Let $M$ be a compact manifold with dimension $\geq 3$. By a theorem of ELIASHBERG,

the cotangent bundle of $M$ admits an integrable complex structure $J$ such that

$(T^*M, J)$ is a stein manifold. My question is: Define the map $f$ by

$$f: T^*M \rightarrow T^*M, (x,v) \rightarrow (x, -v), x \in M, v \in T_x^*M$$. Is it true that $f$ is an anti-holomorphic map with respect to the complex structure $J$ on $T^*M$?

  • $\begingroup$ Isn't it holomorphic? The function $\mathbb{C}^n\times\mathbb{C}^n \to \mathbb{C}^n\times\mathbb{C}^n$ given by $(z, w) \mapsto (z, -w)$ is holomorphic. $\endgroup$ – Michael Albanese Nov 9 '14 at 4:06
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    $\begingroup$ The problem I see here is that the construction is rather abstract, and doesn't give you a concrete complex structure, but rather the isotopy class of $J$. What I think can be asked is whether there is a representative in that class such that the map you ask for is holomorphic/antiholomorphic. $\endgroup$ – Marco Golla Nov 9 '14 at 8:50
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    $\begingroup$ @MichaelAlbanese: When $\dim M$ is odd, the map he defines reverses the orientation, so in general it can't be holomorphic. As for the theorem he mentions, take a look at the recent book by Cieliebak and Eliashberg. The theorem he refers to simply states that Stein structure in dimension $\ge6$ are obstructed by topology (existence of a Morse function with critical points of index less than half of the dimension of the manifold) and obstruction theory (existence of an almost-complex structure). $\endgroup$ – Marco Golla Nov 9 '14 at 8:52
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    $\begingroup$ As you said, it shows there is "an" almost complex structure, so there is most probably a family of such. Some of these may be symmetric and some may not. I belive the steps of proof shows that one can respect the symmetry through the construction. Especially becuase the pluri sub-harmonic function can be chosen to be symmetric. $\endgroup$ – Mohammad Farajzadeh-Tehrani Nov 9 '14 at 14:00
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    $\begingroup$ By the way, if you want to export such complex structure to a neighborhood of lagrangian manifold M in some symplectic manifold X such that X admits an anti- symplectic involution $\phi$ with $Fix(\phi)=M$, via Weinstein neighborhood theorem, some where in my thesis i showed that there is symplectomorphism that respects the anti- symplectic involution. $\endgroup$ – Mohammad Farajzadeh-Tehrani Nov 9 '14 at 14:06

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