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)