Abstract

Sheaf theory can be viewed as the area of geometry that studies the mathematical transition from local properties to global properties. The cohomology of a sheaf tells us about any obstacles that may be preventing us from making this transition. For example, let B be a connected open subset of ℂn. A meromorphic complex function on B is a complex function h defined on B − S, where S is a set of isolated points in B, satisfying the following local property for every b ∈ B − S: there exists an open neighborhood U of b in B and holomorphic complex functions fU, gU on U such that hU=fUgU. This is a direct generalization of Lemma-Definition 4.59. Poincaré showed in 1883 that every meromorphic function h on B is of the form h=fg, where f, g are two functions that are holomorphic on B (global property) when B=ℂ2. This result had previously been shown by Weierstrass in the case where B=ℂ (Corollary 233). One of the key questions studied by the theory of functions in multiple complex variables is which types of open subset B of ℂn still allow this transition from the local to the global property. Any such open sets B are known as solutions of the Poincaré problem (we shall denote this problem by (P) below).

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