*$J$-Hermitian metric on $X$ is a $\rr $-bilinear form $h:TX\times TX\to \cc $ such that*

\(a) $h(Ju,v)=\overline h(v,Ju)=ih(u,v)$;

\(b) $h(u,u)>0$ for $u\not= 0$.

Decompose $h(u,v)=g_h(u,v) -i\omega_h(u,v)$. Then :

Exercise. (a) $g_h$ is a Riemannian metric on $X$ and $\omega_h$ is an exterior 2-form.

\(b) $g_h(u,v)=\omega_h(u,Jv)$ and thus $h(u,v)=\omega_h(u,Jv) -
i\omega_h(u,v)$. More over, one has $\omega_h (Ju,Jv)=\omega_h (u,v)$.

\(c) $\omega_h(u,v)=-g_h(u,Jv)$.

Definition 1.1.4. The form [*$\omega_h$ is called a (1,1)-form associated to $h$.*]{}

Conversely,

Definition 1.1.5.

*An exterior $2$-form $\omega $ on $X$ is called $J$-calibrated if*

\(a) $\omega (Ju,Jv)=\omega (u,v)$;

\(b) $\omega (u,Ju)>0$ for $u\not= 0$.

Thus, $\omega $ clearly defines a $J$-Hermitian metric $h^{\omega }(u,v)=\omega (u,Jv) - i\omega (u,v)$.

A triple $(X,J,h)$, where $h$ is $J$-Hermitian is called an almost-Hermitian manifold. We shall not use this term in these notes.

Example. To explain the coefficient $-i$ chosen in the decomposition $h(u,v)=g(u,Jv) - i\omega (u,v)$ consider $(\rr^{2n}, \omega\st, J\st)$. Here $\omega\st = \sum_{j=1}^ndx_j\wedge dy_j$ is a standard symplectic form in $\rr^{2n}$. The Hermitian metric here is $h\st = \sum_{j=1}^ndz_j\otimes d\bar z_j = \sum_{j=1}^n (dx_j\otimes
dx_j + dy_j\otimes dy_j) - i\sum_{j=1}^n(dx_j\otimes dy_j - dy_j\otimes dx_j)
= g\st -i\omega\st $.

Thus $\omega\st = -\im h\st $ is a canonical expression for a Kähler 