Geometry

By Pfeffer, Riemannian

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Extra resources for Approach To Integration

Example text

40. 1. Assume that we have constructed u1,.. ,uk satisfying the conditions above. 1, f on an open neighbourhood such that a — 0 on an open neighbourhood of Dk. Hence, if we take Taylor's expansion of the coefficients of the centre of D, we may find P a at with polynomial coefficients, such that < I Now we define uk+l and see that uk+1 satisfies the required — P conditions and so the inductive step is completed. The sequence (u,K) has coefficients which converge uniformly on each and so (uk) converges to a continuous (p,0)—form, u.

Applying the theory of §4, we see that the transition function c4'ij of (N splits as a sum where 29. (here we have identified and 8jj: with using the maps j, 3 described in §4). Clearly and are the transition functions for complex vector bundles on M which we shall denote by TM and TM respectively. By our construction we see that TM and TM are complementary complex eubbundles of (M and so we have (M — Moreover J — the "bar" (Hence +i on TM, i — —i on TM and S(Th) - TM notation). have the natural inclusion map j: m projection P: + Since — j411P and j linear the are holoinorphic and so TM has the holomorphic vector bundle.

Proposition For p < 0, Proof. For p 0, is canonically isomorphic of homogeneous polynomials of degree p on to the space consists of the zero section. Suppose p 0. If s we let s1: -. denote the local representatives of a relative to the standard trivialisation of For 0 i n we have S1 and so 44. n+l Hence we may define the holomorphic map S: 0. — by \ are homogeneous of Since the By Hartog's theorem, S degree zero, S is homogeneous of degree p. as an analytic function which we continue to denote by S.