By Klaus Fritzsche

This creation to the speculation of complicated manifolds covers crucial branches and strategies in advanced research of numerous variables whereas thoroughly warding off summary thoughts regarding sheaves, coherence, and higher-dimensional cohomology. purely simple tools equivalent to energy sequence, holomorphic vector bundles, and one-dimensional cocycles are used. every one bankruptcy features a number of examples and routines.

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**Example text**

Here and in the future we will identify the linear map L with the matrix of L. If the rank of the linear map L is s, then there exist 5 linearly independent rows of the corresponding linear equations, and we solve the system with the right-hand side of these s linear equations arbitrarily prescribed. Let us index such a set as follows: If we fix such a set of components, let us consider the remaining ambiguity in our coframes. , L(JC) = yo>then or K = v' + v with v e kerL. As such, and making the initial Lie algebra compatible absorption, we have the remaining possible normalized Lie algebra compatible absorption in the form w = n' + va) with As a result the induced changes on the torsion coefficients become which means that the torsion coefficients are now well defined.

Now since G acts transitively on rv(U x G) it can be identified with the homogeneous space G/GTo. The structure map has constant rank, hence TM '(TO) is a manifold. The point iu(u,e] is on the orbit, hence for each u there is a C(u) e G such that Therefore is a manifold which submerses onto U. A To-modified coframe is a section of T^(TO). The Implicit Function Theorem implies that there is a local section, r(M) = (n,A/(M)), where 38 LECTURE 4 which satisfies Utilizing this map flu : U —> G we can construct a t§-modified coframe by taking the coframe flu^uNote if there is an equivalence O1 with ®I(UQ,SQ) = (VQ,TQ) then With this preparation we can now state the most important and historically worst-treated result in the subject.

The Implicit Function Theorem implies that there is a local section, r(M) = (n,A/(M)), where 38 LECTURE 4 which satisfies Utilizing this map flu : U —> G we can construct a t§-modified coframe by taking the coframe flu^uNote if there is an equivalence O1 with ®I(UQ,SQ) = (VQ,TQ) then With this preparation we can now state the most important and historically worst-treated result in the subject. THEOREM (REDUCTION OF THE STRUCTURE GROUP). > induces a GlQ-equivalence between iQ-modified coframes given by Proof.