Motivated by this question, I began to wonder if there is a global definition of the almost complex structure of a complex manifold. It is (almost) always presented as multiplication by complex $i$ on the tangent space, and then globalized. Using the formulae given earlier $$ \overline{\partial}\omega = \frac{1}{2}(\text{d}\omega + i \text{d}(J\omega)), $$ and $$ \partial \omega = \frac{1}{2}(\text{d}\omega - i \text{d}(J\omega)), $$ it is easy to see that $$ -\frac{i}{2}d\omega = d(J\omega). $$ Thus, $J\omega = -\frac{i}{2}\omega + \omega'$, where $\omega'$ is some closed form. What this $\omega'$ is, however, I cannot see.
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4$\begingroup$ Isn't defining an almost complex structure $J$ as a section of $End(T_M)$ which satisfies $J^2 = -id_{T_M}$ pretty global? Also, to derive your last equation you need to know that $d = \partial + \bar \partial$, which is equivalent to $J$ being integrable. And what is your $\omega$? $\endgroup$– Gunnar Þór MagnússonCommented Feb 22, 2011 at 16:14
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$\begingroup$ @Gunnar For your first question: The definition of an almost complex structure $J$ is certainly global, but the construction of the canonical $J$ for a complex manifold is what I'm interested in. Surely, there is more than one almost complex structure on a complex maniold, ie $J^2 = $id$_{T_M}$ does not define it uniqely. So I suppose my question is how does one identify the cancoical one globally? For your second: As I said just above, I am assuming that my manifold is complex, and so, I certainly have $\text{d}=\partial + \overline{\partial}$. $\endgroup$– John McCarthyCommented Feb 22, 2011 at 16:23
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1$\begingroup$ Indeed it is often the case that there more than one complex structure on a complex manifold. In any case, once you fix a complex structure, the almost complex structure associated to it is multiplication by i. How is this definition not global? $\endgroup$– Andrea FerrettiCommented Feb 22, 2011 at 16:40
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5$\begingroup$ The whole question seems to stem from a misunderstanding. $J$ is an endomorphism of $TM$. It acts on the dual space $T^*M$ by the dual map, and on $k$-forms (possibly complex-valued) by multilinear extension. $\Lambda^{1,0}T^*M$ is by definition the $+i$ eigenspace of $J$ in $T^*M \otimes \mathbb{C}$, and $\Lambda^{0,1}T^*M$ the $-i$ eigenspace. $J$ acts on $\Lambda^{p,q}T^*M$ as multiplication by $i^{p-q}$. The $(1,0)$-part of a 1-form $\alpha$ is $\frac{1}{2}(\alpha - iJ\alpha)$, so $\partial f = \frac{1}{2}(df + iJ(df))$ for a function. The expressions in the question seem incorrect. $\endgroup$– Johannes NordströmCommented Feb 22, 2011 at 18:45
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2$\begingroup$ Ok, I see now what's going on. Thanks a lot Johannes. Sorry for asking a question before I understood waht I was asking about. I think the best thing to do with this question would be to close it. $\endgroup$– John McCarthyCommented Feb 22, 2011 at 20:51
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the construction of the canonical $J$ for a complex manifold is what I'm interested in
Given a complex manifold, you have a bundle of (1,0)-forms within complexified 1-forms which is generated (over $C^\infty$) by differentials of holomorphic functions. This gives a decomposition of 1-forms tensor C into (1,0) and (0,1)-part. Your I is an operator which is equal to $\sqrt -1$ on (1,0)-forms and $-\sqrt -1$ on (0,1)-forms. It is in fact real, hence defines a real endomorphism of TM, squared to -Id.
answered Feb 23, 2011 at 20:57
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$\begingroup$ This definition is of course also local; there do not need to exist nontrivial holomorphic functions. But: complex structures are by definition a local thing, anyway. $\endgroup$ Commented Feb 23, 2011 at 22:06