By Kunihiko Kodaira
Kodaira is a Fields Medal Prize Winner. (In the absence of a Nobel prize in arithmetic, they're considered as the top specialist honour a mathematician can attain.)
Kodaira is an honorary member of the London Mathematical Society.
Affordable softcover version of 1986 classic
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Example text
Then by the Weierstrass preparation theorem, we have the factorization /o(w, Z) = M(W, z)P(w, z), where w is a unit in OQ, and P(w, z) is a distinguished polynomial. Let P(w, z) ==nr=i ^fc(^» ^) t>e the factorization of P{w, z) into irreducible distinguished polynomials in R[w]. Then m /o(w, z) = w n P/c(w, z), with a unit w = M(W, Z) is an irreducible factorization of/o(z) in OQ. Since/O(W, Z) = 0 is minimal, the irreducible polynomials Pfc(w, z), /c = 1 , . . , m are mutually coprime. Consequently k j=\ k9^j is not divisible by any Pfc(w, z).
A connected analytic subset S of M" without singular points is called a complex submanifold of M. ^ S and that L^R(g) ^ U{q) for qe S. } is a locally finite open covering. {p) and Vj for i^iqj). 9) where m = n-Pj is independent of 7. In fact if 5 n L^ n L4 7^ 0 , it is clear that rrij = m^. Then the assertion follows from the connectedness of S. From this result, we see that S is itself a complex manifold. ,zr{p))) is a homeomorphism of Vj onto a polydisk in C". 1. Complex Manifolds 35 is biholomorphic since it is the restriction of TJ^: /^l ^^ -,"\ ^ / ^ l ^"1 (Zfc, .
When we define the product of two automorphisms gi, g2 of W by their composite gig2, the set of all automorphisms of W forms a group, which we denote by ^. The unit of ^ is the identity of W^ and the inverse of g e ^ is the inverse map g~^ of g. Any subgroup of ^ is called a group of automorphisms of W. Let G be a group of automorphisms of W. For peW, the set Gp = {g(p) \geG} is called the orbit of G through p. Two orbits Gp and Gq do not have a common element unless they coincide. Thus W is decomposed into the mutually disjoint orbits of G.