Question

For a Riemannian manifold $(M,g)$, that is also a complex manifold, when does the Levi-Civita $\nabla_g$ connection restrict to a connection on the holomorphic forms $\Omega^{(\cdot,0)}$, and when does it restrict to a connection on the anti-holomorphic forms $\Omega^{(0,\cdot)}$?

I would assume there are some sufficient and neccessary conditions for the interaction of the metric $g$ and the complex structure.

Answer 1

If you assume that $(M,g,J)$ is almost Hermitian, then the Levi-Civita $\nabla$ preserves $\Lambda^{0,p}$ or $\Lambda^{p,0}$ for some $p>0$ iff $J$ is $\nabla$-parallel, i.e. iff $(M,g,J)$ is K\"ahler. This is an easy exercice in Riemannian geometry.