The total space of cotangent bundle of any manifold $M$ is a symplectic manifold.


Is it true/false/unknown that for any $M$, $T^\*M$ has Kähler structure?


Please support your claim with reference or counterexample.



Kähler structure on cotangent bundle?
This is true! I assume $M$ compact.


**Method 1.** Real algebraic geometry. Cf. [Fukaya, Seidel, and Smith - Exact Lagrangian submanifolds in simply-connected cotangent bundles](http://arxiv.org/abs/math/0701783). By a version of the Nash–Tognoli embedding theorem, one can realise $M$ as an real affine algebraic variety $V\_\mathbb{R}$, cut out by polynomials $f\_i \in \mathbb{R}[x\_1,\dotsc,x\_N]$. The complex variety $V\_\mathbb{C}$ will then be smooth in a small neighbourhood $U$ of $V\_\mathbb{R}$, hence Kaehler in that region, with $V\_{\mathbb{R}}$ as a Lagrangian submanifold. But $U$ is diffeomorphic to $T^\ast M$. The resulting symplectic structure on $T^\ast M$ may be non-standard; via the Lagrangian neighbourhood theorem, you can take the symplectic form to be the canonical one if you'll settle for a Kaehler structure only near the zero-section.


**Method 2.** Eliashberg's existence theorem for Stein structures. See Cieliebak–Eliashberg's unfinished book, [Symplectic geometry of Stein manifolds](http://math.stanford.edu/~eliash/Public/Site/Eilenberg_Lectures_files/stein7.pdf), Theorem 9.5. We observe that $T^\ast M$ has an almost complex structure $J$ (one compatible with the canonical 2-form, for instance) and a bounded-below, proper Morse function $\phi$ whose critical points have at most the middle index (namely, the norm-squared plus a small multiple of a Morse function pulled back from $M$). In this situation Eliashberg, via an amazing chain of deformations, finds an integrable complex structure $I$ homotopic to $J$ such that $dd^c \phi$ is non-degenerate. This makes $T^\ast M$ Stein! His theorem only applies in dimensions $\geq 6$ (the paper [Constructing Stein manifolds after Eliashberg](http://arxiv.org/abs/0810.4511) of Gompf explains what you have to check in dimension 4), so without doing those checks or appealing to other methods, the case of $M$ a surface is left out.


I think that the more precise version of Eliashberg's theorem, which may not yet be in the book, would tell us that the Stein structure is homotopic to an easy-to-write-down Weinstein structure on $T^\ast M$ involving its canonical symplectic structure $\omega\_\text{can}$, hence that $dd^c\phi$ is symplectomorphic to $\omega\_\text{can}$.


