Take the variety $X$ to be $\mathbb{C}_\infty \times\mathbb{C}_\infty $ with the points $(0,0)$ and $(\infty,\infty)$ removed. Use coordinates $(z,w)\in\mathbb{C}\times \mathbb{C}$ for one chart of the product of the two Riemann spheres. Then $\xi\in\Omega^{0,1}X$given by \begin{eqnarray*} \xi\,=\,\frac{z\,\mathrm{d}\bar z+w\,\mathrm{d} \bar w}{|z|^2+|w|^2}\ . \end{eqnarray*} has $\bar\partial\xi=0$ so we have $[\xi]\in \mathrm{H}^{0,1}(X;\bar\partial)$. My questions are:

1) What is $ \mathrm{H}^{*,*}(X;\bar\partial)$ ?

2) Is $[\xi]\neq 0$ ?

3) Is there a holomorphic sheaf $S$ on $X$ whose $\mathrm{H}^0(X;S)$ gives $ \mathrm{H}^{0,1}(X;\bar\partial)$ (assuming that the latter is nonzero).

4) What is the place to look for methods used for this sort of problem?

Now the background: First I am obviously not an expert in algebraic geometry, so I apologise to those who are for posing this question. The motivation comes from higher dimensional Lorentz invariant soliton theory, about which almost nothing is known. (Richard Ward's construction of a 2+1 dimensional soliton equation by the mini-twistor method being an exception.) The method I am considering is generalising the inverse scattering method to higher dimensional varieties, and the obvious thing to do is to look for geometric objects 'localised' on singularities of codimension more than one. (In the same sense that a meromorphic function is described by poles 'localised' on a point in dimension one.) The variety $X$ above is an obvious place to start for a 2+1 Lorentz invariant system, as it corresponds to the variety $t^2=x^2+y^2+z^2$, and removing the 2 singular points (not obvious but changing variables checks that other points are regular) for the given 0,1 form.

If anyone would be interested in discussing this possible geometric approach to finding higher dimensional soliton systems I would be very happy to do so. (I use 'soliton system' for the reason that such systems might not be classically integrable systems)