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By Jones G.A., Singerman D.

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Low Dimensional Topology

During this quantity, that's devoted to H. Seifert, are papers according to talks given on the Isle of Thorns convention on low dimensional topology held in 1982.

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A a = minimum number of generators of m over 0 (from Nakayama's lemma again). a a 30 II. I. 8 From our earlier discussion of non-singularity we get the II. J. THEOREM Let X be an analytic space. The set of points at which X is a complex manifold is the complement of an analytic subvariety. From the definition of dimension one sees that dimnXis upper semicontinuous in a, and that Xis non-singular at a just in case climax If = dimcr:<~cr:)/ma(OX/O::)a. is irreducible then dimaXis constant, say at n, since the regular locus is connected.

11. l J : (x, v) -> (x, 11 • •(x)v) ~ If X is a differentiable manifold one gets the notion of a differentiable va:tor bundle by requiring all maps to be differentiable, using the natural differentiable structures on a::n, GL(n, CC:). If Xis an analytic space one defines a holomorphic vector bundle by requiring Y to be an analytic space and all maps to be holomorphic (using the natural n structures on a:: , GL(n, 0:)). A slight modification is required to define algebraic vector bundles over algebraic varieties: Here one requires Y to be an algebraic variety CD to be algebraic.

U. x a::Z i l ar 2 with 71 i algebraic. GL(n, O::) is a Zariski open of a::n and so has a natural algebraic 34 II. 2. 2 structure; one requires the maps II . l] to be algebraic. E_> X is a vector bundle (continuous, differentiable, holomorphic or algebraic) one defines a sheaf of 0 cont' 0 diff' Ohol' or 0 alg - modules r (U,r (Y)) = sections r (Y) -1 of 1fJ : r/> (U) -> U These are sheaves of modules because one has the isomorphisms (or a:zn ar) with which to multiply by functions pointwise and add; the linearity of the patching shows that this is unambiguously defined.