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Let $(M,J_{M})$ be a almost complex manifold and $(N,J_{N})$ be a complex manifold. I want to prove that $F^{*}(\mathcal{O}_{N})\subset\mathcal{O}_{M}$ implies that $F:M\rightarrow N$ is almost complex. $\mathcal{O}_{M}$ denotes the sheaf of holomorphic functions on $M$ and similarily for $\mathcal{O}_{N}$.

$F:M\rightarrow N$ is almost complex means that $dF\circ J_{M}=J_{N}\circ dF$.

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1 Answer 1

$F$ is almost complex iff $dF$ maps the $+i$-eigenspace of $J_M$ to the $+i$-eigenspace of $J_N$, or equivalently (by dualization) iff $F^*$ maps $\Omega^{1,0}_N$ to $\Omega^{1,0}_M$. However, for any $\alpha\in\wedge^{1,0}T^*_xN$ there is a holomorphic function $f$ in a neighourhood of $x$ s.t. $\alpha=d_xf$ (it's enough to consider linear combinations of some local holomorhic coordinates around $x$). As $F^*f\in\mathcal{O}_M$, for any $y\in M$ s.t. $F(y)=x$ we have $F^*\alpha=d_y(F^*f)\in\wedge^{1,0}T^*_xM$, hence $F^*$ maps $\Omega^{1,0}_N$ to $\Omega^{1,0}_M$.

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One thing I don't understand is how to get that for any $\alpha\in\wedge^{1,0}T_{x}^{1,0}N$, there exists a holomorphic function $f$ in a neighbourhood of $x$ such that $\alpha=d_{x}f$. Is this a general theorem or lemma? –  kobebryant Oct 17 '11 at 4:21
    
If $z^i$ are local holomorphic coordinates and $\alpha=\sum a_i\, dz^i$, you can take $f=\sum a_i\,z^i$ (i.e. $f$ is a linear combination of the coordinates) –  user8268 Oct 17 '11 at 19:32

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