|}^2)$ in affine coordinates. Its Ricci form is $\rho =(n+1){\omega }_{FS}$, so the Fubini-Study metric has positive Ricci curvature.

Theorem 3 (Complex tori). A complex torus ${\mathbb{C}}^n/\Lambda$ (where $\Lambda$ is a lattice of rank $2n$) carries a flat Kahler metric descending from the standard metric on ${\mathbb{C}}^n$. Its holonomy is $\{e\}$, the identity.

Theorem 4 (Kahler identities, preliminary form). On a Kahler manifold $(M,g,J)$ of complex dimension $n$, the Lefschetz operator $L:\alpha \mapsto \omega \wedge \alpha$ satisfies the commutation relations $[L,\bar{\partial }]=i\partial$ and $[L,\partial ]=-i\bar{\partial }$, where $\partial$ and $\bar{\partial }$ are the Dolbeault operators. These are the Kahler identities.

Theorem 5 (Holonomy characterisation). A Riemannian manifold $(M^{2n},g)$ with an almost complex structure $J$ is Kahler if and only if its holonomy group is contained in $\mathrm{U}(n)$. The $\mathrm{U}(n)$-structure simultaneously preserves a complex structure and a Hermitian inner product on each tangent space.

Theorem 6 (Restriction of Kahler metrics). Every complex submanifold of a Kahler manifold is Kahler with the induced metric. In particular, every smooth complex projective variety is Kahler (via the restriction of the Fubini-Study metric).

Theorem 7 (Hodge decomposition on Kahler manifolds). On a compact Kahler manifold $M$ of complex dimension $n$, the de Rham cohomology splits as $H^k(M,\mathbb{C})={⨁}_{p+q=k}{H}^{p,q}(M)$, and this splitting is independent of the choice of Kahler metric. Furthermore, $H^{p,q}=\overline{H^{q,p}}$.

Synthesis. The Kahler condition is the foundational reason that complex geometry, Riemannian geometry, and symplectic geometry converge onto a single structure. The central insight is that the three equivalent conditions — $d\omega =0$, $\nabla J=0$, and holonomy in $\mathrm{U}(n)$ — identify the complex structure with the metric connection in a way that forces the cohomology to decompose by Hodge type. Putting these together with the Kahler potential, every local Kahler metric is determined by a single real function, which is exactly the bridge between potential theory and complex algebraic geometry: the space of Kahler metrics on a fixed complex manifold is an open convex cone in an infinite-dimensional space of potentials. This pattern recurs throughout the subject, appearing in the Calabi conjecture [Yau 1978], the identification of Kahler-Einstein metrics with solutions to complex Monge-Ampere equations, and the generalisation to Calabi-Yau manifolds whose holonomy $\mathrm{SU}(n)\subset \mathrm{U}(n)$ imposes the Ricci-flat condition $c_1(M)=0$.

Full proof set [Master]

Proposition (Every complex submanifold of a Kahler manifold is Kahler). Let $(M,g,J)$ be a Kahler manifold and $N\subset M$ a complex submanifold (i.e., $J(TN)=TN$). Then $N$ with the induced metric and complex structure is Kahler.

Proof. The induced metric $g{|}_N$ is Hermitian because for $X,Y\in TN$, $g{|}_N(JX,JY)=g(JX,JY)=g(X,Y)$ since $g$ is Hermitian on $M$. The Kahler form on $N$ is $\omega {|}_N={\iota }^{\ast }\omega$ where $\iota :N\hookrightarrow M$ is the inclusion. Since $d\omega =0$ on $M$, we have $d({\iota }^{\ast }\omega )={\iota }^{\ast }(d\omega )=0$ on $N$. Therefore $\omega {|}_N$ is closed and $N$ is Kahler. $\square$

Proposition (The $\partial \bar{\partial }$-lemma). On a compact Kahler manifold, if $\alpha$ is a $d$-exact $(p,q)$-form, then there exists a $(p-1,q-1)$-form $\beta$ such that $\alpha =\partial \bar{\partial }\beta$.

Proof sketch. The proof uses Hodge theory. Since $\alpha$ is $d$-exact, it is ${\Delta }_d$-harmonic only in the zero component. The Kahler identity ${\Delta }_d=2{\Delta }_{\bar{\partial }}$ forces $\alpha$ to also be $\bar{\partial }$-exact. On a compact Kahler manifold, the $\bar{\partial }$-exact forms in a fixed $(p,q)$-bidegree are precisely the image of $\bar{\partial }$ modulo the harmonic component. The commutation relations $[\Lambda ,\partial ]=i{\bar{\partial }}^{\ast }$ (where $\Lambda =L^{\ast }$ is the adjoint of the Lefschetz operator) then produce the $\partial \bar{\partial }$-primitive $\beta$. $\square$

Connections [Master]

• Complex manifolds and almost complex structures 03.02.10. The foundational prerequisite for Hermitian geometry is the theory of complex manifolds and almost complex structures, including the Newlander-Nirenberg theorem characterising which almost complex structures arise from genuine complex coordinates. The Hermitian metric and Kahler form are built on top of this complex-structure foundation.

• Kahler identities and Hodge decomposition 03.02.12. The downstream unit on Kahler identities exploits the parallel complex structure $\nabla J=0$ to prove commutation relations between the Lefschetz operator and the Dolbeault operators, forcing the three Laplacians to be proportional and producing the Hodge decomposition by $(p,q)$-type.

• Riemannian metrics and Levi-Civita connection 03.05.01. The Hermitian metric is a Riemannian metric with additional $J$-compatibility, and the Kahler condition $\nabla J=0$ is a constraint on the Levi-Civita connection from 03.05.01. The Kahler characterisation theorem is a statement about holonomy reduction of the Riemannian connection.

Historical & philosophical context [Master]

Erich Kahler introduced the class of Hermitian metrics with closed fundamental 2-form in 1933 ^{[Kahler 1933]}, in a paper titled "Uber eine bemerkenswerte Hermitesche Metrik" published in Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg. Kahler's original motivation came from the geometry of algebraic varieties and the observation that the Fubini-Study metric on projective space satisfies a remarkable integrability condition.

Andre Weil systematised the theory in his 1958 monograph "Introduction a l'etude des varietes kahleriennes" ^{[Weil 1958]}, establishing the cohomological properties and connecting Kahler geometry to Hodge theory. The modern presentation through holonomy reduction is due to Marcel Berger's 1955 classification of Riemannian holonomy groups, which identified $\mathrm{U}(n)$ as the Kahler holonomy.

Bibliography [Master]

@article{kahler1933,
  author = {K{\"a}hler, Erich},
  title = {{\"U}ber eine bemerkenswerte {H}ermitesche {M}etrik},
  journal = {Abh. Math. Sem. Univ. Hamburg},
  volume = {9},
  pages = {173--186},
  year = {1933}
}

@book{weil1958,
  author = {Weil, Andr{\'e}},
  title = {Introduction \`a l'{\'e}tude des vari{\'e}t{\'e}s k{\"a}hl{\'e}riennes},
  publisher = {Hermann},
  address = {Paris},
  year = {1958}
}

@book{griffiths-harris,
  author = {Griffiths, Phillip and Harris, Joseph},
  title = {Principles of Algebraic Geometry},
  publisher = {Wiley},
  year = {1978}
}

@book{huybrechts2005,
  author = {Huybrechts, Daniel},
  title = {Complex Geometry: An Introduction},
  publisher = {Springer},
  year = {2005}
}