Let $M$ be a Riemann surface and $[\alpha] \in H^{0,1}(M; T^{1,0} M) \simeq H^0(M;K^2)$. Considering $\alpha$ as a map $T^{0,1} M \to T^{1,0} M$, the bundle $$ \{v + \alpha(v) \mid v \in T^{0,1} M\} \subset T_{\mathbb{C}} M $$ forms the (0,1) part of another complex structure on (the underlying real manifold of) $M$. Is it true that every complex structure (modulo diffeomorphism) arises in this way?

I'm pretty sure this is true and very standard, so a good reference would be great.


I should add that $\alpha$ must be sufficiently small so that its graph inside $T_{\mathbb C} M$ does not intersect the real tangent bundle $TM$.


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