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Several Complex Variables (VI): Implicit Mapping Theorem

To conclude our discussion of the analysis aspect of the several complex variables theory, we prove the holomorphic versions of the implicit and inverse mapping theorems.

Implicit function theorem: Let UU be an open subset of Cn\bb C^n, and let ff be a holomorphic function on UU with a zero at λ\lambda. Suppose that fzn(λ)0\frac{\del f}{\del z_n}(\lambda)\neq0. Then there is a polydisc

Δ(λ,r)=Δ(λ,r)×Δ(λn,rn)Cn1×C,\Delta(\lambda,r)=\Delta(\lambda',r')\times\Delta(\lambda_n,r_n)\subseteq\bb C^{n-1}\times\bb C,

and a holomorphic g:Δ(λ,r)Δ(λn,rn)g:\Delta(\lambda’,r’)\to\Delta(\lambda_n,r_n), such that for z=(z,zn)Δ(λ,r)z=(z’,z_n)\in\Delta(\lambda,r),

f(z)=0    g(z)=zn.f(z)=0\quad\iff\quad g(z')=z_n.

Proof: This is an immediate consequence of the Weierstrass preparation theorem: by the given conditions, ff has vanishing order 1 in znz_n at λ\lambda. Thus there is a polydisc Δ(z,r)\Delta(z,r) and a factorisation f=uhf=uh, where uu is holomorphic and nonzero, and hh is a Weierstrass polynomial of degree 1, ie.

h(z)=zng(z)h(z)=z_n-g(z')

for some holomorphic gg with g(λ)=0g(\lambda’)=0. Now the result follows from the fact that ff and hh have the same roots on Δ(λ,r)\Delta(\lambda,r).  \qed

We will now generalise this result to holomorphic F:UCmF:U\to\bb C^m. Any such map can be decomposed into coordinate functions F(z)=(f1(z),,fm(z))F(z)=(f_1(z),\ldots,f_m(z)). Just as in (real) multivariable calculus, the non-degeneracy condition on the derivative generalises to a non-degeneracy condition on the m×nm\times n Jacobian matrix

JF(z)(fizj(z))i,j=1m,n.J_F(z)\coloneqq\left(\frac{\del f_i}{\del z_j}(z)\right)_{i,j=1}^{m,n}.

Implicit mapping theorem: Let UU be an open subset of Cn\bb C^n, and let F:UCmF:U\to\bb C^m be a holomorphic map, with coordinate functions f1,,fmf_1,\ldots,f_m and a zero at λ\lambda. Suppose that the last mm columns of Jf(z)J_f(z) form a non-singular m×mm\times m matrix. Then there is a polydisc

Δ(λ,r)=Δ(λ,r)×Δ(λ,r)Cnm×Cm,\Delta(\lambda,r)=\Delta(\lambda',r')\times\Delta(\lambda'',r'')\subseteq\bb C^{n-m}\times\bb C^m,

and a holomorphic G:Δ(λ,r)Δ(λ,r)G:\Delta(\lambda’,r’)\to\Delta(\lambda^{\prime\prime},r^{\prime\prime}), such that for z=(z,z)Δ(λ,r)z=(z’,z^{\prime\prime})\in\Delta(\lambda,r),

F(z)=0    g(z)=z.F(z)=0\quad\iff\quad g(z')=z''.

Proof: By induction on mm. The case m=1m=1 is exactly the implicit function theorem.

We now prove the statement for mm from the statement for m1m-1. First, split JF=(JF,JF)J_F=(J_F’,J_F^{\prime\prime}) into its first nmn-m and last mm columns, so JF(λ)J_F^{\prime\prime}(\lambda) is non-singular. By a change of coordinates of the range Cm\bb C^m, we may assume that JF(λ)J_F^{\prime\prime}(\lambda) is the m×mm\times m identity matrix.

Now fmzn(λ)=1\frac{\del f_m}{\del z_n}(\lambda)=1, so by the implicit function theorem there exists a polydisc Δ(λ,r)\Delta(\lambda,r) and a holomorphic map

h:Δ((λ1,,λn1),(r1,,rn1))Δ(λn,rn)h:\Delta((\lambda_1,\ldots,\lambda_{n-1}),(r_1,\ldots,r_{n-1}))\to\Delta(\lambda_n,r_n)

such that for zΔ(λ,r)z\in\Delta(\lambda,r),

fm(z)=0    zn=h(z1,,zn1).f_m(z)=0\quad\iff\quad z_n=h(z_1,\ldots,z_{n-1}).

The idea is to restrict to the zero set of the last coordinate fmf_m, which is locally described by the hypersurface (z1,,zn1,h(z1,,zn1))(z_1,\ldots,z_{n-1},h(z_1,\ldots,z_{n-1})), and apply the induction hypothesis. More specifically, define the map

F:Δ((λ1,,λn1),(r1,,rn1))Cm1F':\Delta((\lambda_1,\ldots,\lambda_{n-1}),(r_1,\ldots,r_{n-1}))\to\bb C^{m-1}

by defining its coordinate functions $f’1,\ldots,f’{m-1}$ as

fi(z1,,zn1)=fi(z1,,zn1,h(z1,,zn1)).f'_i(z_1,\ldots,z_{n-1})=f_i(z_1,\ldots,z_{n-1},h(z_1,\ldots,z_{n-1})).

Then at the point (λ1,,λn1)(\lambda_1,\ldots,\lambda_{n-1}), we have F=0F’=0, and

fizj=fizj+fiznhzj={1,j=i+nm0,else, \begin{aligned} \frac{\del f_i'}{\del z_j}&=\frac{\del f_i}{\del z_j}+\frac{\del f_i}{\del z_n}\frac{\del h}{\del z_j}\\ &=\begin{cases}1,&j=i+n-m\\0,&\text{else},\end{cases} \end{aligned}

since fizn=0\frac{\del f_i}{\del z_n}=0. Hence the last m1m-1 columns of JFJ_{F’} form a (m1)×(m1)(m-1)\times(m-1) identity matrix, so by induction hypothesis there exists (after possibly restricting to a smaller polydisc) a holomorphic map

G:Δ(λ,r)Δ((λnm+1,,λn1),(rnm+1,,rn1))G':\Delta(\lambda',r')\to\Delta((\lambda_{n-m+1},\ldots,\lambda_{n-1}),(r_{n-m+1},\ldots,r_{n-1}))

such that for (z1,,zn1)Δ((λ1,,λn1),(r1,,rn1))(z_1,\ldots,z_{n-1})\in\Delta((\lambda_1,\ldots,\lambda_{n-1}),(r_1,\ldots,r_{n-1})),

F(z1,,zn1)=0    G(z)=(znm+1,,zn1).F'(z_1,\ldots,z_{n-1})=0\quad\iff\quad G'(z')=(z_{n-m+1},\ldots,z_{n-1}).

Hence if we define

G(z)(G(z),h(z,G(z))),G(z')\coloneqq(G'(z'),h(z',G'(z'))),

then for zΔ(λ,r)z\in\Delta(\lambda,r) we have

F(z)=0    F(z1,,zn1)=0and zn=h(z1,,zn1)    G(z)=(znm+1,,zn1)and zn=h(z1,,znm,G(z))    G(z)=z, \begin{aligned} F(z)=0&\iff F'(z_1,\ldots,z_{n-1})=0\\ &\qquad\text{and }z_n=h(z_1,\ldots,z_{n-1})\\ &\iff G'(z')=(z_{n-m+1},\ldots,z_{n-1})\\ &\qquad\text{and }z_n=h(z_1,\ldots,z_{n-m},G'(z'))\\ &\iff G(z')=z'', \end{aligned}

so GG satisfies the given conditions, and induction is complete.  \qed

Now we give some important consequences of this theorem.

Inverse mapping theorem: Let UU be an open subset of Cn\bb C^n, and let H:UCnH:U\to\bb C^n be a holomorphic map. Let λU\lambda\in U, and suppose that JH(λ)J_H(\lambda) is non-singular. Then there is a neighbourhood VUV\subseteq U of λ\lambda such that HH is a biholomorphic mapping from VV onto some neighbourhood of H(λ)H(\lambda).

Proof: By the implicit function theorem applied to the function F:Cn×UCnF:\bb C^n\times U\to\bb C^n given by F(z,z)=H(z)zF(z’,z^{\prime\prime})=H(z^{\prime\prime})-z’, we get a function GG on some polydisc with

G(z)=z    F(z,z)=0    H(z)=z,G(z')=z''\quad\iff\quad F(z',z'')=0\quad\iff\quad H(z'')=z',

so GG is an inverse for HH on this polydisc.  \qed

More generally, we have the following:

Constant rank theorem: Let UU be an open subset of Cn\bb C^n, and let F:UCmF:U\to\bb C^m be a holomorphic map. Suppose that JfJ_f has constant rank kk in UU. Then for each λU\lambda\in U, there is a neighbourhood UλU_\lambda of λ\lambda in which FF is biholomorphically equivalent to the standard projection

(z1,,zn)(z1,,zk,0,,0)(z_1,\ldots,z_n)\mapsto(z_1,\ldots,z_k,0,\ldots,0)

from a neighbourhood of 0Cn0\in\bb C^n to a neighbourhood of 0Cm0\in\bb C^m.

Proof: We may assume that λ=0\lambda=0 and F(λ)=0F(\lambda)=0, and after a change of coordinates that the upper left k×kk\times k submatrix of JF(z)J_F(z) is non-singular at 0, and hence in a neighbourhood UU’ of 0Cn0\in\bb C^n. Define G:UCnG:U’\to\bb C^n by

G(z1,,zn)=(f1(z1,,zn),,fk(z1,,zn),zk+1,,zn).G(z_1,\ldots,z_n)=(f_1(z_1,\ldots,z_n),\ldots,f_k(z_1,\ldots,z_n),z_{k+1},\ldots,z_n).

Then JGJ_G is non-singular in UU’, so by inverse mapping theorem GG is a biholomorphic map from some neighbourhood UUU^{\prime\prime}\subseteq U’ of 0Cn0\in\bb C^n to another.

Now FG1F\circ G^{-1} has the form (z1,,zn)(z1,,zk,fk+1,,fm)(z_1,\ldots,z_n)\mapsto(z_1,\ldots,z_k,f_{k+1}’,\ldots,f_m’). Since its Jacobian also has rank kk in UU^{\prime\prime}, we have fjzi\frac{\del f_j’}{\del z_i} are identically zero for i>ki>k, ie. fjf_j’ are functions in z1,,zkz_1,\ldots,z_k alone. Now define

H(z1,,zm)=(z1,,zk,zk+1fk+1,,zmfm).H(z_1,\ldots,z_m)=(z_1,\ldots,z_k,z_{k+1}-f_{k+1}',\ldots,z_m-f_m').

Note that HH has inverse

H1(z1,,zm)=(z1,,zk,zk+1+fk+1,,zm+fm),H^{-1}(z_1,\ldots,z_m)=(z_1,\ldots,z_k,z_{k+1}+f_{k+1}',\ldots,z_m+f_m'),

so HH is biholomorphic on some neighbourhood of 0Cm0\in\bb C^m. Now

HFG1(z1,,zn)=(z1,,zk,0,,0)H\circ F\circ G^{-1}(z_1,\ldots,z_n)=(z_1,\ldots,z_k,0,\ldots,0)

on a neighbourhood UU of 0Cm0\in\bb C^m, as desired.  \qed

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