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)



Cohomology of a projective variety with points removed
If I understand correctly, you are trying to compute the Dolbeault cohomology of your $X$. Also, I am assuming that by $\mathbb C\_\infty$ you mean the Riemann sphere, which in algebraic geometry would be the projective line $\mathbb P^1\_{\mathbb C}$. Of course, this is just rephrasing what you are saying, mainly for my own sake.


Let me actually answer in sort of a general way. Let's say that $\bar X$ is a smooth projective variety, $Z\subseteq \bar X$ is a closed subvariety and $X=\bar X\setminus Z$. In your case $\bar X=\mathbb P^1\_{\mathbb C}\times \mathbb P^1\_{\mathbb C}$ and $Z=\{P,Q\}$ with $P\neq Q\in \bar X$.


Anyway, the simple answer to your first question is that 
$$ H^{p,q}(X, \bar\partial)\simeq H^q(X,\Omega\_X^p),$$
where $\Omega\_X^p$ is the sheaf of holomorphic $p$-forms on $X$. I realize that you may already know this, so here is a little more. 


It might turn out that you can easily compute $H^q(\bar X,\Omega\_{\bar X}^p)$ (which is true in your case: this is very easy on $\mathbb P^1\_{\mathbb C}$, and then use the Künneth formula). If that's the case, then you can use the following long exact sequence:
$$
\dots\to H^q\_Z(\bar X,\Omega\_{\bar X}^p)\to H^q(\bar X,\Omega\_{\bar X}^p) \to H^q(X,\Omega\_X^p) \to H^{q+1}\_Z(\bar X,\Omega\_{\bar X}^p)\to\dots
$$
where $H^q\_Z(\bar X,\Omega\_{\bar X}^p)$ is the local cohomology supported at $Z$. 
Of course, now you need to be able to compute $H^q\_Z(\bar X,\Omega\_{\bar X}^p)$. In general this may or may not be easy. In your case it is:
Since your $Z$ is the disjoint union of two points, you only need to compute the same with $Z$ replaced by a single point. 


Next replace $\bar X= \mathbb P^1\_{\mathbb C}\times \mathbb P^1\_{\mathbb C}$ by $\mathbb A^2\_{\mathbb C}$. The local cohomology will be the same, because it only depends on a neighbourhood of $Z$. $\Omega \_{\mathbb A^2\_{\mathbb C}}^p$ is trivial, so what you get is that you only need to compute the local cohomology of the polynomial ring $k[x,y]$ at the maximal ideal $(x,y)$. You can probably do this yourself or find it in a book as an example. 


So, now you can bootstrap back to your original cohomology group. You will probably have to write down some maps explicitly, but $H^q(\bar X,\Omega\_{\bar X}^p)$ is small, so most of your cohomology will come from the local cohomology contribution. Once you do all this, I am sure you will be able to decide whether $[\xi]\neq 0$.


As far as your third question is concerned, I am not entirely sure what you consider a "holomorphic sheaf". Do you mean *coherent*? If so, then the answer is probably "no". This will be a pretty big group. From the long exact cohomology sequence you can see that most of the cohomology is supported at the missing points. You can write down a (double) skyscraper sheaf on $\bar X$ whose sections will give you that part of the cohomology, but I am not sure that that's what you want.


For #4, I'd say that as you see above, probably the best way to deal with this is via local cohomology.


