Real (1, 1)-form

A differential form of type (p, q) $\phi$ of type $(1,1)$ is real when $\bar{\phi }=\phi$. Locally this is equivalent tofor a Hermitian matrix $(h_{j\bar{k}})$.

A real $(1,1)$-form $\phi$ is positive whenfor every nonzero tangent vector $\xi$ of type $(1,0)$. Equivalently, the local Hermitian matrix in the representation $\phi =i{h}_{j\bar{k}}d{z}^j\wedge d{\bar{z}}^k$ is positive definite.

If $\phi$ and $\psi$ are positive real $(1,1)$-forms and $\xi ,\eta$ are linearly independent tangent vectors of type $(1,0)$, thenAfter simultaneous diagonalization by congruence, with positive diagonal entries $a_j,b_j$, the left side iswhich is positive because some $2\times 2$ minor is nonzero.