# The Kähler Condition

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

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

Definition 3.5

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

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

Proposition 3.4

Let $M$ be a 2n-dimensional manifold with a closed non-degenerate 2-form $\omega$ . 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 $\omega ^{n}$ being everywhere non-zero, and hence a volume form. If this form were d$\gamma$ for an (n-1)-form $\gamma$ then by Stoke's theorem the integral of $\omega ^{n}$ over $M$ 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 $[\omega ^{n}]=[\omega ]^{n}$ , in particular we must have $[\omega ]^{k}\neq 0$ for each $k$ . We have a non-zero element of $H^{2k}(M)$ for $k=1,..,n$ .