TITLE: Penrose transform and general wave equations
QUESTION [14 upvotes]: In the late 1960's Penrose developed twistor theory, which (amongst other things) led to an exceptional description for solutions to the wave equation on Minkowski space via the so-called Penrose transform;
If 
\begin{equation}
u(x,y,z,t) = \frac {1} {2 \pi i} \oint_{\Gamma \subset \mathbb{C} \mathbb{P}^{1}} f(-(x+iy) + \lambda (t-z), (t+z) + \lambda (-x + i y), \lambda ) d \lambda, \,\,\,\,\,\,\,\,\,\, (1)
\end{equation}
where $\Gamma \subset \mathbb{C} \mathbb{P}^{1}$ is a closed contour and $f$ is holomorphic on $\mathbb{C} \mathbb{P}^{1}$ except at some number of poles, then $u$ satisfies the Minkowski wave (Laplace-Beltrami) equation $\square_{\eta} u = 0$.
I am aware that there are a number of works in the literature describing twistor theory on curved manifolds, but have not seen explicit constructions along the lines of (1) such that the function $u$ satisfies a wave equation of the form $\square_{g} u = 0$ for (Lorentzian) metric $\boldsymbol{g}$. 
Is it known how to explicitly construct contour integrals similar to $(1)$ for some class of metrics $\boldsymbol{g}$? What about when $\boldsymbol{g}$ is Einstein (e.g. Schwarzschild), in particular? Are there topological obstructions in spacetimes $I \times \Sigma$? What about de-Sitter space?

REPLY [3 votes]: I believe you can express the Penrose transform using a contour integral by
expressing Serre duality in terms of Dolbeault cohomology. For example, IIRC,
on a half-conformally-flat
4-manifold $M$ with twistor space $Z$, functions solving the Laplacian equation correspond to elements
of the sheaf cohomology group $H^1(Z, \mathcal{O}(-2))$ and the evaluation at a point
$x \in M$, with corresponding twistor line $P_x \subset Z$, is to restrict to:
$$
  H^1(P_x, \mathcal{O}(-2)) \simeq H^0(P_x, \mathcal{O})^* \simeq \mathbb{C}.
$$
If we represent our element of $H^1(P_x, \mathcal{O}(-2))$ by a form in
$H^{0,1}(P_x, \mathcal{O}(-2))$,
then choosing a contour that does not meet the singularities from the section $\mathcal{O}(-2)$, the Serre duality map above yields an contour integral by Stokes's theorem.