Opposite Hermitian Metric

Question

Let $M$ be a complex manifold, and let $h$ be an Hermitian metric for $M$. Is $h$ still an Hermitian matric with respect to the opposite complex structure on $M$? Also, does the associated fundamental form remain the same?

Answer 1

Let $M$ be a complex manifold, then there is an associated almost complex structure $J$ which is integrable by definition. The opposite complex structure is the complex structure given by the integrable almost complex structure $-J$. To see that $-J$ is necessarily integrable, note that

\begin{align*} N_{-J}(X, Y) &= [X, Y] -J[-JX, Y] -J[X, -JY] - [-JX, -JY]\\ &= [X, Y] + J[JX, Y] + J[X, JY] - [JX, JY]\\ &= N_J(X, Y). \end{align*}

Let $h$ be a hermitian metric on $(M, J)$, i.e. a Riemannian metric such that $h(JX, JY) = h(X, Y)$. As $h(-JX, -JY) = h(JX, JY)$, $h$ is also a hermitian metric on $(M, -J)$. However, $\omega(X, Y) = h(JX, Y) = -h(-JX, Y) = -\omega'(X, Y)$ where $\omega'$ is the fundamental two-form associated to the opposite complex structure.