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Automorphisms of Complex Manifolds

Let M be a complex manifold and ℌ(M) the group of holomorphic transformations of M. In general, ℌ(M) can be infinite dimensional. For instance, ℌ(C n ) is not a Lie group if n≧2. To see this, consider transformations of C2 of the form $$ \begin{array}{*{20}{c}} {z' = z} \\ {w' = w + f\left( z \right)} \\ \end{array} \left( {z,w} \right) \in {{C}^{2}} $$ where f(z) is an entire function in z, e. g., a polynomial of any degree in z. The fact that ℌ(C2) contains these transformations shows that ℌ(C2) cannot be finite dimensional. Similarly, for ℌ(C2) with n≧2. On the other hand, ℌ(C) is the group of orientation preserving conformal transformations and, as we shall see later, it is a Lie group. The purpose of this section is to give conditions on M which imply that ℌ(M) is a Lie group.

Automorphisms مجموعه Manifolds مجتمع

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