# The Kähler Condition

Let ${\displaystyle (M,J)}$ be an almost complex manifold and suppose in addition we have a Riemannian metric ${\displaystyle g}$ such that ${\displaystyle J}$ is an orthogonal transformation with respect to ${\displaystyle g}$. In symbols: For any vector fields ${\displaystyle X}$ and ${\displaystyle Y}$ we have ${\displaystyle g(X,Y)=g(JX,JY)}$. In this case the triple ${\displaystyle (M,g,J)}$ is called a Hermitian Manifold.

Now we define a non-degenerate 2-form ${\displaystyle \omega (X,Y)=g(JX,Y)}$ (${\displaystyle \omega }$ is called non-degenerate if ${\displaystyle \omega (X,Y)=0}$${\displaystyle \forall X\Rightarrow Y=0}$). This allows us to state the Kähler condition:

Definition 3.5

Suppose that ${\displaystyle (M,g,J)}$ is a Hermitian manifold if the associated 2-form ${\displaystyle \omega }$ is closed then we call ${\displaystyle M}$ a Kähler manifold.

For Kähler (and most generally symplectic) manifolds the existance of such an ${\displaystyle \omega }$ imposes topological restrictions on even-dimensional (orientable) manifolds admitting such structures.

Proposition 3.4

Let ${\displaystyle M}$ be a 2n-dimensional manifold with a closed non-degenerate 2-form ${\displaystyle \omega }$. Then the even dimensional de Rham cohomology groups have strictly positive dimension.

Proof

The 2-form omega is non-degenerate, which is equivalent to the form ${\displaystyle \omega ^{n}}$ being everywhere non-zero, and hence a volume form. If this form were d${\displaystyle \gamma }$ for an (n-1)-form ${\displaystyle \gamma }$ then by Stoke's theorem the integral of ${\displaystyle \omega ^{n}}$ over ${\displaystyle M}$ would be 0, hence this form represents a non-trivial cohomology class. Recall that the wedge map on forms descends to a product of cohomology groups so that ${\displaystyle [\omega ^{n}]=[\omega ]^{n}}$, in particular we must have ${\displaystyle [\omega ]^{k}\neq 0}$ for each ${\displaystyle k}$. We have a non-zero element of ${\displaystyle H^{2k}(M)}$ for ${\displaystyle k=1,..,n}$.