By Takeo Ohsawa

ISBN-10: 0821820982

ISBN-13: 9780821820988

One of many methods to the research of capabilities of a number of advanced variables is to take advantage of tools originating in actual research. during this concise booklet, the writer offers a lucid presentation of ways those equipment produce numerous worldwide life theorems within the thought of features (based at the characterization of holomorphic services as vulnerable options of the Cauchy-Riemann equations). Emphasis is on fresh effects, together with an $L^2$ extension theorem for holomorphic services, that experience introduced a deeper knowing of pseudoconvexity and plurisubharmonic capabilities. in accordance with Oka's theorems and his schema for the grouping of difficulties, the e-book covers subject matters on the intersection of the speculation of analytic capabilities of numerous variables and mathematical research. it really is assumed that the reader has a easy wisdom of advanced research on the undergraduate point. The booklet may make a very good supplementary textual content for a graduate-level path on advanced research.

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**Extra resources for Analysis of Several Complex Variables**

**Example text**

V,,) _ E zjvj. j=1 Let. {e1, , en} be the standard basis of V. In terms of a. 7 ak(eil A ... A eik) = L (-1)3a(ei, )ei1 A ... Aeik j=1 and by linearity. Then the following exact sequence is obtained: 0-,nV an, n1Vck- ... AV V 3This will be described in detail starting in the next chapter. 2. FUNCTION RINGS AND $ COHOMOLOGY 32 If we extend the range of definition for the operator 8 to vector-valued differential forms, noting that ak commutes with the operation of 8, then 8 cohomology groups HM(Ker ak) with coefficients in Kerak are defined by HP,q(Kerak) := (CM(0) ® Ker ak) fl Ker B/8 (Cp,q-1(12) (9 Ker ak) .

4 (b)), we immediately obtain a deep result on the Lelong number. 17 (Siu's theorem). 1 and V be as defined above. Then, given a positive number c, the set Ec(') :_ {z E 12 I v(Vi, z) >_ c} is an analytic subset of Q. PROOF. 16 (b), Ec(') = n Ec-n/m(Wr) m> 1 However, since Ec-n/m(1m) _{zI (a)

Q(fl n L), which is also called the restriction homomorphism. ) 01 for 0 j 5 n. 5. q(fl) = {0} for all q with 1 <_ q <- n - m. This is called Serre's criterion. For the proof, we need one lemma. 6. The restriction homomorphism a : HP,q(fl) -, HP,q(fl n L1) is a surjection if Hp,q+l(fl) = {0}. In particular. {0} if HP,q+k(fl) = {0} for all k = 0,1. n Ln_1) _ ZIm overlaps the notation for the imaginary part of a complex number, but there should not be any confusion. 28 2. FUNCTION RINGS AND 8 COHOMOLOGY PROOF.

### Analysis of Several Complex Variables by Takeo Ohsawa

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