References for some analogs of the Picard group. Let $X$ be a compact complex manifold. By definition,
$Pic(X)={\rm H^1}(X,\mathcal{O}^\times)$. We know a lot 
about this group. What is known about the groups
${\rm H^n}(X,\mathcal{O}^\times)$ for $n\ge 2$? 
A bit more specialized question. It is well known that for a 
nonsingular projective complex variety $X$ the natural map 
$${\rm H^1}(X,\mathcal{O}^\times)\to{\rm H^1}(X,\mathcal{M}^\times)$$
is trivial. What is known about the kernel of the same map
for $n=2$ or $n=3$? (Here $\mathcal{M}^\times$ is the sheaf of 
nonzero meromorphic functions, and the topology is the strong one).
 A: First of all, it probably depends on how you define $H^1(X, \mathcal{O}^\times)$. I don't see any reason why derived functor cohomology should agree here with Cech cohomology. 
I think that $H^i(X, \mathcal{O}^\times)$ is a functor of order $i+1$ in the sense of Mumford "Abelian Varieties" (2.6, Remark preceding the proof of the theorem of the cube), at least for complex projective varieties. That is, there is a higher analogue of the theorem of the cube for $H^i(X, \mathcal{O}^\times)$. For this, we look at the exponential sequence as in the aforementioned Remark. 
A: Here is a reference: Grothendieck's three exposés in Dix Exposés sur la Cohomologie des Schémas (and the references therein). One can find there e.g. computation of $H^i_{ét}({\rm Spec}\text{ } \mathcal{O}_K, \mathbb{G}_m)$ for spectra of rings of integers in number fields.
MR0244269 (39 #5586a) Grothendieck, Alexander Le groupe de Brauer. I. Algèbres d'Azumaya et interprétations diverses. (French) 1968 Dix Exposés sur la Cohomologie des Schémas pp. 46–66 North-Holland, Amsterdam; Masson, Paris, 14.55
MR0244270 (39 #5586b) Grothendieck, Alexander Le groupe de Brauer. II. Théorie cohomologique. (French) 1968 Dix Exposés sur la Cohomologie des Schémas pp. 67–87 North-Holland, Amsterdam; Masson, Paris, 14.55
MR0244271 (39 #5586c) Grothendieck, Alexander Le groupe de Brauer. III. Exemples et compléments. (French) 1968 Dix Exposés sur la Cohomologie des Schémas pp. 88–188 North-Holland, Amsterdam; Masson, Paris (Reviewer: J. S. Milne), 14.55
