Kähler structure on cotangent bundle? 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.
 A: MR1131444 (93e:32018)
Guillemin, Victor(1-MIT); Stenzel, Matthew(1-MIT)
Grauert tubes and the homogeneous Monge-Ampère equation.
J. Differential Geom. 34 (1991), no. 2, 561–570.
32F07 (32E10)
A: In a paper by Goldman, Kapovich, and Leeb, it is pointed out that a fuchsian (surface) group embedded into the isometries of complex hyperbolic space has quotient the tangent bundle to the surface. Since the tangent and cotangent bundles are diffeomorphic (e.g., they may be identified using a Riemannian metric), the cotangent bundle will admit a Kahler structure. However, I'm not sure if this is compatible with the natural symplectic form on the cotangent bundle (which I'm guessing is implicitly required in your question). 
There are restrictions on the fundamental groups of closed Kahler manifolds, but I do not know of restrictions on open Kahler manifolds (however I am far from being an expert). 
A: 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. 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, 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 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 $\omega_\text{can}$.
A: I think it is false, in general. I have heard in a talk that $T^*M$ of Riemannian manifolds with non-constant curvature are "standard" examples of strictly almost Kahler manifolds. Quick google search gives me Porosniuc - A locally symmetric Kaehler Einstein structure on the cotangent bundle of a space form whose theorem 3 seems to give an answer.
A: The tangent bundle $TM$ of a Riemannian manifold M has a natural Kähler structure with the Kähler form agreeing with the canonical symplectic form of $TM$ coming from the cotangent bundle.
To see this, pick local coordinates $\mathbf{x}=(x_1,\ldots,x_n)$ on M and let the metric be given by a positive definite matrix A
$$g = d\mathbf{x}^TAd\mathbf{x}$$
Introduce complex coordinates $\mathbf{z}=\mathbf{x}+i\ d{\mathbf{x}}$ and lift the the metric to a Hermitian metric $h$ on $TM$
$$h = d\mathbf{z}^*Ad\mathbf{z} = (d\mathbf{x}-i\ d^2\mathbf{x})^T\ A\ (d\mathbf{x}+i\ d^2\mathbf{x}) $$
(Here $d^2\mathbf{x}=(d^2x_1,\ldots,d^2x_n)$ are coordinates on the second order tangent space.)
The Kähler form is
$$ \Omega = -\text{Im}\ h(d\mathbf{z}_1, d\mathbf{z}_2) = d^2\mathbf{x}_1^T\ A\ d\mathbf{x}_2 - d\mathbf{x}_1^T\ A\ d^2\mathbf{x}_2 $$
and since the momentums (cotangent coordinates) are $\mathbf{p}=d\mathbf{x}^TA$, the Kähler form becomes
$$ \Omega = d\mathbf{p}_1\ d\mathbf{x}_2 - d\mathbf{p}_2\ d\mathbf{x}_1$$
which is the canonical symplectic form of $T^*M$.
A: In the reference mentioned by Zemisch, Guillemin and Stenzel prove:
Theorem. For an analytic manifold $L$ and analytic metric $g$ on $L$,
there is a $\sigma$-invariant neighborhood ($\sigma(x,v)=(x,-v)$) of $L\subset T^*L$ with a unique complex structure such that
i- $\sigma$ is an anti-holomorphic involution
ii- The one-form $\operatorname{Im} \bar\partial h$, where $h=|v|^2$ is the square of length of $v$ with respect to $g$, is the standard one-form $\sum y_i dx^i$. (This would imply $\sqrt{-1}\partial \bar\partial h$ is the standard Kähler form).
This is indeed an impressive result.
