Kähler manifold

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

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 $(1,1)$.

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

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

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

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