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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 $n\ge 2$? Another question: when $X$ is an Abelian variety, are there some explicit constructions similar to theta bundles?

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)functions, and the topology is the strong one).

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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 $n\ge 2$? Another question: when $X$ is an Abelian variety, are there some explicit constructions similar to theta bundles?

A bit more specialized question. It is well known that for a nonsingular projective 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).

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# 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 $n\ge 2$? Another question: when $X$ is an Abelian variety, are there some explicit constructions similar to theta bundles?