Several Complex Variables (VI): Implicit Mapping Theorem
10 Sep 2018
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 U be an open subset of Cn, and let f be a holomorphic function on U with a zero at λ. Suppose that ∂zn∂f(λ)=0. Then there is a polydisc
Δ(λ,r)=Δ(λ′,r′)×Δ(λn,rn)⊆Cn−1×C,
and a holomorphic g:Δ(λ′,r′)→Δ(λn,rn), such that for z=(z′,zn)∈Δ(λ,r),
f(z)=0⟺g(z′)=zn.
Proof: This is an immediate consequence of the Weierstrass preparation theorem: by the given conditions, f has vanishing order 1 in zn at λ. Thus there is a polydisc Δ(z,r) and a factorisation f=uh, where u is holomorphic and nonzero, and h is a Weierstrass polynomial of degree 1, ie.
h(z)=zn−g(z′)
for some holomorphic g with g(λ′)=0. Now the result follows from the fact that f and h have the same roots on Δ(λ,r). □
We will now generalise this result to holomorphic F:U→Cm. Any such map can be decomposed into coordinate functions F(z)=(f1(z),…,fm(z)). Just as in (real) multivariable calculus, the non-degeneracy condition on the derivative generalises to a non-degeneracy condition on the m×n Jacobian matrix
JF(z):=(∂zj∂fi(z))i,j=1m,n.
Implicit mapping theorem: Let U be an open subset of Cn, and let F:U→Cm be a holomorphic map, with coordinate functions f1,…,fm and a zero at λ. Suppose that the last m columns of Jf(z) form a non-singular m×m matrix. Then there is a polydisc
Δ(λ,r)=Δ(λ′,r′)×Δ(λ′′,r′′)⊆Cn−m×Cm,
and a holomorphic G:Δ(λ′,r′)→Δ(λ′′,r′′), such that for z=(z′,z′′)∈Δ(λ,r),
F(z)=0⟺g(z′)=z′′.
Proof: By induction on m. The case m=1 is exactly the implicit function theorem.
We now prove the statement for m from the statement for m−1. First, split JF=(JF′,JF′′) into its first n−m and last m columns, so JF′′(λ) is non-singular. By a change of coordinates of the range Cm, we may assume that JF′′(λ) is the m×m identity matrix.
Now ∂zn∂fm(λ)=1, so by the implicit function theorem there exists a polydisc Δ(λ,r) and a holomorphic map
h:Δ((λ1,…,λn−1),(r1,…,rn−1))→Δ(λn,rn)
such that for z∈Δ(λ,r),
fm(z)=0⟺zn=h(z1,…,zn−1).
The idea is to restrict to the zero set of the last coordinate fm, which is locally described by the hypersurface (z1,…,zn−1,h(z1,…,zn−1)), and apply the induction hypothesis. More specifically, define the map
F′:Δ((λ1,…,λn−1),(r1,…,rn−1))→Cm−1
by defining its coordinate functions $f’1,\ldots,f’{m-1}$ as
fi′(z1,…,zn−1)=fi(z1,…,zn−1,h(z1,…,zn−1)).
Then at the point (λ1,…,λn−1), we have F′=0, and
∂zj∂fi′=∂zj∂fi+∂zn∂fi∂zj∂h={1,0,j=i+n−melse,
since ∂zn∂fi=0. Hence the last m−1 columns of JF′ form a (m−1)×(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((λn−m+1,…,λn−1),(rn−m+1,…,rn−1))
such that for (z1,…,zn−1)∈Δ((λ1,…,λn−1),(r1,…,rn−1)),
F′(z1,…,zn−1)=0⟺G′(z′)=(zn−m+1,…,zn−1).
Hence if we define
G(z′):=(G′(z′),h(z′,G′(z′))),
then for z∈Δ(λ,r) we have
F(z)=0⟺F′(z1,…,zn−1)=0and zn=h(z1,…,zn−1)⟺G′(z′)=(zn−m+1,…,zn−1)and zn=h(z1,…,zn−m,G′(z′))⟺G(z′)=z′′,
so G satisfies the given conditions, and induction is complete. □
Now we give some important consequences of this theorem.
Inverse mapping theorem: Let U be an open subset of Cn, and let H:U→Cn be a holomorphic map. Let λ∈U, and suppose that JH(λ) is non-singular. Then there is a neighbourhood V⊆U of λ such that H is a biholomorphic mapping from V onto some neighbourhood of H(λ).
Proof: By the implicit function theorem applied to the function F:Cn×U→Cn given by F(z′,z′′)=H(z′′)−z′, we get a function G on some polydisc with
G(z′)=z′′⟺F(z′,z′′)=0⟺H(z′′)=z′,
so G is an inverse for H on this polydisc. □
More generally, we have the following:
Constant rank theorem: Let U be an open subset of Cn, and let F:U→Cm be a holomorphic map. Suppose that Jf has constant rank k in U. Then for each λ∈U, there is a neighbourhood Uλ of λ in which F is biholomorphically equivalent to the standard projection
(z1,…,zn)↦(z1,…,zk,0,…,0)
from a neighbourhood of 0∈Cn to a neighbourhood of 0∈Cm.
Proof: We may assume that λ=0 and F(λ)=0, and after a change of coordinates that the upper left k×k submatrix of JF(z) is non-singular at 0, and hence in a neighbourhood U′ of 0∈Cn. Define G:U′→Cn by
G(z1,…,zn)=(f1(z1,…,zn),…,fk(z1,…,zn),zk+1,…,zn).
Then JG is non-singular in U′, so by inverse mapping theorem G is a biholomorphic map from some neighbourhood U′′⊆U′ of 0∈Cn to another.
Now F∘G−1 has the form (z1,…,zn)↦(z1,…,zk,fk+1′,…,fm′). Since its Jacobian also has rank k in U′′, we have ∂zi∂fj′ are identically zero for i>k, ie. fj′ are functions in z1,…,zk alone. Now define
H(z1,…,zm)=(z1,…,zk,zk+1−fk+1′,…,zm−fm′).
Note that H has inverse
H−1(z1,…,zm)=(z1,…,zk,zk+1+fk+1′,…,zm+fm′),
so H is biholomorphic on some neighbourhood of 0∈Cm. Now
H∘F∘G−1(z1,…,zn)=(z1,…,zk,0,…,0)
on a neighbourhood U of 0∈Cm, as desired. □
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