Holomorphic mapping

A mapping $ f: D \rightarrow D ^ { \prime } $ of a domain $ D \subset \mathbf C ^ {n} $ into a domain $ D ^ { \prime } \subset \mathbf C ^ {m} $ under which

$$ z = ( z _ {1} \dots z _ {n} ) \rightarrow \ ( f _ {1} ( z) \dots f _ {m} ( z)), $$

where all coordinate functions $ f _ {1} \dots f _ {m} $ are holomorphic in $ D $. If $ m = 1 $, a holomorphic mapping coincides with a holomorphic function (cf. Analytic function).

A holomorphic mapping is called non-degenerate at a point $ z \in D $ if the rank of the Jacobian matrix $ \| \partial f / \partial z \| $ is maximal at $ z $( and hence equals $ \min ( n, m) $). A holomorphic mapping is said to be non-degenerate in the domain $ D $ if it is non-degenerate at all points $ z \in D $. If $ m = n $, the non-degeneracy of $ f $ is equivalent to the condition

$$ \mathop{\rm det} \left \| \frac{\partial f }{\partial z } \ \right \| \neq 0. $$

If $ n = m = 1 $, a non-degenerate holomorphic mapping is a conformal mapping. If $ n = m \geq 2 $, a non-degenerate holomorphic mapping does not, in general, preserve angles between directions. If a holomorphic mapping $ f $ is non-degenerate at a point $ a \in D $ and if $ m = n $, then $ f $ is locally invertible, i.e., then there exist neighbourhoods $ U $, $ U ^ { \prime } $, $ a \in U \subset D $, $ f( a) \in U ^ { \prime } \subset D ^ { \prime } $, and a holomorphic mapping $ f ^ { - 1 } : U ^ { \prime } \rightarrow U $ such that $ f ^ { - 1 } \circ f( z) = z $ for all $ z \in U $. If a holomorphic mapping $ f $ maps $ D $ onto $ f( D) $ in a one-to-one correspondence and if $ m = n $, then $ f $ is non-degenerate in $ D $; if $ m > n $, this is not true, e.g. $ z \rightarrow ( z ^ {2} , z ^ {3} ) $, $ D = \mathbf C $, $ D ^ { \prime } = \mathbf C ^ {2} $. If $ m \leq n $ and if $ f $ is non-degenerate in $ D $, then the image of $ D $ is also a domain in $ \mathbf C ^ {m} $; if $ m > 1 $, the principle of invariance of domain does not hold for mappings that are degenerate at certain points, e.g. $ ( z _ {1} , z _ {2} ) \rightarrow ( z _ {1} , z _ {1} z _ {2} ) $, $ D = D ^ { \prime } = \mathbf C ^ {2} $.

If $ M $ and $ M ^ { \prime } $ are complex manifolds, $ \{ ( U _ \alpha , \phi _ \alpha ) \} $ and $ \{ ( U _ \beta ^ { \prime } , \phi _ \beta ^ \prime ) \} $ are atlases of their local coordinate systems ( $ \phi _ \alpha : U _ \alpha \rightarrow D _ \alpha \subset \mathbf C ^ {n} $, $ \phi _ \beta ^ \prime : U _ \beta ^ { \prime } \rightarrow D _ \beta ^ { \prime } \subset \mathbf C ^ {m} $ are homeomorphisms; cf. Manifold), then a mapping $ f: M \rightarrow M ^ { \prime } $ is said to be holomorphic if $ \phi _ \beta ^ \prime \circ f \circ \phi _ \alpha ^ {-} 1 : D _ \alpha \rightarrow D _ \beta ^ { \prime } $ is a holomorphic mapping for all $ \alpha $ and $ \beta $. Holomorphic mappings of complex spaces are defined in a similar manner (cf. Analytic mapping). See also Biholomorphic mapping.

References

[1]	R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965)

Comments

A non-degenerate mapping is also called non-singular.

References

[a1]	W. Rudin, "Function theory in the unit ball in " , Springer (1980) pp. Chapt. 15
[a2]	S.G. Krantz, "Function theory of several complex variables" , Wiley (1982)