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Just to follow up on Eric's correct answer: when you have an almost complex structure $J$, you can decompose $1$-forms into type $(1,0)$ and $(0,1)$. Locally, you can find a local basis $e^1, \ldots, …

Kevin Lin gave a great technical answer to this question (which should be accepted) but I would like to add some more "philosophical" reasons for this: [1] algebraic geometry methods are easier to ap …

Of course, it depends on the Riemannian metric you put on $\mathbb O \mathbb P^1 \cong S^8$. For the round metric you do indeed get $\mathrm{SO}(8)$ holonomy. A priori, one could imagine other "natura …

I decided to make my comment into a more detailed answer. When $M$ has an almost complex structure $J$, then one can talk about smooth complex-valued differential forms of type $(p,q)$ in the usual wa …

There are several different "definitions" of Calabi-Yau manifolds, not all equivalent, and not all contained in one general definition. A good discussion of some of these inequivalent definitions can …