By Luis A. Santaló

Crucial geometry originated with difficulties on geometrical chance and convex our bodies. Its later advancements, even though, have proved to be beneficial in different fields starting from natural arithmetic (measure idea, non-stop teams) to technical and utilized disciplines (pattern acceptance, stereology). This ebook is a scientific exposition of the idea and a compilation of the most leads to the sector. the quantity can be utilized for a one-semester undergraduate direction in chance and differential geometry or as a supplement to classical classes on differential geometry, Lie teams, or chance.

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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.