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 expert honour a mathematician can attain.)

Kodaira is an honorary member of the London Mathematical Society.

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Put , M =U (X( = {(z, ()E C" XlPn-llzE n. Let (Zto ... , zn) be the standard coordinates on en, and «(to ... , (n) the homogeneous coordinates on lPn-I. In terms of these coordinates, we denote a point (z, () by (Zto ... , Zn, (to ... , (n). Then Z E ( if (jZk - Zj(k = 0 for j, k = I, ... , n. 18) j, k= I, ... , n. is covered by n pieces of coordinate neighbourhoods ~ = ... ,n,C"xlPn-1 is covered by en X Vis. To verify that M is a submanifold enxlPn-t, consider MI=MIIC"XVI. Let (W2, ••• , Wk, ••• , wn ) be the inhomogeneous coordinates on VI where Wk = (kill.

Z~(p)) with centre q. Take r>O such that the closed polydisk {(z~, ... , z~) Ilz~1 ~ r, k = 1, ... , n} is contained in the range of Zq, and put Ur(q) = {p Ilz~(p)1 < r, ... , Iz~(p)1 < r}. Given q E W, if we choose r sufficiently small, we have g( Ur(q)) n Ur(q) = 0 for any g E G except the identity. In fact, if otherwise, there is an element gn E G, gn ¥- 1, for each n = 1,2, ... , such that gn( Un) nUn ¥- 0, where Un = Ur/n(q). Then gn(U,)n U,¥-0 for any n. Since [U,] is compact, and Gis properly discontinuous, {g" ...

If Tjk is biholomorphic for any j, k such that ~ n Uk ~ 0, each Zj: p ~ Zj (p) is called local complex coordinates defined on ~, and the collection {z" ... , Zj, ... } is called a system of local complex coordinates on L. 1. If a system of local complex coordinates {z" ... , Zj, ... 1. Complex Manifolds is defined on :l. A connected Hausdorff space is called a complex manifold if a complex structure is defined on it. We denote a complex manifold by the letters M, N, etc. The system oflocal complex coordinates {Zh ...

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