By S. G. Gindikin, S. G. Gindikin, G. M. Khenkin

This quantity of the EMS includes 4 survey articles on analytic areas. they're first-class introductions to every respective sector. ranging from simple rules in numerous advanced variables each one article stretches out to present developments in learn. Graduate scholars and researchers will discover a priceless addition within the broad bibliography on the finish of every article.

**Read Online or Download Several Complex Variables IV: Algebraic Aspects of Complex Analysis PDF**

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**Extra resources for Several Complex Variables IV: Algebraic Aspects of Complex Analysis **

**Sample text**

In the case when X is reduced it suffices to apply Theorem of ideals 9 c 0x consisting of the stalks 9x = {f E Ox,,1f,h, h E T(X, J&*) is the given meromorphic function on X. For the see [l]. We recall that the group of divisor classes CD(X) of a complex viewed as a subgroup of Pit X. Theorem H’(X, Z). 14. 5). Applying Theorem A to the sheaf of germs of sections $ of the bundle E, we can construct a holomorphic section s of the bundle E over all of X such that s(x) # 0 on a dense set of points x in X.

We consider one application of Theorem B. 1. Let X be a Stein space, 3 a coherent sheaf of ideas, and Y c X an analytic subspace defined by the sheaf 9. , any holomorphic function on Y extends to a holomorphic function on X. The theorem follows sequence of sheaves 0 + The proof of Theorem B holds is a Stein space. 1 shows that More precisely, equation H’(X, y) = 0 and the exact 0. any complex space for which Theorem the following result is true. 2. A complex space X with a coutable basis is a Stein space if and only ifH’(X, 9) = 0 for any coherent sheaf of ideals 3 c 0, with a discrete zero set.

6 [64]. Letf : Y -+ X be a holomorphic mapping with X a Stein space. Assume that every point in X has a neighborhood U such that for any connected component W of the set f -l(U) the mapping f : W -+ U is finite. Then Y is a Stein space. In the article [ 1011it is proven that any Stein analytic subspaceof a complex space X has a Stein neighborhood in X. L. Let(x,yi)fori= 1,2 ,... be a sequence of germs of analytic sets in various Cni, all irreducible components of which have the same dimension k 3 1.