Complex Manifolds and Deformation of Complex Structures by Kunihiko Kodaira

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.

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Complex Manifolds and Deformation of Complex Structures

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 reach. ) Kodaira is an honorary member of the London Mathematical Society. cheap softcover variation of 1986 vintage

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Put (o^„ = m(o-^n, and define mn/ P{z) is a meromorphic function on C, called the Weierstrass P-function. P(z) is G-invariant, that is, P(z + w^„) = P(z) for any comn- Hence P(z) is considered as a meromorphic function on C = C/G. P{z) has a pole of order 2 at each w^„, and holomorphic elsewhere. Therefore as a meromorphic function on C, P(z) has a pole of order 2 at the point 0 e C corresponding to OG C, and holomorphic elsewhere. 15) where the coefficients g2 and g3 are given by g2 = 60 I ^-, g3=140 I ^ .

Complex Manifolds for pe U{q). , z-(p),fi,(p),... 19. , z'^{p),f\ip),... ,r,ip)), m = n~v, as local coordinates with centre q. In terms of these, we have SnU{q) = {peU{q)\z-^\p) = ''' = z''^{p)==0}. 5. 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.

Makes a system of local complex coordinates of M". Hence there are infinitely many choices of systems of local complex coordinates for one and the same complex manifold M". In view of this fact we may define a complex manifold as follows: first let two systems of local complex coordinates {zj} = { z i , . . } and {W),} = { w , , . . } be given on a connected Hausdorff space 2 , Uj the domain of z^, and W^ the domain of WA. ), pe UjnW^, is a homeomorphism from the open set %), = {Zj{p)\p^ Ujn W^} onto the open set Wj^j = {Wj(p) \peW^n Uj}.

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