INITIAL MONOMIAL INVARIANTS OF HOLOMORPHIC MAPS

arXiv:1502.03434v3 [math.CV] 9 Sep 2015

DUSTY GRUNDMEIER AND JIŘÍ LEBL
Abstract. We study a new biholomorphic invariant of holomorphic maps between domains in different dimensions based on generic initial ideals. We start with the standard
generic monomial ideals to find invariants for rational maps of spheres and hyperquadrics,
giving a readily computable invariant in this important case. For example, the generic initial
monomials distinguish all four inequivalent rational proper maps from the two to the three
dimensional ball. Next, we associate to each subspace X ⊂ O(U ) a generic initial monomial subspace, which is invariant under biholomorphic transformations and multiplication
by nonzero functions. The generic initial monomial subspace is a biholomorphic invariant for holomorphic maps if the target automorphism is linear fractional as in the case of
automorphisms of spheres or hyperquadrics.

1. Introduction
Let U ⊂ Cn and V ⊂ Cm be domains. Denote by O(U, V ) the set of holomorphic maps
f : U → V.

(1)

Write O(U) = O(U, C) as usual. A fundamental problem in several complex variables is to
understand maps in O(U, V ) up to automorphisms; that is, f : U → V and g : U → V are
equivalent if there exist biholomorphisms τ ∈ Aut(U) and χ ∈ Aut(V ) such that
f ◦ τ = χ ◦ g.

(2)

A particularly important case is when the maps are proper. A map f : U → V is proper
if for every compact K ⊂⊂ V , the set f −1 (K) is compact. If f extends continuously to
the boundary, then the proper map f takes boundary to boundary. The map f restricted
to the boundary gives a CR map, and therefore, we have a problem in CR geometry. It
is important to understand those situations where V has a large automorphism group, and
thus we focus most of our attention on the unit ball of a certain signature. For a pair of
integers (a, b), a ≥ 1, the unit ball of signature b is given by
Ba+b
b

b
a+b
o
n
X
X
2
a+b
|zj |2 < 1 .
= z = (z1 , . . . , za+b ) ∈ C
:−
|zj | +
j=1

(3)

j=1+b

is the
The ball Bn0 with signature 0 is the standard unit ball Bn . The boundary of Ba+b
b
hyperquadric
b
a+b
o
n
X
X
2
a+b
|zj |2 = 1 .
(4)
Q(a, b) = z ∈ C
:−
|zj | +
j=1

j=1+b

Date: September 9, 2015.
The second author was in part supported by NSF grant DMS-1362337 and Oklahoma State University’s
DIG and ASR grants.
1

2

DUSTY GRUNDMEIER AND JIŘÍ LEBL

Hyperquadrics are the model hypersurfaces for Levi-nondegenerate surfaces. The hyperquadrics are also the flat models in CR geometry from the point of view of Riemannian
geometry.
The CR geometry problem of maps into hyperquadrics has a long history beginning with
Webster who in 1978 [17] proved that every algebraic hypersurface can be embedded into
a hyperquadric for large enough a and b. On the other hand Forstnerič [8] proved that in
general a CR submanifold will not map to a finite dimensional hyperquadric. Thus not every
CR submanifold can be realized as a submanifold of the flat model (i.e. the hyperquadric).
The mapping problem has been studied extensively for proper maps between balls, where
the related CR question is to classify the CR maps between spheres. That is, suppose
f : Bn → BN is a proper holomorphic map. Two such maps f and g are spherically equivalent
if there exist automorphisms τ ∈ Aut(Bn ) and χ ∈ Aut(BN ) such that f ◦ τ = χ ◦ g.
There has been considerable progress in the classification of such maps, especially for small
codimensional cases; see for example [1, 3, 4, 6, 9, 14] and the many references within.
In particular, Forstnerič [9] proved that given sufficient boundary regularity, the map
is rational and the degree bounded in terms of the dimensions only. The degree is an
invariant under spherical equivalence and the sharp bounds for the degree were conjectured
by D’Angelo [5].
In this article we introduce a new biholomorphic invariant that is far more general than
degree. When the map is rational and the domain and target are balls (possibly of nonzero
signature), we obtain invariants via generic initial ideals from commutative algebra. For
arbitrary domains and maps, we generalize the smallest degree part of the ideal.
The techniques center on the idea of generic initial monomial ideals; see e.g. Green [12].
Given a homogoeneous ideal I, the monomial ideal in(I) is generated by the initial monomials of elements of I (see section 3 for precise definitions). The generic initial ideal, denoted
gin(I), is an initial monomial ideal after precomposing I with a generic invertible linear
map. Grauert was the first to introduce generic initial ideals to several complex variables
(see [10]), and they have been used extensively for understanding singularities of varieties.
For rational maps of balls of the form fg , we homogenize f and g, and we then look at
the ideal generated by the components. We prove that the generic initial ideal generated by
the homogenizations of f and g is invariant under spherical equivalence. More precisely, the
first main result is the following.
Theorem 1.1. Suppose f1 : Bn → BN and f2 : Bn → BN are rational proper maps that are
spherically equivalent. Let F1 and F2 be the respective homogenizations. Then


gin I(F1 ) = gin I(F2 ) .
(5)

In section 3, we prove this result, and compute this new invariant for a number of wellknown examples. In particular we show how this invariant distinguishes many of these maps.
In section 4, we obtain a further invariant by considering the holomorphic decomposition of
the quotient
kf (z)k2 − |g(z)|2
.
(6)
kzk2 − 1
To generalize these results to arbitrary holomorphic maps we improve a result proved in
Grundmeier-Lebl-Vivas [13] that itself is a version of Galligo’s theorem for vector subspaces
of O(U). Given a subspace X ⊂ O(U), we precompose with a generic affine map τ , and
define the generic initial monomial subspace as the space spanned by the initial monomials of

INITIAL MONOMIAL INVARIANTS OF HOLOMORPHIC MAPS

3

elements of X ◦ τ , denoted gin(X). Let X be the affine span of the components of a holomorphic map F : U → CN , denoted by affine-span F . See section 5 for precise definitions. We
obtain an invariant under biholomorphic transformations of U and invertible linear fractional
transformations of CN . By using affine rather than just linear maps we obtain invariants of
the map without having a distinguished point. The second main result says this is, indeed,
an invariant.
Theorem 1.2. Let M, M ′ ⊂ Cn be connected real-analytic CR submanifolds. Suppose
F : M → Q(a, b) and G : M ′ → Q(a, b) are real-analytic CR maps equivalent in the sense
that there exists a real-analytic CR isomorphism τ : M ′ → M and a linear fractional automorphism χ of Q(a, b) such that
F ◦ τ = χ ◦ G.
(7)
Then

gin(affine-span F ) = gin(affine-span G).

(8)

In sections 5 and 6, we give precise definitions and lemmas. In particular we prove the
extended version of Galligo’s theorem for our setting. In section 7, we prove the second major
theorem of this paper, and we give examples where we compute this invariant. Finally, in
section 8 we define the generic initial subspace for the analogue of the quotient (6) for CR
maps between spheres and hyperquadrics.
The authors would like to acknowledge John D’Angelo for many conversations on the
subject.
2. The projective setting
Before we work with arbitrary maps, we consider the special case of rational maps between
hyperquadrics and spheres. In this case we directly apply the standard theory of generic
initial ideals to obtain invariants.
For z, w ∈ Ca+b define
def

hz, wib = hIb z, wi = −

b
X

zj w̄j +

j=1

a+b
X

zj w̄j

and

j=b+1

def

kzk2b = hz, zib .

(9)

By Ib we mean the (a + b) × (a + b) diagonal matrix with b (−1)’s and a 1’s on the diagonal.
We use the same definition for the homogeneous case when the index on the variables starts
with a zero. In this case the subscript still refers to the number of negatives. That is, for
Z = (Z0 , . . . , Za+b ) ∈ Ca+b+1 we write
def

hZ, W ib+1 = hIb+1 Z, W i = −

b
X
j=0

Zj W̄j +

a+b
X

j=b+1

Zj W̄j

and

def

kZk2b+1 = hZ, Zib+1. (10)

Let us homogenize Q(a, b). We add a variable Z0 and work with the homogeneous coordinates [Z0 , Z1 , . . . , Zn ] in Pn . Homogenize the equation above to obtain
def 
(11)
HQ(a, b + 1) = Z ∈ Pa+b : kZk2b+1 = 0 .

If we think of Ca+b ⊂ Pa+b , then Q(a, b) is a subset of Pa+b and HQ(a, b + 1) is the closure
of Q(a, b) in Pa+b .

4

DUSTY GRUNDMEIER AND JIŘÍ LEBL

Automorphisms of Pa+b can be represented as invertible linear maps on Ca+b+1 . The
automorphisms of HQ(a, b + 1) are those linear maps T that preserve the form in (11) up
to a real scalar λ 6= 0, that is
(12)
kT Zk2b+1 = λ kZk2b+1 .
p
If we represent T as a matrix, then the condition above is T ∗ Ib+1 T = µIb , where µ is ± |λ|.
If a 6= b + 1, then λ > 0. If λ < 0, then the automorphism swaps the sides of HQ(a,
b + 1).

a+b
Write the set of the corresponding automorphisms of P
as Aut HQ(a, b + 1) . As usual
the point Z ∈ Pa+b is an equivalence class of points of Ca+b+1 up to complex multiple. Thus
(as long as a 6= b + 1) the group Aut HQ(a, b + 1) is the group SU(a, b + 1)/K where K is
the subgroup of matrices ζI where ζ a+b+1 = 1. If a = b + 1, we need to include the matrix
that switches sides. The important fact for us is that automorphisms are represented by
matrices.
Fix (a, b) and (A, B). A rational map F : Pa+b 99K PA+B is represented by a homogeneous
polynomial map of Ca+b+1 to CA+B+1 . The equivalence we wish to consider is the following.
a+b
Two rational maps
99K PA+B and G : Pa+b 99K PA+B are equivalent if there exists
 F: P
a+b+1
τ ∈ Aut P
and χ ∈ Aut PA+B+1 such that
F ◦ τ = χ ◦ G.

(13)

In the applications we will have τ ∈ Aut HQ(a, b + 1) and χ ∈ Aut HQ(A, B + 1) .
If f : Bn → BN is a rational proper map of balls, its homogenization F : Pn 99K PN takes
HQ(n, 1) to HQ(N,
1)
 1). The equivalence on such maps F using the groups Aut HQ(n,
n
and Aut HQ(N, 1) is precisely the standard spherical equivalence on f ; that is, f : B → BN
and g : Bn → BN are spherically equivalent if there exist automorphisms τ and χ of Bn and
BN respectively such that f ◦ τ = χ ◦ g.


3. Generic initial ideals and rational maps
In this section we briefly introduce the relevant definitions and results from commutative
algebra. We use the setup from Green [12], and in a later section we use the techniques
developed by the authors in [13]. To be consistent with these two papers, we use the slightly
unusual monomial ordering as used by Green.
Let Z0 , Z1 , . . . , Zn denote our variables. Given a multi-index α ∈ Nn+1
we write Z α to
0
mean Z0α0 Z1α1 · · · Znαn as usual, and let |α| = α0 + · · · + αn denote the total degree. A
multiplicative monomial order is a total ordering on all monomials Z α , such that
(i) Z0 > Z1 > · · · > Zn ,
(ii) Z α > Z β ⇒ Z γ Z α > Z γ Z β ,
(iii) |α| < |β| ⇒ Z α > Z β .
Such orderings are not unique. Using the so-called graded reverse lexicographic ordering
when n = 2, we obtain
1 > Z0 > Z1 > Z2 > Z02 > Z0 Z1 > Z12 > Z0 Z2 > Z1 Z2 > Z22 > · · · .

(14)

Fix a certain monomial order. Given a set S of monomials, then the initial monomial is
the maximal monomial in S according to the ordering. For a homogeneous polynomial P ,
write
def
in(P ) = initial monomial of P .
(15)

INITIAL MONOMIAL INVARIANTS OF HOLOMORPHIC MAPS

5

Let I be a homogeneous ideal. Define the initial monomial ideal in(I) as the smallest ideal
such that z α ∈ in(I) whenever z α = in(P ) for some P ∈ I.
Galligo’s theorem implies that for an open dense set of linear maps T , in(I) = in(I ◦ T ).
Let T be such a generic linear map and define
def

gin(I) = in(I ◦ T ).

(16)

I(F ) = ideal generated by components of F in homogeneous coordinates.

(17)

F ◦ τ = χ ◦ G.

(18)

n

Finally, for a rational map F : P 99K P

N

define

Proposition 3.1. Suppose
F : Pn 99K PNand G : Pn 99K PN are equivalent in the sense that

n
there exist τ ∈ Aut P and χ ∈ Aut PN such that
Then



gin I(F ) = gin I(G) .

(19)




Proof. By definition of the generic initial ideal, gin I(F ◦ τ ) = gin I(F ) , as τ is an
invertible linear map. The ideal generated by linear combinations of G is the
 same as the
 one
generated by G, and χ is also an invertible linear map. So, gin I(χ ◦ G) = gin I(G) . 
In the important special case of rational proper maps of balls we get the first main theorem
from the introduction.

Corollary 3.2. Suppose f : Bn → BN and g : Bn → BN are rational proper maps that are
spherically equivalent. Let F and G be the respective homogenizations. Then


gin I(F ) = gin I(G) .
(20)

We close this section with several examples. The following computations are done with
Macaulay2 [11] with the GenericInitialIdeal package using the standard graded reverse
lex ordering unless stated otherwise. One advantage of using gins as invariants is that they
are very simple to compute using computer algebra systems.

Example 3.3. Generic initial ideals distinguish all the maps from B2 to B3 . Faran [6] proved
that all maps sufficiently smooth up to the boundary are spherically equivalent to one of the
following 4 maps:
(i) (z1 , z2 ) 7→ (z1 , z2 , 0),
(ii) (z1 , z2 ) 7→ (z1 , z√1 z2 , z22 ),
(iii) (z1 , z2 ) 7→ (z12 , √2z1 z2 , z22 ),
(iv) (z1 , z2 ) 7→ (z13 , 3z1 z2 , z23 ).
Let us list the results in a table. The first column gives the map itself. The second column
gives the map homogenized by adding the Z0 variable. The last column gives the generators
of the generic initial ideal.
Map
Homogenized
gin
(z1 , z2 , 0)
(Z0 , Z1 , Z2 , 0)
(Z0 , Z1 , Z2 )
2
2
2
(z1 , z√1 z2 , z2 )
(Z0 , Z1 Z√
(Z02 , Z0 Z1 , Z12 , Z0 Z2 , Z1 Z22 )
0 , Z1 Z2 , Z2 )
(z12 , √2z1 z2 , z22 ) (Z02 , Z12 , √2Z1 Z2 , Z22 )
(Z02 , Z0 Z1 , Z12 , Z0 Z2 , Z23 )
3
3
3
3
3
(z1 , 3z1 z2 , z2 ) (Z0 , Z1 , 3Z1 Z2 Z0 , Z2 ) (Z03 , Z02 Z1 , Z0 Z12 , Z02 Z2 ,
Z14 , Z13 Z2 , Z0 Z1 Z22 , Z12 Z22 ,
Z0 Z24 , Z1 Z24 , Z25 )

6

DUSTY GRUNDMEIER AND JIŘÍ LEBL

Notice that the generic initial ideals are all different. In particular, we distinguish the two
very similar second degree maps using the third degree part of the ideal.
Example 3.4. In [7] it was proved that the proper map of B2 to B4 given by

q

2
3
z
1
−
|a|
√
− a) 2

(z1 , z2 ) 7→ z12 , 2z1 z2 ,
,
1 − āz1
1 − āz1


z22 (z1

(21)

is spherically equivalent to a polynomial map if and only if a = 0. This map is the homogeneous second degree map from Faran’s theorem above with the last component tensored by
an automorphism of the ball B2 taking a to 0. If a = 0, then the generic initial ideal is

(Z03 , Z02 Z1 , Z0 Z12 , Z13 , Z02 Z2 , Z0 Z1 Z22 , Z12 Z22 , Z0 Z24 ).

(22)

If a = 21 , we obtain a different generic initial ideal

(Z03 , Z02 Z1 , Z0 Z12 , Z13 , Z02 Z2 , Z0 Z1 Z22 , Z12 Z22 , Z0 Z23 ).

(23)

Example 3.5. In [15] the second author classified all the maps from the two sphere, Q(2, 0),
to the hyperquadric Q(2, 1) (see also Reiter [16] for a different approach to the classification).
There are 7 equivalence classes of maps:
(i) (z1 , z2 ) 7→ (0, z1 , z2√),
(ii) (z1 , z2 ) 7→ 
(z22 , z12 , 2z2 ),
z2
(iii) (z1 , z2 ) 7→ zz22 , z11 , z22 ,
 12 √ 1 2

√
√
z − 3 z z +z −z z 2 + 3 z z +z 2 −z z 2 +z − 3 z −1
(iv) (z1 , z2 ) 7→ 1z 2 +z +1 √23 z 2−1 1 , 1z 2 +z +1 √23 z 2−1 1 , z22 +z1 +√3 z2 −1 ,
2
2
2
1
2
2 1
 √4 2 1
√
√
2(z1 z2 +iz1 ) 4 2(z1 z2 −iz1 ) z22 − 2 iz2 +1
√
√
√
(v) (z1 , z2 ) 7→ z 2 + 2 iz +1 , z 2 + 2 iz +1 , z 2 + 2 iz +1 ,
2
2
2
2
√2
2
z2 z12 −z2
2z23
z13 +3z1
(vi) (z1 , z2 ) 7→
3 3z 2 +1 , 3z 2 +1 , 3z 2 +1 ,
1
1
1
(vii) (z1 , z2 ) 7→ g(z1 , z2 ), g(z1 , z2 ), 1 for an arbitrary holomorphic function g.
Note that the maps are written slightly differently from [15] as we are ordering the negative
components first.
We list the generic initial ideal for each of the first 6 classes of maps. The final 7th class
is already distinguished as it is not transversal; that is, it maps an open neighborhood into
Q(2, 1).

INITIAL MONOMIAL INVARIANTS OF HOLOMORPHIC MAPS

Map
(i)
(ii)
(iii)
(iv)

(v)

Homogenized
(Z0 , 0, Z1 , Z2√
)
2
2
2
(Z0 , Z2 , Z1 , 2Z2 Z0 )
2
(Z12 , Z2 Z0 , Z1Z√
0 , Z2 )
(Z22 + Z
3Z2 Z0 − Z02 ,
1 Z0 +
√
Z12 − √3Z1 Z2 + Z22 − Z1 Z0 ,
Z12 + 3Z1 Z2√
+ Z22 − Z1 Z0 ,
2
Z22 + Z
√1 Z0 − 3Z22Z0 − Z0 )
2
(Z
2iZ2 Z0 + Z0 ,
2 +
√
4
2(Z1 Z2 + iZ1 Z0 ),
√
4
2(Z√
1 Z2 − iZ1 Z0 ),
2
2
Z2 − 2iZ2 Z√
0 + Z0 )
(3Z12 Z0 + Z03 , 3(Z2 Z12 − Z2 Z02 ),
2Z23 , Z13 + 3Z1 Z02 )

7

gin
(Z0 , Z1 , Z2 )
(Z02 , Z0 Z1 , Z12 , Z0 Z2 , Z1 Z22 , Z23 )
(Z02 , Z0 Z1 , Z12 , Z0 Z2 , Z1 Z22 )
(Z02 , Z0 Z1 , Z12 , Z0 Z2 , Z1 Z22 )

(Z02 , Z0 Z1 , Z12 , Z0 Z2 , Z1 Z22 )

(Z03 , Z02 Z1 , Z0 Z12 , Z02 Z2 ,
Z14 , Z0 Z1 Z22 , Z12 Z22 , Z13 Z2 ,
Z0 Z24 , Z1 Z24 , Z25 )
We distinguish some maps, but not all. It is to be expected that not all maps can be
distinguished. After all, computing a generic initial ideal throws away much information
about the map. It should be noted that we only looked at small degree examples, where the
number of different possible gins is limited.
(vi)

Example 3.6. The particular monomial ordering may make a difference, even in simple
situations. It might be possible to tell some ideals (and hence maps) apart using one ordering,
but not using another. Let us give a very simple example. In P2 , the ideal generated by
(Z02 , Z1 Z2 ) has the gin
(Z02 , Z0 Z1 , Z13 )

in graded reverse lex ordering,

(Z02 , Z0 Z1 , Z0 Z22 , Z14 )

in graded lex ordering.

(24)

However, the ideal generated by (Z02 , Z12 ) has the same gin in both orderings:
(Z02 , Z0Z1 , Z13 ).

(25)

4. Invariants of the quotient for rational maps
Let F : Pa+b 99K PA+B be a rational map that takes HQ(a, b + 1) to HQ(A, B + 1), where
defined. Identify F with the homogeneous polynomial map taking Ca+b+1 to CA+B+1 . Write
kF (Z)k2B+1 = kZk2b+1 q(Z, Z̄),

(26)

q(Z, Z̄) = kh+ (Z)k2 − kh− (Z)k2 ,

(27)

H(q) = {h+ , h− }

(28)

where the quotient q is a bihomogeneous polynomial in Z. Find the holomorphic decomposition (see p. 101 of [2]) of q as
where h+ and h− are homogeneous holomorphic polynomial maps. Write
for the set of functions in the holomorphic decomposition of q. The holomorphic decomposition is not unique. However, the linear span of H(q) is unique,
and therefore the ideal

generated by H(q) is unique.
We thus study the ideal I H(q) and furthermore the generic

initial ideal gin I H(q) .

8

DUSTY GRUNDMEIER AND JIŘÍ LEBL

The reason for looking at q is that it may reveal further information about the map that
is not immediately visible from F . For example the quotient q was critically used in [5] for
the degree estimates problem. The number of functions in the decomposition of q is often
larger than the number of components of F as many of these components may cancel out
once multiplied with kZk2b+1 .
Suppose F : Pa+b 99K PA+B and G : Pa+b 99K PA+B are equivalent as before; i.e. F ◦τ = χ◦
G, where τ and χ are automorphisms preserving HQ(a, b+1) and HQ(A, B +1) respectively.
Let qF and qG be the corresponding quotients of F and G respectively. We regard τ and χ
as linear maps, which we also rescale, such that kτ Zk2b+1 = ± kZk2b+1 for Z ∈ Ca+b+1 and
kχW k2B+1 = ± kW k2B+1 for W ∈ CA+B+1 . The ± is there in case a = b + 1 or A = B + 1
and the sides are swapped, otherwise it would be a +.
As F and G are equivalent, kF (τ Z)k2B+1 = kχ ◦ G(Z)k2B+1 . Then
kZk2b+1 qF (τ Z, τ Z) = ± kτ Zk2b+1 qF (τ Z, τ Z) = ± kF (τ Z)k2B+1 =

= ± kχ ◦ Gk2B+1 = ± kG(Z)k2B+1 = ± kZk2b+1 qG (Z, Z̄). (29)

In other words qF (τ Z, τ Z) = ±qG (Z, Z̄).
a+b
Proposition 4.1. Suppose F : Pa+b 99K PA+B
99K PA+B are equivalent in the
 and G : P
sense that there exist τ ∈ Aut HQ(a, b + 1) and χ ∈ Aut HQ(A, B + 1) such that

F ◦ τ = χ ◦ G.

Let qF and qG be the corresponding quotients. Then




gin I H(qF ) = gin I H(qG ) .

(30)

(31)

Proof. Above we proved qF (τ Z, τ Z) = ±qG (Z, Z̄). As we are talking about gins, the τ is not
relevant. The ± does not change the components of the holomorphic decomposition.

Again, in the important special case of rational proper maps of balls we get:
Corollary 4.2. Suppose f : Bn → BN and g : Bn → BN are rational proper maps that are
spherically equivalent. Let F and G be the respective homogenizations and let qF and qG be
the corresponding quotients. Then




gin I H(qF ) = gin I H(qG ) .
(32)

√
Example 4.3. Let us consider the Faran map (z1 , z2 ) 7→ (z13 , 3z1 z2 , z23 ) that takes B2 to
B3 . We compute the quotient
√
2
2
2
|z13 | + 3z1 z2 + |z23 | − 1
2
2
(33)
= z12 + z22 − |z1 z2 |2 + |z1 |2 + |z2 |2 + 1.
2
2
|z1 | + |z2 | − 1

Bihomogenizing the quotient, we obtain all the degree-two monomials in H(q). Therefore
the gin is generated by all the degree-two monomials:


gin I H(q) = (Z02 , Z0 Z1 , Z12 , Z0Z2 , Z1 Z2 , Z22 ).
(34)

INITIAL MONOMIAL INVARIANTS OF HOLOMORPHIC MAPS

9

5. Affine spans and automorphisms of the target
To deal with nonrational maps we switch to the affine setting. We work in CN and treat
it as a subset of PN . From now on z = (z1 , . . . , zN ) will be the inhomogeneous coordinates
on PN , that is, coordinates on CN . In other words, we fix a specific embedding ι : CN → PN .
For equivalence of maps F : U → CN we consider the target automorphisms to be the linear
fractional automorphisms of Aut(PN ) and consider CN ⊂ PN as above. Take χ ∈ Aut(PN ),
F : U → CN , and G : U → CN . When we write an equation such as χ ◦ G = F , we mean
χ ◦ ι ◦ G = ι ◦ F , where ι is the embedding above. That is, we consider F to be valued in PN
using the embedding ι : CN → PN . Note that after composing with automorphisms of PN
given as linear fractional maps, it may happen that the new map has poles in U. In fact,
when looking at linear fractional automorphisms of Q(a, b), there will in general be poles
that intersect Q(a, b).
The reason for not simply working in projective space is that the components in nonrational
holomorphic maps F : U → CN cannot be homogenized; the degree is unbounded. For this
same reason, we have to work with affine span instead of just linear span.
Definition 5.1. Given a collection of holomorphic functions F , the affine span of F is

affine-span F = span F ∪ {1} .
(35)
If F is a map, then affine-span F is the affine span of the components of F . Let X ⊂ O(U)
be a vector subspace and ϕ ∈ O(U). Denote by ϕX the vector subspace obtained by
multiplying every element in X by ϕ.

Lemma 5.2. Suppose F : U ⊂ Cn → CN and G : U ⊂ Cn → CN are such that there exists
χ ∈ Aut(PN ) such that F = χ ◦ G. Then there exists ϕ ∈ affine-span G such that
affine-span G = ϕ affine-span F.

(36)

P (w)
◦G(z)
Proof. Let χ(w) = Q(w)
be the linear fractional automorphism such that F (z) = PQ◦G(z)
for
N
N
N
z ∈ U. Here P : C → C and Q : C → C are affine maps. Therefore (Q ◦ G)F = P ◦ G.
Therefore the components of (Q ◦ G)F are in the affine span of G. Also Q ◦ G is in the affine
span of G. The span of the components of (Q ◦ G)F and the function (Q ◦ G) is exactly
(Q ◦ G) affine-span F , and therefore (Q ◦ G) affine-span F ⊂ affine-span G.
Clearly dim affine-span F ≤ dim affine-span G. By inverting χ and applying the argument
in reverse we get dim affine-span G ≤ dim affine-span F . Therefore, (Q ◦ G) affine-span F =
affine-span G.


The point of the lemma is to eliminate the target automorphism by looking at the vector
space generated by F1 , . . . , FN and 1, up to multiplication by a scalar-valued function. To find
invariants of the map F we need to find invariants of this new object under biholomorphisms
of the source.
6. Generic initial vector space
In this section we define the gin of a vector subspace of holomorphic maps. The idea is to
generalize the generic initial ideals to the setting of vector spaces of holomorphic functions
using affine maps instead of linear maps. We generalize the setup from Green [12], using the
techniques developed by the authors in [13].

10

DUSTY GRUNDMEIER AND JIŘÍ LEBL

Let monomial order be defined on z1 , . . . , zn in the same way as above. That is, given a
multi-index α ∈ Nn0 we write z α to mean z1α1 z2α2 · · · znαn , and |α| = α1 + · · · + αn is the total
degree. We require the order to be multiplicative:
(i) z1 > z2 > · · · > zn ,
(ii) z α > z β ⇒ z γ z α > z γ z β ,
(iii) |α| < |β| ⇒ z α > z β .
In the following definitions we fix a point. We assume this point is 0 ∈ Cn , although any
point can be used after translation.
Definition 6.1. Fix a monomial order. By an initial monomial from a collection we mean
the largest monomial in the order. Given a Taylor series T , the initial monomial of T is
the largest monomial in T with a nonzero coefficient. Suppose U ⊂ Cn and 0 ∈ U. For
f ∈ O(U), let T0 f be the Taylor series for f at 0 and
def

in(f ) = initial monomial of T0 f .

(37)

Given X ⊂ O(U) a vector subspace, define the initial monomial subspace
def

in(X) = span{z α : z α = in(f ) for some f ∈ X}.

(38)

A subspace X ⊂ O(U) is a monomial subspace if X admits an algebraic basis consisting of
monomials.
If X is a monomial subspace then the basis of monomials defining it is unique. Therefore,
there is a one-to-one equivalence between monomial subspaces and subsets of the set of all
monomials.
A set S of monomials is affine-Borel-fixed if whenever z α ∈ S and zj |z α , then for all ℓ < j,
the monomials z α z1j and z α zzjℓ are in S. If X ⊂ O(U) is a monomial subspace, then we say
that X is affine-Borel-fixed if the basis of monomials that generates X is affine-Borel-fixed.
Take an invertible affine self map τ of Cn such that τ (0) ∈ U. For a subspace X ⊂ O(U),
define a subspace X ◦ τ ⊂ O(U ′ ) where U ′ = τ −1 (U). Notice that 0 ∈ U ′ and we may again
talk about initial monomials as above.
The initial monomial space in(X) is not always preserved under such precomposition with
affine maps. For example, let X be the span of {1, z2 }, so X = in(X). For a generic choice of
an affine map τ we have in(X ◦ τ ) = span{1, z1 }. Notice that {1, z2 } is not affine-Borel-fixed,
but in(X ◦ τ ) is.
We need to prove an analogue to Galligo’s Theorem (see Theorem 1.27 in [12]). The finite
dimensional version of the result is proved in [13].
2
The set Aff(n) of affine self maps of Cn can be parametrized as Mn (C) × Cn , or Cn × Cn .
We use the standard topology on this set.
Theorem 6.2. Suppose U ⊂ Cn and 0 ∈ U, and let X ⊂ O(U) be a vector subspace.
There is some neighbourhood N of the identity in Aff(n) and A ⊂ N of second category
(countable intersection of open dense subsets of N ), such that for τ ∈ A, the space in(X ◦ τ )
is affine-Borel-fixed. Furthermore, in(X ◦ τ ) = in(X ◦ τ ′ ) for any two affine τ and τ ′ in A.
Finally, if X is finite dimensional, A is open and dense in N .
So after a generic affine transformation the initial monomial subspace is always the same
space. It will also not be necessary to assume 0 ∈ U. We pick a generic affine τ such that

INITIAL MONOMIAL INVARIANTS OF HOLOMORPHIC MAPS

11

τ (0) ∈ U. Then 0 ∈ τ −1 (U), and so it makes sense to take in(X ◦ τ ) for X ⊂ O(U). Before
we prove the theorem, let us make the following definition.
Definition 6.3. Let U ⊂ Cn and let X ⊂ O(U) be a subspace. Take a generic affine τ such
that τ (0) ∈ U. Define the generic initial monomial subspace
gin(X) = in(X ◦ τ ).

(39)

Remark 6.4. While the proof of the theorem may seem formal, we are using affine transformations and precomposing formal power series with affine transformations may not make
sense.
Proof of Theorem 6.2. The finite dimensional version of the theorem is proved in [13] as
Proposition 1 (or Proposition 5.5 on arXiv).
Suppose X is not finite dimensional. The space O(U) is a separable Frechét space. For
the subspace X ⊂ O(U), pick a countable set {fk }∞
k=1 in X such that if
Xm = span{f1 , f2 , . . . , fm },

(40)

[

(41)

then
m

Xm ⊂ X ⊂ X =

[

Xm .

m

Apply the finite dimensional result to Xm for each m. The τ varies over countably many
open dense sets and the intersection of those sets is a second category set as claimed.
Given any k, the initial k monomials in {f1 , f2 , . . . , fm } must stabilize as m grows. That
is, the first k monomials are the same for all large enough m. We therefore obtain a sequence
of initial monomials. To obtain the first k monomials, simply go far enough
S until the initial
k monomials stabilize. Then z α is in this sequence if and only if z α ∈ in m Xm .
It is left to show that
[


[

Xm = in
Xm .
(42)
in X = in
m

m


α
α
Suppose this is not
true.
Find
the
first
(maximal)
monomial
z
such
that
z
∈
in
X
, but

S
α
α
z 6∈ in m Xm . Suppose z is the kth monomial in the ordering given. Let πk be the
projection of O(U) onto the space spanned by the
 first k monomials. By Cauchy’s formula,
SM
πk is continous. The dimension of πk m=1 Xm stabilizes as M grows, and hence for large
enough M
[

∞


πk XM = πk
(43)
Xm = πk X .
m=1

α

There exists an f ∈ X with z = in(f ). By the above equality there must exist a g ∈ XM
such that z α = in(g). That is a contradiction.

We now generalize the gins to biholomorphic maps. Let X ⊂ O(U) be a subspace and
let f : U ′ → U be a holomorphic map. Define X ◦ f to be the vector subspace of O(U ′ )
consisting of all F ◦ f for F ∈ X.
Lemma 6.5. Let X be a vector subspace of O(U) and let f : U ′ → U be a biholomorphic
map. Then gin(X) = gin(X ◦ f ).

12

DUSTY GRUNDMEIER AND JIŘÍ LEBL

Proof. Suppose 0 is in both U ′ and U, and suppose f (0) = 0. Composing f with an invertible
linear map does not change the gin (we must change U appropriately). Hence, assume f ′ (0)
is the identity. Let
f (z) = z + E(z),
(44)
where E is of order 2 and higher. We only need to show that in(X) = in(X ◦ f ).
Suppose z α ∈ in(X), that is, there exists an element of X of the form
X
cβ z β .
(45)
g(z) = z α +
β<α

In (g ◦ f ), the terms from E create monomials of degree strictly higher than |α|. So
X
dβ z β .
(g ◦ f )(z) = z α +

(46)

β<α

α

for some dβ . Therefore z ∈ in(X ◦ f ).
Since f −1 (z) = z + F (z) for some F or higher order we obtain by symmetry that if
α
z ∈ in(X ◦ f ) then if z α ∈ in(X ◦ f ◦ f −1 ) = in(X).

Lemma 6.6. Let X be a vector subspace of O(U), 0 ∈ U, and let ϕ : U → C be a holomorphic
function that is not identically zero. Then gin(X) = gin(ϕX).
Proof. As we are talking about gins, we precompose with a generic affine map, which also
precomposes ϕ, and therefore we assume that ϕ(0) 6= 0. In fact, without loss of generality
we assume ϕ(0) = 1. Suppose z α ∈ in(X), that is, there is a g ∈ X of the form
X
g(z) = z α +
cβ z β .
(47)
β<α

Then
ϕ(z)g(z) = z α +

X

dβ z β ,

(48)

β<α

for some dβ by multiplicativity of the monomial ordering. Therefore z α ∈ in(ϕX). By
symmetry, in(X) = in(ϕX).

7. Gins as invariants of maps
If X ⊂ O(U) is a subspace and U ′ ⊂ U is an open set then the restriction X|U ′ (the space
of restrictions of maps from X to U ′ ) clearly has the same gin as X. Let us extend gins to
complex manifolds. Let X ⊂ O(U) be a subspace where U is a connected complex manifold
of dimension n. As gin is invariant under biholomorphic transformations, it is well-defined
on every chart for U. If two (connected) charts overlap, then the gin must be the same
(compute the gin on the intersection). Therefore, gin(X) is well-defined.
Theorem 7.1. Let U, W be connected complex manifolds of dimension n. Suppose F : U →
CN and G : W → CN are equivalent in the sense that there exists a biholomorphic map
τ : W → U and a linear fractional automorphism χ ∈ Aut(PN ) such that
Then

F ◦ τ = χ ◦ G.

(49)

gin(affine-span F ) = gin(affine-span G).

(50)

INITIAL MONOMIAL INVARIANTS OF HOLOMORPHIC MAPS

13

Proof. As noted above, we take two charts of U and W , and therefore, without loss of
generality we assume that U and W are domains in Cn .
As F ◦ τ and G are equivalent via a target automorphism, Lemma 5.2 says
affine-span(F ◦ τ ) = ϕ affine-span G.

As affine-span(F ◦ τ ) = (affine-span F ) ◦ τ , Lemma 6.5 says

gin affine-span(F ◦ τ ) = gin(affine-span F ).

(51)
(52)

Via Lemma 6.6, we get

gin(ϕ affine-span G) = gin(affine-span G).
The result follows.

(53)


The same proof is used for the following CR version. In the CR version we start with realanalytic CR maps. Since real-analytic CR maps extend to holomorphic maps, we therefore
work with the extended holomorphic maps when taking gins and affine spans.
Corollary 7.2. Let M, M ′ ⊂ Cn be connected real-analytic CR submanifolds. Suppose
F : M → Q(a, b) and G : M ′ → Q(a, b) are real-analytic CR maps equivalent in the sense
that there exists a real-analytic CR isomorphism τ : M ′ → M and a linear fractional automorphism χ of Q(a, b) such that
F ◦ τ = χ ◦ G.
(54)
Then
gin(affine-span F ) = gin(affine-span G).
(55)
Again we may take M and M ′ to be submanifolds of a complex manifold of dimension n
instead of Cn .
For examples we need only look at rational maps. If the map is rational we homogenize
with Z0 as before. The generic initial monomial vector space is simply the lowest degree part
of the generic initial ideal generated by the components of the homogenized map. That is
because the lowest degree part of the ideal is the linear span of the components of the map.
We must be careful that the ordering is compatible. In particular, the ordering must respect
the “grading” above, that is in each degree if we set Z0 = 1 we must still get a multiplicative
monomial ordering. For example the standard reverse lex ordering in each degree will not
work, although the lex order will.
√
3
As an example
take
the
Faran
map
F
(z)
=
(z
,
3z1 z2 , z23 ). We add 1 and homogenize to
1
√
3
3
3
get (Z0 , Z1 , 3Z0 Z1 Z2 , Z2 ). We compute the gin in the graded lex order. The degree 3 part
of the gin is generated by
Z03 , Z02 Z1 , Z02 Z2 , Z0 Z12 .
(56)
Therefore,
gin(affine-span F ) = span{1, z1 , z2 , z12 }.
(57)
8. Invariants of the quotient of a hyperquadric map
When the source is either the ball Ba+b
in the holomorphic case or Q(a, b) in the CR case,
b
we obtain further invariants by considering the quotient of the function composed with the
defining function of the target divided by the defining function of the source as we did in
the rational case.

14

DUSTY GRUNDMEIER AND JIŘÍ LEBL

P
P
2
Recall the definition kzk2b = − bj=1 |zj |2 + a+b
j=b+1 |zj | . The defining function for Q(a, b)
is then kzk2b − 1 = 0. Suppose F : U ⊂ Q(a, b) → Q(A, B) is a real-analytic CR map. A
real-analytic CR map is a holomorphic map of a neighborhood of U, and so we will identify
F with this holomorphic map. Define the quotient q via

kF (z)k2B − 1 = kzk2b − 1 q(z, z̄).
(58)
Near some point find the holomorphic decomposition of q:

q(z, z̄) = kh+ (z)k2 − kh− (z)k2 ,

(59)

H(q) = {h+ , h− }

(60)

where h+ and h− are possibly ℓ2 valued holomorphic maps. Write
for the set of functions in the holomorphic decomposition of q. The holomorphic decomposition is not unique and depends on the point. However we do have the following lemma.
Lemma 8.1. Suppose q : U ⊂ Cn → R is real-analytic and U is a connected open set. If
H1 (q) and H2 (q) are two holomorphic decompositions at two points of U, then


(61)
gin span H1 (q) = gin span H2 (q) .
Proof. By a standard connectedness argument using a path between the two points we only
need to consider the case where the domains of convergence of H1 and H2 overlap. Assume
that we work on this intersection. So we only need to show that if
kh+ (z)k2 − kh− (z)k2 = h′+ (z)

2

− h′− (z)

2

(62)

′
′
for maps h+ , h− , h′+ , h′− converging on a fixed open set, with {h+ , h
 − } and {h+ , h− } lin′
′
early independent sets, then gin span{h+ , h− } = gin span{h+ , h− } . From (62) we have
span{h+ , h− } = span{h′+ , h′− }, and so the result follows.


Thus, we do not need to specify which point and which decomposition is used.

Theorem 8.2. Suppose F : U ⊂ Q(a, b) → Q(A, B) and G : V ⊂ Q(a, b) → Q(A, B) are
real-analytic CR maps equivalent in the sense that there exists a linear fractional automorphism τ of Q(a, b) and a linear fractional automorphism χ of Q(A, B) such that
Define the quotients qF and qG via

kF (z)k2B − 1 = kzk2b − 1 qF (z, z̄)

F ◦ τ = χ ◦ G.

kG(z)k2B − 1 = kzk2b − 1 qG (z, z̄).

and

(63)
(64)

Then taking holomorphic decompositions of qF and qG at any point in U and V respectively


gin span H(qF ) = gin span H(qG ) .
(65)

Proof. Write the linear fractional maps τ and χ as
τ′
χ′
τ = ′′
and
χ = ′′ .
τ
χ
′
′′
′
′′
for affine maps τ , τ , χ , and χ . Then
2

so

2

kτ ′ (z)kb − |τ ′′ (z)| = ±(kzk2b − 1)
2

|τ ′′ (z)| (kτ (z)k2b − 1) = ±(kzk2b − 1).

(66)

(67)
(68)

INITIAL MONOMIAL INVARIANTS OF HOLOMORPHIC MAPS

15

The ± is there again in case τ switches the sides of Q(a, b). We have the same equation for
w and χ with A, B instead of a, b.
We have kF (τ (z))k2B + 1 = kχ ◦ G(z)k2B + 1. Then



2
(kzk2b − 1)qF τ (z), τ (z) = ± |τ ′′ (z)| kτ (z)k2b − 1 qF τ (z), τ (z)

2
= ± |τ ′′ (z)| kF (τ (z))k2B − 1

2
= ± |τ ′′ (z)| kχ ◦ G(z)k2B − 1
τ ′′ (z)

= ± ′′
χ G(z)

In other words

τ ′′ (z)

= ± ′′
χ G(z)

2

kG(z)k2B − 1



(69)

2


kzk2b − 1 qG (z, z̄).
2

τ ′′ (z)
 qG (z, z̄).
qF τ (z), τ (z) = ± ′′
(70)
χ G(z)
Multiplying by absolute value squared of a holomorphic function multiplies the elements
of the holomorphic decomposition by that function and so does not change the gin by
Lemma 6.6. Similarly composing with τ does not change the gin either by Lemma 6.5. 


There is an equivalent theorem for holomorphic maps, when the automorhphism maps on
the source are simply the linear fractional automorphisms of Ba+b
b , that is, automorphisms of
a+b
a+b
a+b
P
preserving the closure of Bb in P . We lose no generality if we also allow swapping
sides (when a = b + 1) and hence we consider all linear fractional automorphisms of Q(a, b),
that is, self maps of Pa+b preserving HQ(a, b + 1).
Theorem 8.3. Suppose F : U ⊂ Ca+b → CA+B and G : V ⊂ Ca+b → CA+B are holomorphic
maps equivalent in the sense that there exists a linear fractional automorphism τ of Q(a, b),
where τ (V ) = U, and a linear fractional automorphism χ of Q(A, B) such that
F ◦ τ = χ ◦ G.

Define the quotients qF and qG as before:

and
kF (z)k2B − 1 = kzk2b − 1 qF (z, z̄)


kG(z)k2B − 1 = kzk2b − 1 qG (z, z̄).

(71)
(72)

Then taking holomorphic decompositions of qF and qG at any point in U and V respectively


gin span H(qF ) = gin span H(qG ) .
(73)
9. Gins of real-analytic functions

We end the article with a remark that Lemma 8.1 may be of independent interest, not
only for mapping questions. Let us consider the gin of a holomorphic decomposition of realanalytic functions. That is, for a domain U ⊂ Cn , consider two real-analytic functions of realanalytic functions r1 : U → R and r2 : U → R. Then say the functions are biholomorphically
equivalent if there exists a biholomorphism F : U → U, such that r1 = r2 ◦ F .
If we take H(rj ) to be the holomorphic decomposition of rj at any point of U, then
Lemma 8.1 says that if r1 and r2 are equivalent as above, then


gin affine-span H(r1 ) = gin affine-span H(r2 ) .
(74)

16

DUSTY GRUNDMEIER AND JIŘÍ LEBL

The gin as defined above is not a pointwise invariant. That is, a real-analytic function
defined on a connected open set has the same gin near every point. By the same argument as
before we can also define the gin of a real-analytic function on a connected complex manifold
U by noting that the gin is already well-defined if we take any connected chart.
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Department of Mathematics, Ball State University, Muncie, IN 47306, USA
Current address: Department of Mathematics, Harvard University, Cambridge MA 02138, USA
E-mail address: deg@math.harvard.edu
Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA
E-mail address: lebl@math.okstate.edu

