By Takeo Ohsawa

One of many ways to the research of capabilities of a number of complicated variables is to exploit tools originating in actual research. during this concise ebook, the writer provides a lucid presentation of the way those tools produce numerous worldwide life theorems within the conception of capabilities (based at the characterization of holomorphic services as susceptible ideas of the Cauchy-Riemann equations). Emphasis is on contemporary effects, together with an $L^2$ extension theorem for holomorphic capabilities, that experience introduced a deeper realizing of pseudoconvexity and plurisubharmonic capabilities. in line with Oka's theorems and his schema for the grouping of difficulties, the ebook covers issues on the intersection of the idea of analytic capabilities of a number of variables and mathematical research. it really is assumed that the reader has a uncomplicated wisdom of advanced research on the undergraduate point. The ebook might make a very good supplementary textual content for a graduate-level direction on advanced research.

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A of subharmonic functions that are bounded from above uniformly on compact sets in Q. _) Si. 12) A real-valued function 0 of class C2 on Si is subharmonic if and only if the following differential inequality holds everywhere: a2 i. 13) If a sequence {j} I of subharmonic functions on I? is monotone decreasing, then lim bj is subharmonic. 12) and often applied is that if V is subharmonic for any is subharmonic and of class C2. then increasing convex function A of class C2 on R. 13), for instance, it is easily seen that log E I f j 12 and Ul I fj I2j=1 (a ?

2) such that n E (zj - aj)gj(z) = 1. 13. R that satisfies Serre's condition is closed in Spec,,, A(f2) with respect to the weak topology. 12. R) by it a(vi ... 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) .

PROOF. 16 (b), Ec(') = n Ec-n/m(Wr) m> 1 However, since Ec-n/m(1m) _{zI (a)