Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 18 / 1 / 1 / Solution

A holomorphic atlas on a real smooth manifold of dimension $2n$ consists of compatible smooth charts ${\phi }_a:U_a\to V_a\subseteq {\mathbb{C}}^n$ whose transition maps ${\phi }_b{\phi }_a^{-1}$ are biholomorphic on their domains. An almost complex structure is a smooth real vector bundle endomorphism $J:TX\to TX$ with $J^2=-\mathrm{id}$. In each chart defineThe differential of a holomorphic map is a complex-linear map, so the transition differentials commute with multiplication by $i$. Consequently the local definitions agree on overlaps. They are smooth and square to $-\mathrm{id}$, yielding the natural almost complex structure induced by a complex atlas. This construction uses the atlas, rather than an arbitrary choice of coordinates on the underlying real smooth manifold.