Let $X$ be a compact Kähler manifold of complex dimension $n$, and let $[\omega]\in H^2(X;\mathbb R)$ be the cohomology class of a Kähler form on $X$. For each pair of integers $a,b$, let $A^{a,b}(X)$ denote the complex [vector space](/page/Vector%20Space) of smooth complex-valued differential forms of type $(a,b)$ on $X$, with $A^{a,b}(X)=0$ if $a<0$, $b<0$, $a>n$, or $b>n$. For every integer $k$, define the real Lefschetz operator $L_k:H^k(X;\mathbb R)\to H^{k+2}(X;\mathbb R)$ by

\begin{align*} L_k(\alpha)=[\omega]\smile \alpha. \end{align*}

For every integer $m\geq 0$, let $L^m$ denote the $m$-fold iterated Lefschetz map on cohomology, with $L^0$ equal to the identity map on the relevant cohomology group.

Then the graded real cohomology algebra $H^*(X;\mathbb R)$, together with the distinguished class $[\omega]$, satisfies the following properties.

For every integer $k$ with $0\leq k\leq n$, the map

\begin{align*} L^{n-k}:H^k(X;\mathbb R)\to H^{2n-k}(X;\mathbb R) \end{align*}

is an isomorphism.

For every integer $k\geq 0$, the complex cohomology group admits a [Hodge decomposition](/theorems/2745)

\begin{align*} H^k(X;\mathbb C)=\bigoplus_{p+q=k}H^{p,q}(X). \end{align*}

Here $H^{p,q}(X)$ denotes the subspace of $H^{p+q}(X;\mathbb C)$ consisting of de Rham cohomology classes represented by closed forms in $A^{p,q}(X)$, equivalently the $(p,q)$-summand supplied by the compact Kähler [Hodge decomposition](/theorems/3941).

For all integers $p,q\geq 0$, complex conjugation on $H^{p+q}(X;\mathbb C)$ satisfies

\begin{align*} \overline{H^{p,q}(X)}=H^{q,p}(X). \end{align*}

For all integers $p,q,r,s\geq 0$, the cup product satisfies

\begin{align*} H^{p,q}(X)\smile H^{r,s}(X)\subset H^{p+r,q+s}(X), \end{align*}

with the convention that $H^{a,b}(X)=0$ if $a<0$, $b<0$, $a>n$, or $b>n$.