By H. Grauert, K. Fritzsche
The current booklet grew out of introductory lectures at the thought offunctions of numerous variables. Its motive is to make the reader regular, by means of the dialogue of examples and targeted situations, with an important branches and techniques of this thought, between them, e.g., the issues of holomorphic continuation, the algebraic therapy of energy sequence, sheaf and cohomology idea, and the genuine equipment which stem from elliptic partial differential equations. within the first bankruptcy we commence with the definition of holomorphic services of a number of variables, their illustration by way of the Cauchy imperative, and their strength sequence enlargement on Reinhardt domain names. It seems that, in l:ontrast ~ 2 there exist domain names G, G c en to the idea of a unmarried variable, for n with G c G and G "# G such that every functionality holomorphic in G has a continuation on G. domain names G for which this sort of G doesn't exist are known as domain names of holomorphy. In bankruptcy 2 we provide a number of characterizations of those domain names of holomorphy (theorem of Cartan-Thullen, Levi's problem). We eventually build the holomorphic hull H(G} for every area G, that's the biggest (not unavoidably schlicht) area over en into which every functionality holomorphic on G should be continued.
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Additional resources for Several Complex Variables
Gog -11V = id v is a holomorphic mapping. · gA,w" + L gv,z; . JbW) IDl. 0 ~ ( 0_" gn,w" Since det 9)1g =1= 0 there is only the trivial solution: gA, w" = 0 for all A and all fl. This holds in all of V. Therefore the Cauchy-Riemann differential D equations are satisfied and g-l is holomorphic. 5. Let Been be a region, g = (gb ... ,gn) holomorphic and oneto-one in B. Then Mg =1= 0 throughout B. 27 1. Holomorphic Functions This theorem is wrong in the real case: for example y = x 3 is one-to-one, but the derivative y' = 3x 2 vanishes at the origin.
If B is holomorphically convex then there exists a normal exhausting (Kv) of B with the property that Kv = Rv for every v EN. Let (Kv) be any normal exhaustion of B. Then for all v, Kv c c B and as B is holomorphically convex, it follows that Rv c c B. Rv is therefore a compact subset of B. We now construct a subsequence of the Rv. LetK~: = R 1 • . PROOF. 49 II. Domains of Holomorphy Suppose Ki, ... , K~-1 have been constructed (K~-1 compact and K~-1 = K~-l). (v). (v) Clearly the K~ are compact subsets of B with K~ = K~.
In particular, 3~ then lies in G - :F and by case (a) we can join it with 32' Ifal, 32 E U then both points can be connected with a point 30 E (; - :F and therefore with one another. Figure 11. 5. 2. Let n: e x en -1 -4 en -1 be the projection onto the second component. Thennl:F::F -4 Gisa topological mapping with (nl:F)-1 = gandn(:F n U) is an open neighborhood V of 30: = n(30). Let h(zJ, ... , zn): = (idG,(ZI), h2(Z2),"" h~(zn)) with h~(zv): = (zv - z~)j (z~ Zv - 1) for v = 2, ... ,n. h: P -4 P is a biholomorphic mapping with h(O) = (0,30)' Set ql: = q and choose qv with < qv < 1 for v = 2, ...
Several Complex Variables by H. Grauert, K. Fritzsche