Download From Holomorphic Functions to Complex Manifolds by Klaus Fritzsche PDF

By Klaus Fritzsche

This advent to the idea of complicated manifolds covers an important branches and techniques in advanced research of a number of variables whereas thoroughly heading off summary suggestions regarding sheaves, coherence, and higher-dimensional cohomology. merely simple tools comparable to energy sequence, holomorphic vector bundles, and one-dimensional cocycles are used. each one bankruptcy includes a number of examples and routines.

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Extra resources for From Holomorphic Functions to Complex Manifolds

Example text

For v = 1 , . . ' n - 1 let Tv be the automorphism of D defined by = := Then h : pn ---+ pn with h(z1 , . . , zn ) := (T1 (z2) , . . , Tn - 1 ( zn ) , z! ) is holo­ morphic, h(O) (z�, O) , and h({z E pn : ! z1 l > q}) C {w E pn : lwn l > q } . 3. Extending a holomorphic function across a hypersurface 1. We define q1 := The Continuity Theorem 47 q, and for v = 2, . . , n choose qv such that h (D X Dq2 (0) X · · · X Dq, (O)) C U' X D. 2, . . 3). Since P n G' G' is connected, the proposition follows from the continuity theorem.

Let f(z) = :Z::: v >O avz v and g (z) = l: v >O bv z v be two convergent power series in a neighborhood U = U(O) c en . Ifthere is a neigborhood V(O) c U with f l v = 9 l v, then a, = b, fo r all v . PROOF: We know that f and g are holomorphic. Then D v f (0) = D v g ( O ) for all v , and successive differentiation gives Dv f ( O )= v! av and Dv g ( O ) = v! b,. • 4 . 8 Cor ollary. Let G c e n be a domain, Zo E G a point, and f : B --+ e a holomorphic fu nction. If f ( z) = :Z::: v >o av (z - zo) v is the (uniquely deter­ m ined) power series expansion near z0 -E G, then a, = � .

D.. " (zo) t , := w . (D.. ' (z o)) t + w · (D.. '' (zo)) t . " (zo ) ll , it follows that f is differentiable at z0 with derivative L. • I. Holomorphic Functions 28 Wir tin ger ' s Calculus Definition . Let f : B representation --+ e be real differentiable at zo . If we have a f(z) = f(zo) + (z - zo) . �' (z) t + (z - zo) . �" (z) t , with �' and � " continuous at z0, then the uniquely determined numbers and of Ozv (zo) = fzv (zo) := ev - �"(zo) t are called the Wi rtinger derivatives of f at zo.

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