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This is cross posted from math.SE: https://math.stackexchange.com/q/876432/9

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.


EDIT

I should a 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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  • $\begingroup$ You forgot to dualize: $H^{0,1}(M, T^{1,0}M)$ is isomorphic to the dual vector space, $H^0(M,K^2)^\vee$, not $H^0(M,K^2)$. $\endgroup$ Jul 25, 2014 at 2:46
  • $\begingroup$ I'm not sure to understand, but $\alpha$ looks like the beltrami differential associated to a quasi conformal diffeo $(M,\sigma_1)\to (M,\sigma_2)$ $\endgroup$
    – user126154
    Jul 25, 2014 at 14:39
  • $\begingroup$ Infinitesimally, I would say yes since there is no obstruction to the deformation of the complex structure for the case of complex dimension 1. For large scale, I could not see why the statement is true. $\endgroup$ Oct 9, 2014 at 16:28

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