Download Complex Manifolds and Deformation of Complex Structures by Kunihiko Kodaira PDF

By Kunihiko Kodaira

ISBN-10: 1461385903

ISBN-13: 9781461385905

ISBN-10: 146138592X

ISBN-13: 9781461385929

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.

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Since clearly G is fixed point free, W= W / G is a complex manifold and has a group structure as the quotient group of W by G. Thus W is a complex Lie group. S. A complex vector space C" is a complex Lie group with respect to the usual addition. Take 2n vectors Wj = (wj, ... , wj) E C" for j = 1, ... ,2n, such that these Wj are linearly independent over IR. Then Wj generate a discrete subgroup of C". Since a fundamental domain of G is compact, T" = C" / G is a compact commutative complex Lie group, which we call a complex torus.

I Corollary. Let f(z) be a holomorphic function in a domain of em, and S the analytic hypersurface defined by the equation f( z) = O. We assume 0 E S. Supppose fo( z) = 0 is a minimal equation of S at o. c (z) = 0 is a minimal equation of S at c for every c E S with Icl < E. 14. Let U be a domain where f( z) is defined. f( z) = 0 is called a minimal equation of S in U if i (z) = 0 is a minimal equation of S at every c E S (\ U. 17, if f(z) = 0 and g(z) = 0 are both minimal equations of S in U, u(z) = f(z)/g(z) is a non-vanishing holomorphic function in U.

We may assume that OUj is a polydisk with centre Cj = Zj(q). Put g(w) = jj(c] + w(z] - c]), ... , cj+ w(zj - cj)). Then for (z], ... , zj) E ~,g( w) is a holomorphic function of won Iwl < 1 + 8 if 8 is sufficiently small, and Ig( w)1 attains its maximum at w = o. Consequently, by the maximum principle, g(w) is a constant. Thus jj(p) is a constant on ~, and, by the analytic continuation, one sees that f( p) is a constant on all of M. I In this section we give several examples of compact complex manifolds.

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Complex Manifolds and Deformation of Complex Structures by Kunihiko Kodaira

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