PARUSIŃSKI’S “KEY LEMMA” VIA ALGEBRAIC GEOMETRY

Z. REICHSTEIN AND B. YOUSSIN

Abstract. We give a new algebro-geometric proof of the “Key Lemma” of Parusiński.

Contents

1. Introduction 1 2. Preliminaries 3 Notational conventions 3 “Naive” vs. “honest” stabilizers 3 Semi-linear representations and skew group rings 4 3. Reduction to an algebro-geometric problem 5 4. G-varieties in standard form 8 The results of [RY] 8 A local coordinate system 9 Property (*) 11 Proof of Theorem ?? 13 References 13

1. Introduction

The purpose of this paper is to give a new proof of the ”Key Lemma” of Parusiński; see [Pa, pp. 202–203] and Theorem 1.1 below.

In order to place this result in its proper context, we begin with a brief introduction to the subject of Lipschitz stratifications. Two germs of varieties X, X′ at a point O in ℝ^{n} are called Lipschitz equivalent if there is a germ of a Lipschitz homeomorphism between two neighborhoods of O in ℝ^{n} which transforms X into X′. Siebenmann and Sullivan [SS, p. 511] asked whether or not there are only countably many equivalence classes of singular points of analytic varieties up to Lipschitz equivalence. Intuitively, the question is, whether the classification of analytic singularities up to Lipschitz equivalence has continuous moduli.

1991 Mathematics Subject Classification. Primary 14E15, 14F10, 14L30, 32B10, 58A40.

Secondary: 16S35.

Z. Reichstein was partially supported by NSF grant DMS-9801675.

Research at MSRI is supported in part by NSF grant DMS-9701755.

Lipschitz equivalence appeared for the following reasons. On the one hand, it is well-known that the analytic equivalence has continuous moduli: for example, the union of four curves in the plane intersecting at a given point x, has a continuous invariant, the double ratio of the tangent lines at x. For this reason, analytic equivalence is too delicate for classification purposes. On the other hand, there are no moduli of singularities up to homeomorphism: any family of singularities can be Whitney stratified [Wh], and Thom’s Isotopy Lemma [Ma] shows that the singularities at any two points of the same Whitney stratum are homeomorphic.

However, the homeomorphism classification is very coarse, as it captures only the topological properties of the singular set. It was hoped that a Lipschitz classification, while capturing some of the analytic properties, would have no continuous moduli. The analog of a Whitney stratification is a Lipschitz stratification, i.e., a stratification with the property that for any point of a stratum S there is a neighborhood which has a Lipschitz product structure along S.

It turned out that Lipschitz stratifications are also important in the study of L^{2} and L^{p} cohomology: if a stratification is Lipschitz then L^{2} and L^{p} cohomology are locally constant along each stratum [C] (see also [Y]). T. Ohsawa told the second author in November 1993 that he could fix all the difficulties in his unsuccessful attempt [Oh] at the Cheeger—Goresky—MacPherson conjecture, with the exception of the existence of stratifications satisfying certain even stronger Lipschitz properties, which are not yet known to exist.

The existence of a complex-analytic Lipschitz stratification for a germ of a complex-analytic space was proved by Mostowski [Most]; however, this trailblazing paper is so difficult, that many experts do not understand all of its details. Parusiński [Pa] proved the existence of Lipschitz stratifications in the class of semianalytic sets; his arguments are simpler than those of [Most]. Parusiński’s approach relies on the following result, which he calls the “Key Lemma”.

Theorem 1.1. (Parusiński [Pa, pp. 202–203]) For any positive integer n, there is a finite set of homogeneous symmetric polynomials W_{1}, …, W_{N} ∈ ℤ[x_{1}, …, x_{n}] and a constant M > 0 such that

|\frac{dx_{i}}{x_{i}} (p, v)| ≤ M max_{j=1,...,N}|\frac{dW_{j}}{W_{j}} (p, v)|  (1.1)

for all p ∈ ℂ^{n} and v ∈ T_{p}ℂ^{n} for which both sides are defined. Here, for any P ∈ ℂ[x_{1}, …, x_{n}] we view the meromorphic differential form \frac{dP}{P} on ℂ^{n} as a densely defined function on the total space of the tangent bundle T ℂ^{n}.

The proof of the “Key Lemma” given in [Pa, Section 6] is quite difficult, being apparently the hardest part of [Pa]. The purpose of this paper is to show that Theorem 1.1, in spite of its analytic appearance, has a natural proof in the framework of algebraic geometry. Our argument is an application of the results of [RY] about resolving group actions on algebraic varieties; these results, in turn, rely on canonical resolution of singularities. The necessary background material from [RY] is summarized at the beginning of Section 4.

We remark that Parusiński proves the inequality (1.1) under the the additional assumption that dV (p, v) = 0 if V (p) = 0 for every V belonging to finite set V of polynomials. Since this additional requirement does not affect a dense Zariski open subset of T ℂ^{m} (given by V (p) ≠ 0 for every V ∈ V), it can be dropped. We also note that the statement of the Key Lemma in [Pa] only asserts the existence of polynomials W_{1}, …, W_{N} with real coefficients; however, the construction given in the proof there produces polynomials over ℤ. Thus, while Theorem 1.1 appears to be stronger than the “Key Lemma” in [Pa], the two are, in fact, equivalent.

2. Preliminaries

Notational conventions. All algebraic varieties considered in this paper, are assumed to be irreducible and definied over a field k of characteristic 0. The base field k is not assumed to be algebraically closed; the two cases of interest to us are k = ℚ and k = ℂ. By a point of a variety x ∈ X, we shall always understand a closed point; the local ring of x will be denoted by O_{x,X}, its maximal ideal by 𝔪_{x}, its residue field by k(x) = O_{x,X}/𝔪_{x}, and its tangent space by T_{x}(X) = (𝔪_{x}/(𝔪_{x})^{2})^{∗}. The ring of regular functions on a Zariski open subset U ⊂ X will be denoted by O_{X}(U). The sheaf of regular functions on X will be denoted by O_{X}. Given an embedding k ⊂ ℂ, we define X_{ℂ} = X ⊗_{k} ℂ; for any rational function f on X, we shall denote the corresponding rational function on X_{ℂ} by f as well.

Throughout this paper G will be a finite group. A G-variety X is a variety with a regular action of G, G × X ⟶ X, where G × X is understood as the disjoint union of |G| copies of X. We will always assume that the G-action is faithful, i.e., every nonidentity element of G acts nontrivially. By a morphism (respectively, rational map, birational isomorphism) of Gvarieties we shall mean a G-equivariant morphism (respectively, rational map, birational isomorphism).

“Naive” vs. “honest” stabilizers. For a point x in a G-variety X, we define its “naive” stabilizer NStab(x) as the set of all g ∈ G which preserve x. If k is not algebraically closed, the residue field k(x) may be a nontrivial finite extension of k, and G may act on it nontrivially; we define the “honest” stabilizer Stab(x) as the set of all g ∈ NStab(x) that act on k(x) trivially. In other words, Stab(x) is the stabilizer of x in the scheme-theoretic sense; cf. [MFK, Definition 0.4]. Both NStab(x) and Stab(x) are subgroups of G; they are sometimes called the decomposition group and the inertia group respectively.

If \overline{k} is the algebraic closure of k, then x is represented by a set of “conjugate” points of the variety X\overline{k} = X ⊗_{k} \overline{k} (“geometric points” of X), one for each embedding k(x) ↪ \overline{k}. The “honest” stabilizer Stab(x) fixes each of these points while NStab(x) permutes them. The case where Stab(x) is strictly smaller than NStab(x), is quite common. For example, consider the action of G = ℤ/2ℤ on the affine line 𝔸_{ℚ}^{1} = Spec ℚ[t]: the nontrivial element of this group acts by t ↦ −t. For any x ∈ 𝔸_{ℚ}^{1} − {0}, Stab(x) = {1}. On the other hand, NStab(x) = G iff x corresponds to the ideal in ℚ[t] generated by an irreducible polynomial of the form q(t^{2}) (e.g., t^{2} + 1).

This phenomenon is entirely arithmetic; we are concerned with it here because the symmetric polynomials W_{1}, …, W_{N} in Theorem 1.1 are asserted to have integer coefficients. A reader who is only interested in the existence of such polynomials in ℂ[x_{1}, …, x_{n}] may skip the rest of this section and assume that “naive” stabilizers always coincide with “honest” ones in the sequel.

Semi-linear representations and skew group rings. Let X be a Gvariety and let x ∈ X. The “naive stabilizer” NStab(x) acts upon T_{x}(X)^{∗} = 𝔪_{x}/𝔪_{x}^{2}. However, if NStab(x) is strictly larger than Stab(x) then this action is not linear over k(x) but rather “semi-linear” in the following sense.

Definition 2.1. Suppose a finite group H acts by automorphisms on a field K. A semi-linear representation of H over K is a K-vector space V with a K^{H}-linear action of H on V having the property g(λv) = g(λ)g(v) for any g ∈ H, λ ∈ K and v ∈ V.

For the rest of this section we shall assume that K is a field, K^{∗} is the multiplicative group of K, H is a finite group acting on K by automorphisms, and H′ is the kernel of this action. In the subsequent applications we will take K = k(x), H = NStab(x) and H′ = Stab(x).

Recall that skew group algebra K ∗ H is generated by K and H with the relations ha = h(a)h for any h ∈ H and a ∈ K; alternatively, it is the space of all formal sums ∑_{h∈H} a_{h}h where the coefficients a_{h} are elements of K and multiplication is defined by the formula (∑_{h∈H} a_{h}h)(∑_{h∈H} b_{h}h) = ∑_{h∈H} c_{h}h where c_{h} = ∑_{h_{1}h_{2}=h} a_{h_{1}} · h_{1}(b_{h_{2}}).

It is immediate from Definition 2.1 that a semi-linear representation of H is the same thing as a (K ∗ H)-module. (All modules in this paper are understood to be left modules.)

Remark 2.2. Note that V = K has a natural structure of a (K ∗ H)- module. This module contains a vector 1 ∈ K which is fixed by H.

Recall that by the Wedderburn Structure Theorem every semisimple ring R is a direct product of simple rings, called the simple components of R; see, e.g., [Bo, Theorem VIII.5.1].

Lemma 2.3. K ∗ H is a semisimple K^{H}-algebra with at most |H′| simple components. (Here H′ is the kernel of the H-action on K, as above.)

Proof. Semisimplicity of K ∗ H is proved by the same averaging argument as the usual Maschke’s theorem; for details see, e.g., [Mont, Theorem 0.1 and Corollary 0.2].

Denote the simple components of K ∗H by S_{1}, …, S_{m}. Then Z(K ∗H) = Z(S_{1}) × ⋯ × Z(S_{m}), where Z(A) denotes the center of A. Here m is the number of simple components of Z(K ∗ H); we will prove that m ≤ |H′| by showing that Z(K ∗ H) is a direct product of ≤ |H′| fields. It is easy to see that

Z(K ∗ H) ⊂ { ∑_{h′∈H′} a_{h′}h′ | a_{h′} ∈ K} = K[H′] ,

where Z(A) denotes the center of the ring A, and the (usual) group algebra K[H′] is naturally embedded in K ∗H. (Indeed, for every s ∈ K ∗H −K[H′] we can choose a ∈ K so that sa ≠ as.) Consequently,

Z(K ∗ H) ⊂ Z(K[H′]) .  (2.1)

Recall that the group ring K[H′] is a semisimple K-algebra of dimension |H′|. Thus K[H′] has ≤ |H′| simple components and, consequently, Z(K[H′]) is a direct product of ≤ |H′| fields. Since Z(K ∗ H) is a direct product of m fields, it has m mutually orthogonal idempotents and, in view of (2.1), so does Z(K[H′]). This is only possible if m ≤ |H′|, as claimed.

We are now ready to describe the particular type of skew group rings we will encounter in the sequel.

Proposition 2.4. Suppose that H′ is an abelian group of exponent e, K contains a primitive eth root of unity and χ_{1}, ..., χ_{m} is a set of generators for the dual group (H′)^{∗} = Hom(H′, K^{∗}). Assume further that for each i there is a one-dimensinal semi-linear representation V_{i} of H over K such that h′(v) = χ_{i}(h′)v for every v ∈ V_{i} and every h′ ∈ H′. Then:

(a) For every χ ∈ (H′)^{∗}, there exists a unique semi-linear representation V_{χ} of H such that dim_{K}(V_{χ}) = 1 and h′(v) = χ(h′)v for every v ∈ V_{χ} and every h′ ∈ H′.

(b) Every simple (K ∗ H)-module is isomorphic to V_{χ} for some χ ∈ (H′)^{∗}.

Proof. Note that if V_{1} and V_{2} are semi-linear representations of H over K then so is V_{1} ⊗_{K} V_{2}. Indeed,

h(λv_{1}) ⊗ h(v_{2}) = h(λ) · (h(v_{1}) ⊗ h(v_{2})) = h(v_{1}) ⊗ h(λv_{2}) .

We can thus construct V_{χ} as follows. Write χ ∈ (H′)^{∗} as χ = χ_{1}^{l_{1}} … χ_{m}^{l_{m}} for some nonnegative integers l_{1}, …, l_{m}. Now set

V_{χ} = V_{1}^{⊗l_{1}} ⊗_{K} ⋯ ⊗_{K} V_{m}^{⊗l_{m}} .

The subgroup H′ acts on V_{χ} by the character χ, as desired. By our construction dim_{K}(V_{χ}) = 1; consequently, V_{χ} is an simple (K ∗ H)-module. Note that the (K ∗ H)-modules V_{χ} are pairwise nonisomorphic because H′ acts on them by different characters.

The isomorphism classes of simple (K∗H)-modules are in 1—1 correspondence with the simple components of K ∗H; see [Bo, Proposition VIII.5.11].

Thus Lemma 2.3 implies that K ∗ H has ≤ |H′| nonisomorphic simple modules. On the other hand, we have constructed |H′| nonisomorphic simple modules V_{χ}. This proves (b) and the uniqueness of V_{χ} in (a).

Remark 2.5. One can show that, under the assumptions of Proposition 2.4, H is a semidirect product of H′ and H/H′, where the the action of H/H′ on H′ is given by embedding H′ into (K^{∗})^{m} via h′ ⟶ (χ_{1}(h′), …, χ_{m}(h′)).

3. Reduction to an algebro-geometric problem

Reduction 3.1. In the setting of Theorem 1.1 it is enough to construct symmetric polynomials W_{1}, …, W_{N} with rational coefficients so that (1.1) is satisfied.

Indeed, after multiplying W_{1}, …, W_{N} ∈ ℚ[x_{1}, …, x_{n}] by a suitable integer constant, we will obtain a desired collection of symmetric polynomials with integer coefficients.

Reduction 3.2. In the setting of Theorem 1.1 it is enough to choose W_{1}, …, W_{N} so that inequality (1.1) holds for i = 1 only.

To see this, note that the natural S_{n}-action transitively permutes the differentials dx_{i}/x_{i} and leaves the differentials dW_{j}/W_{j} invariant. Thus if inequality (1.1) holds for i = 1 then it holds for every i (with the same M and W_{1}, …, W_{N}).

Next we would like to restate (1.1) as an inequality involving densely defined functions on the tangent bundle of ℙ_{ℂ}^{n−1} rather than the tangent bundle of ℂ^{n}. The reason this will be helpful in the sequel is that ℙ_{ℂ}^{n−1} is compact in the metric topology; this will allow us to pass from local to global estimates. Note that inequality (1.1) cannot be “projectivised” directly because dP/P is not a pullback of any differential on ℙ_{ℂ}^{n−1}, even if P is a homogeneous polynomial in x_{1}, …, x_{n}. On the other hand, if P and Q are homogeneous polynomials of the same degree then the difference dP/P − dQ/Q equals dr/r, where r = P/Q is a well-defined rational function on ℙ_{ℂ}^{n−1}. If we choose Q to be symmetric, then for the purposes of proving Theorem 1.1, replacing dP/P by dr/r will not cause any harm. The argument of Reduction 3.4 below makes this observation precise.

Proposition 3.3. Let X be a projective G-variety over a field k ⊂ ℂ, and let f be a rational function on X. Then there exist G-invariant rational functions β_{1}, …, β_{m} on X and a constant K > 0 such that

|\frac{df}{f} (p, v)| ≤ K max_{j=1,...,m}|\frac{dβ_{j}}{β_{j}} (p, v)|  (3.1)

for any (p, v) ∈ T (X_{ℂ}) such that p is a smooth point of X_{ℂ} and does not lie on the divisors of f, β_{1}, …, β_{m}.

Reduction 3.4. Proposition 3.3 ⟹ Theorem 1.1.

Indeed, apply Proposition 3.3 to the case where X = ℙ_{ℚ}^{n−1} (so that k = ℚ), G = S_{n} acts on ℙ_{ℚ}^{n−1} by permutations of the homogeneous coordinates x_{1}, …, x_{n}, f = x_{1}/s_{1} and s_{1} = x_{1} + ⋯ + x_{n}. Write each β_{j} as a quotient of two homogeneous polynomials (of the same degree) in x_{1}, …, x_{n}:

β_{1} = \frac{W_{1}}{W_{2}} , …, β_{m} = \frac{W_{2m−1}}{W_{2m}} .

We claim that the polynomials W_{1}, …, W_{2m}, W_{2m+1} =^{def} s_{1} have the property asserted in Theorem 1.1. Indeed, since df/f = dx_{1}/x_{1}−ds_{1}/s_{1} and dβ_{i}/β_{i} = dW_{2i−1}/W_{2i−1} − dW_{2i}/W_{2i}, inequality (3.1) translates into

|(\frac{dx_{1}}{x_{1}} − \frac{ds_{1}}{s_{1}} ) (p, v)| ≤ K max_{i=1,...,m}|(\frac{dW_{2i−1}}{W_{2i−1}} − \frac{dW_{2i}}{W_{2i}} ) (p, v)|

≤ 2K max_{j=1,...,2m}|\frac{dW_{j}}{W_{j}} (p, v)|

Consequently,

|\frac{dx_{1}}{x_{1}} (p, v)| ≤ 2K max_{j=1,...,2m}|\frac{dW_{j}}{W_{j}} (p, v)| + |\frac{ds_{1}}{s_{1}} (p, v)|

≤ (2K + 1) max_{j=1,...,2m+1}|\frac{dW_{j}}{W_{j}} (p, v)| .

This means that (1.1) holds for i = 1, with M = 2K + 1 and N = 2m + 1. In view of Reduction 3.2 this proves Theorem 1.1.

Reduction 3.5. Note that the statement of Proposition 3.3 involves only rational functions on X. This suggests that in the course of proving Proposition 3.3 we may replace X by a birationally equivalent G-variety. More precisely,

Suppose X and X′ are birationally isomorphic G-varieties over k ⊂ ℂ. If Proposition 3.3 holds for X and f ∈ k(X) then it holds for X′ and the same f ∈ k(X′) = k(X).

Indeed, X and X′ have isomorphic Zariski-open subsets U and U′. After passing to smaller subsets if necessary, we may assume that U and U′ are smooth and do not intersect the divisors of f, β_{1}, …, β_{n} on X and X′ respectively. Thus if inequality (3.1) holds for every (p, v) such that p ∈ U_{ℂ} and v ∈ T_{p}(X_{ℂ}) then

|\frac{df}{f} (p′, v′)| ≤ K max_{j=1,...,m}|\frac{dβ_{j}}{β_{j}} (p′, v′)|

for every (p′, v′) such that p′ ∈ U′_{ℂ} and v′ ∈ T_{p′}(X′_{ℂ}). The subset U′_{ℂ} is dense in X′_{ℂ} with respect to metric topology; hence, by continuity the same inequality (with the same β_{j} and the same K) holds for every (p′, v′) ∈ T (X′_{ℂ}) such that p′ is a smooth point of X′_{ℂ} and does not lie in the union of divisors of f, β_{1}, …, β_{m}. This means that Proposition 3.3 holds for X′ as claimed.

Definition 3.6. Let X be a projective G-variety and f be a rational function on X. We shall say that the pair (X, f) has property (*) if there is a Zariski open covering X = ⋃_{i} U_{i} and, for each i, rational functions β_{i1}, …, β_{i,q_{i}} ∈ k(X)^{G} and regular functions γ_{i1}, …, γ_{i,q_{i}} ∈ O_{X}(U_{i}) such that

\frac{df}{f} = γ_{i1} \frac{dβ_{i1}}{β_{i1}} + … γ_{i,q_{i}} \frac{dβ_{i,q_{i}}}{β_{i,q_{i}}} .  (3.2)

In other words, the pair (X, f) has property (*) if df/f is a global section of the sheaf of differentials on X generated over O_{X} by dβ/β, as β ranges over some finite subset of k(X)^{G} (or, equivalently, as β ranges over all of k(X)^{G}).

Reduction 3.7. Proposition 3.3 holds, assuming the pair (X, f) that appears there, has property (*).

Indeed, we have an open covering X_{ℂ} = ⋃_{i} U_{i,ℂ} (with respect to both Zariski and metric topology), and the equality (3.2) holds in U_{i,ℂ} where γ_{ij} can be viewed as continuous functions on U_{i,ℂ} with respect to metric topology.

It follows that any point x ∈ X (whether smooth or not) has a neighborhood U_{x} (in the metric topology) such that U_{x} ⊂ U_{i_{x},ℂ} for some i_{x}, and

|\frac{df}{f} (p, v)| ≤ K_{x} max_{j=1,...,q_{ix}} |\frac{dβ_{i_{x},j}}{β_{i_{x},j}} (p, v)|

whenever v ∈ T_{p}(X) and p is a smooth point of U_{x} which does not lie on the divisors of f, β_{i_{x},1}, …, β_{i_{x},q_{ix}}. The open sets U_{x} form a cover of X. Since X compact in the metric topology, we can choose a finite subcover U_{x_{1}}, …, U_{x_{r}}. Now if K > K_{x_{1}}, …, K_{x_{r}} then

|\frac{df}{f} (p, v)| ≤ K max_{i,j} |\frac{dβ_{ij}}{β_{ij}} (p, v)| .

This shows that Proposition 3.3 holds.

We have thus shown that Theorem 1.1 is a consequence of the following, purely algebraic statement (see Reductions 3.4, 3.5 and 3.7).

Proposition 3.8. Let G be a finite group, X a projective G-variety, and f ∈ k(X). Then there exists a (regular) birational morphism π: X′ ⟶ X of G-varieties such that the pair (X′, π^{∗}(f)) has property (*) (see Definition 3.6).

A proof of Proposition 3.8 (and thus of Theorem 1.1) will be given in the next section. The idea of the proof can be seen from the simplest example where X = 𝔸^{1} = Spec k[t], k contains the mth roots of unity, G = ℤ/mℤ acts on 𝔸^{1} linearly by a faithful character, and f = t. In this case we can take β = t^{m} ∈ k(X)^{G}, and the equality df/f = \frac{1}{m}dβ/β shows that (X, f) has property (*).

4. G-varieties in standard form

The results of [RY].

Definition 4.1. ([RY, Definition 3.1]) We say that a generically free Gvariety X is in standard form with respect to a divisor Y if

(a) X is smooth and Y is a normal crossing divisor on X,

(b) the G-action on X − Y is free, and

(c) for every g ∈ G and for every irreducible component Y_{0} of Y either g(Y_{0}) = Y_{0} or g(Y_{0}) ∩ Y_{0} = ∅.

Theorem 4.2. ([RY, Corollary 3.6]) Let X be a G-variety and Y ⊂ X be a Zariski closed G-invariant subvariety such that the action of G on X − Y is free. Then there is a sequence of blowups

π: X_{n} ⟶^{π_{n}} X_{n−1} ⋯ ⟶^{π_{2}} X_{1} ⟶^{π_{1}} X_{0} = X  (4.1)

with smooth G-invariant centers C_{i} ⊂ X_{i} such that X_{n} is in standard form with respect to a divisor Ỹ containing π^{−1}(Y).

Theorem 4.3. Let X be a G-variety in standard form with respect to a divisor Y, let x be a point of X, let Y_{1}, …, Y_{m} be the irreducible components of Y passing though x, and let W = Y_{1} ∩ ⋯ ∩ Y_{m}. Then

(a) ([RY, Theorem 4.1]) Stab(x) is commutative.

(b) ([RY, Remark 4.4]) The action of Stab(x) in the normal space to W at x is faithful and decomposes into the sum of one-dimensional representations as follows:

T_{x}(X)/T_{x}(W) = ⨁_{i=1}^{m} \frac{T_{x}(Y_{1}) ∩ ⋯ ∩ { \atop T_{x}(Y_{i})} ∩ ⋯ ∩ T_{x}(Y_{m})}{T_{x}(W)} . (4.2)

(c) ([RY, Remark 4.5]) Let e be the exponent of Stab(x). Then the residue field k(x) of x contains the eth roots of unity.

Remark 4.4. Under the assumptions of Theorem 4.3, set H = NStab(x) and H′ = Stab(x). Recall that H acts on T_{x}(X) semi-linearly; see Definition 2.1. Property (c) of Definition 4.1 implies that Y_{i} is preserved by the action of H, and hence, the subspace T_{x}(Y_{i}) is H-invariant for each i. It follows that all spaces appearing in (4.2), are (k(x) ∗ H)-modules.

The conormal space (T_{x}(X)/T_{x}(Y_{i}))^{∗} is dual to the ith summand in (4.2); it is a (k(x)∗H)-module of dimension 1 over k(x), and H′ acts on it by a character. Denote this character by ξ_{i}. Theorem 4.3(b) implies that the characters ξ_{1}, …, ξ_{m} generate the dual group (H′)^{∗}. In view of Theorem 4.3(c), this shows that Proposition 2.4 applies in this setting.

A local coordinate system. Suppose that X is an algebraic variety and x is a point of X. Recall that u_{1}, …, u_{n} ∈ 𝔪_{x} are said to form a local coordinate system on X at x if their classes modulo 𝔪_{x}^{2} form a basis of 𝔪_{x}/𝔪_{x}^{2} as a k(x)-vector space.

Proposition 4.5. Let X be a quasiprojective G-variety in standard form with respect to a divisor Y. Suppose Y_{1}, …, Y_{m} are the irreducible components of Y passing though a point x of X. Then there exists a local coordinate system u_{1}, …, u_{m}, v_{1}, …, v_{l} at x with the following properties:

(a) Let \overline{u}_{i} = u_{i} mod 𝔪_{x}^{2} and \overline{v}_{j} = v_{j} mod 𝔪_{x}^{2}. Then each of \overline{u}_{1}, …, \overline{u}_{m}, \overline{v}_{1}, …, \overline{v}_{l} generates a one-dimensional H-invariant k(x)- subspace of 𝔪_{x}/𝔪_{x}^{2}.

(b) For every i = 1, …, m and every g ∈ G, u_{i} is a local equation of g(Y_{i}) at gx.

(c) For every j = 1, …, l there are integers e_{j1}, …, e_{jm} ≥ 0 such that h_{j} = u_{1}^{e_{j1}} … u_{m}^{e_{jm}}v_{j} is a G-invariant rational function on X.

Proof. We begin by constructing u_{1}, …, u_{m}. Consider the divisor D_{i} = ∑ g(Y_{i}), where each summand of the form g(Y_{i}) (for some g ∈ G) appears in this sum exactly once. Since X is quasiprojective, the divisor D_{i} can be “moved off” any finite set of points (see [Sh, Theorem III.1.1]); in particular it can be “moved off” the orbit Gx. In other words, for every i = 1, …, m there is a rational function u_{i} on X such that the support of the divisor D_{i} − (u_{i}) does not intersect Gx. It is now easy to see that u_{1}, …, u_{m} satisfy (a) and (b).

Next we turn to the construction of v_{1}, …, v_{l}. Each \overline{u}_{i} generates a onedimensional k(x)-subspace <\overline{u}_{i}> = (T_{x}(X)/T_{x}(Y_{i}))^{∗} ⊂ 𝔪_{x}/𝔪_{x}^{2}. We have seen in Remark 4.4 that <\overline{u}_{i}> is H-invariant; H′ acts on it by the character ξ_{i}. In view of Lemma 2.3 and Proposition 2.4, 𝔪_{x}/𝔪_{x}^{2} can be decomposed into a direct sum of (k(x) ∗ H)-submodules which are one-dimensional over k(x) and include <\overline{u}_{1}>, …, <\overline{u}_{m}>. Let \overline{v}_{1}, …, \overline{v}_{l} ∈ 𝔪_{x}/𝔪_{x}^{2} be generators of the other submodules of this decomposition. Denote the character of H′ associated to \overline{v}_{j} by η_{j}.

By Remark 4.4, ξ_{1}, …, ξ_{m} generate the dual group (H′)^{∗}; consequently, each η_{j} can be written in the form

η_{j} = ξ_{1}^{−e_{j1}} … ξ_{m}^{−e_{jm}}

for some integers e_{j1}, …, e_{jm}. We may assume without loss of generality that each e_{ji} ≥ 0. Note that \overline{h}_{j} = \overline{u}_{1}^{e_{j1}} … \overline{u}_{m}^{e_{jm}} \overline{v}_{j} is an H′-invariant element of 𝔪_{x}^{d_{j}}/𝔪_{x}^{d_{j}+1}, where d_{j} = e_{j1} + ⋯ + e_{jm} + 1. Clearly, \overline{h}_{j} generates an Hinvariant one-dimensional k(x)-subspace <\overline{h}_{j}> ⊂ 𝔪_{x}^{d_{j}}/𝔪_{x}^{d_{j}+1} on which H′ acts trivially. By the uniqueness statement in Proposition 2.4(a), <\overline{h}_{j}> =^{∼} k(x) as (k(x) ∗ H)-modules, and by Remark 2.2, after replacing \overline{v}_{j} by λ\overline{v}_{j} for some λ ∈ k(x), we may assume

\overline{h}_{j} = \overline{u}_{1}^{e_{j1}} … \overline{u}_{m}^{e_{jm}} \overline{v}_{j} is an H-invariant element of 𝔪_{x}^{d_{j}}/𝔪_{x}^{d_{j}+1} . (4.3)

We claim that we can choose v_{1}, …, v_{l} ∈ 𝔪_{x} so that \overline{v}_{j} = v_{j} mod 𝔪_{x}^{2} and each h_{j} = u_{1}^{e_{j1}} … u_{m}^{e_{jm}}v_{j} is G-invariant. If we can do this, then u_{1}, …, u_{m}, v_{1}, …, v_{l} will clearly satisfy the requirements of the proposition.

To prove the claim, let R = ⋂_{y∈Gx} O_{y,X} be the ring of rational functions on X that are well-defined on Gx and let I_{Gx} = ⋂_{y∈Gx} 𝔪_{y} be a G-invariant ideal in R consisting of all elements that vanish on Gx. By our choice of u_{1}, …, u_{m},

g^{∗}(u_{i})/u_{i} is defined and invertible at every point of Gx  (4.4)

for every g ∈ G, and every i = 1, …, m. Consequently, u_{1}^{e_{j1}} … u_{m}^{e_{jm}}I_{Gx} is a G-invariant ideal of R.

For each y ∈ Gx, the functions u_{1}, …, u_{m} vanish at y, and hence, u_{1}^{e_{j1}} … u_{m}^{e_{jm}}I_{Gx} ⊂ 𝔪_{y}^{d_{j}}. Consider the G-equivariant projection map

ψ: u_{1}^{e_{j1}} … u_{m}^{e_{jm}}I_{Gx} ⟶ ⨁_{y∈Gx} 𝔪_{y}^{d_{j}}/𝔪_{y}^{d_{j}+1} .

If v ∈ I_{Gx} then ψ(u_{1}^{e_{j1}} … u_{m}^{e_{jm}}v) depends only on the image of v in ⨁_{y∈Gx} 𝔪_{y}/𝔪_{y}^{2}. Moreover, since X is quasiprojective, the finite set Gx lies in an affine open subset of X and hence, by the Chinese Remainder Theorem, the projection map I_{Gx} ⟶ ⨁_{y∈Gx} 𝔪_{y}/𝔪_{y}^{2} is surjective. Thus

Im(ψ) = ⨁_{y∈Gx} (u_{1}^{e_{j1}} … u_{m}^{e_{jm}}𝔪_{y}) mod 𝔪_{y}^{d_{j}+1} ⊂ ⨁_{y∈Gx} 𝔪_{y}^{d_{j}}/𝔪_{y}^{d_{j}+1} .

We shall denote elements of Im(ψ) by a = (a_{y} | y ∈ Gx), where a_{y} ∈ (u_{1}^{e_{j1}} … u_{m}^{e_{jm}}𝔪_{y}) mod 𝔪_{y}^{d_{j}+1}. Recall that by (4.3), \overline{h}_{j} is a nonzero

H-invariant element of (u_{1}^{e_{j1}} … u_{m}^{e_{jm}}𝔪_{x}) mod 𝔪_{x}^{d_{j}+1}. Let a_{j} be the element of Im(ψ) such that (a_{j})_{y} = (g^{−1})^{∗}(\overline{h}_{j}), where y = gx; in view of (4.4), we have here

(g^{−1})^{∗}[(u_{1}^{e_{j1}} … u_{m}^{e_{jm}}𝔪_{x}) mod 𝔪_{x}^{d_{j}+1}] = (u_{1}^{e_{j1}} … u_{m}^{e_{jm}}𝔪_{y}) mod 𝔪_{y}^{d_{j}+1} ,

and hence, (a_{j})_{y} ∈ (u_{1}^{e_{j1}} … u_{m}^{e_{jm}}𝔪_{y}) mod 𝔪_{y}^{d_{j}+1}. Note that since \overline{h}_{j} is Hinvariant, (a_{j})_{y} is independent of the choice of g. By our construction a_{j} is G-invariant and (a_{j})_{x} = \overline{h}_{j} ∈ 𝔪_{x}^{d_{j}}/𝔪_{x}^{d_{j}+1}.

Recall that in characteristic 0, every morphism of (not necessarily finitedimensional) G-vector spaces over k splits. Thus ψ has a G-equivariant k-linear splitting and consequently, there exists a G-invariant element h_{j} = u_{1}^{e_{j1}} … u_{m}^{e_{jm}}v_{j} ∈ u_{1}^{e_{j1}} … u_{m}^{e_{jm}}I_{Gx} such that ψ(h_{j}) = a_{j}. In particular, \overline{h}_{j} = \overline{u}_{1}^{e_{j1}} … \overline{u}_{m}^{e_{jm}} \overline{v}_{j} = h_{j} mod 𝔪_{x}^{d_{j}+1} and hence \overline{v}_{j} = v_{j} mod 𝔪_{x}^{2}. This proves the claim and thus shows that u_{1}, …, u_{m}, v_{1}, …, v_{l} have the required properties.

Property (*). We are now ready to examine property (*) of Definition 3.6 more closely.

Lemma 4.6. Suppose X be a quasiprojective G-variety in standard form with respect to a divisor Y, x ∈ X, u_{1}, …, u_{m}, v_{1}, …, v_{l}, and h_{1}, …, h_{l}, are as in Proposition 4.5, and w_{i} = ∏_{g∈G} g^{∗}(u_{i}). Let f be a rational function on X whose divisor is supported on Y. Then

\frac{df}{f} ∈ M ,

where M is the O_{x,X}-module generated by dw_{i}/w_{i} and dh_{j}/h_{j} with i = 1, …, m and j = 1, …, l.

Note that w_{i} and h_{j} are G-invariant rational functions on X for every i = 1, …, m and j = 1, …, l.

Proof. Let (Ω_{X}^{1})_{x} be the O_{x,X}-module of germs at x of regular differential forms on X. Since X is smooth, (Ω_{X}^{1})_{x} is a free O_{x,X}-module generated by du_{1}, …, du_{m}, dv_{1}, …, dv_{l}. Let M′ be the O_{x,X}-module generated by du_{i}/u_{i}, and dv_{j}/v_{j}, where i = 1, …, m and j = 1, …, l. Clearly, (Ω_{X}^{1})_{x} ⊂ 𝔪_{x}M′.

Since the divisor of f is supported on Y, locally near x it is a union of smooth hypersurfaces of the form {u_{i} = 0}. This means that f = au_{1}^{e_{1}} … u_{m}^{e_{m}} for some a ∈ O_{x,X} − 𝔪_{x} and e_{1}, …, e_{m} ≥ 0 and, hence,

\frac{df}{f} = \frac{da}{a} + e_{1} \frac{du_{1}}{u_{1}} + ⋯ + e_{m} \frac{du_{m}}{u_{m}} .

Since da/a ∈ (Ω_{X}^{1})_{x}, we conclude that df/f is an O_{x,X}-linear combination of du_{i}/u_{i} and dv_{j}; in particular, df/f ∈ M′.

We claim that M = M′; this would imply that df/f ∈ M, as desired. It is clear that M ⊂ M′. It remains to prove the opposite inclusion, i.e., it remains to prove that dw_{i}/w_{i} and dh_{j}/h_{j} generate M′, as i ranges from 1 to m and j ranges from 1 to l.

Note that by our choice of u_{1}, …, u_{m}, we can write w_{i} = a_{i}u_{i}^{|G|} for some a_{i} ∈ O_{X,x} − 𝔪_{x}; see (4.4). Thus

\frac{dw_{i}}{w_{i}} = \frac{da_{i}}{a_{i}} + |G|\frac{du_{i}}{u_{i}} ≡ |G| \frac{du_{i}}{u_{i}} (mod (Ω_{X}^{1})_{x}) .

In particular, since (Ω_{X}^{1})_{x} ⊂ 𝔪_{x}M′, we conclude that

\frac{dw_{i}}{w_{i}} ≡ |G| \frac{du_{i}}{u_{i}} (mod 𝔪_{x}M′) .  (4.5)

On the other hand, since h_{j} = u_{1}^{e_{j1}} … u_{m}^{e_{jm}}v_{j}, we have

\frac{dh_{j}}{h_{j}} = \frac{dv_{j}}{v_{j}} + ∑_{i} e_{ji} \frac{du_{i}}{u_{i}} .  (4.6)

Examining (4.5) and (4.6), we see that dw_{i}/w_{i} (i = 1, …, m), and dh_{j}/h_{j} (j = 1, …, l) generate M′/𝔪_{x}M′. Consequently, by Nakayama’s lemma these elements generate M′. Thus M′ = M, as claimed.

Proposition 4.7. Let X be a projective variety in standard form with respect to a divisor Y and let f be a rational function on X whose divisor is supported on Y. Then the pair (X, f) has property (*); see Definition 3.6.

Proof. By Zariski compactness, it is enough to show that for any x ∈ X, there exist β_{1}, …, β_{q} ∈ k(X)^{G} and γ_{1}, …, γ_{q} ∈ O_{x,X} such that

\frac{df}{f} = γ_{1} \frac{dβ_{1}}{β_{1}} + … γ_{q} \frac{dβ_{q}}{β_{q}} .

In other words, df/f lies in the O_{x,X}-module generated by dβ/β, as β ranges over some finite subset of k(X)^{G}. The last assertion is immediate from Lemma 4.6.

Proof of Theorem 1.1. As we showed in Section 3, Theorem 1.1 follows from Proposition 3.8. We thus proceed with the proof of Proposition 3.8,

Let X be a projective G-variety and f ∈ k(X). Let X_{0} be the subvariety of all points in X with nontrivial stabilizers. Let Y be the union of X_{0} and (the supports of) the divisors of g^{∗}(f) for every g ∈ G; it is a G-invariant Zariski closed subvariety of X. By Theorem 4.2 there exists a birational morphism π: X′ ⟶ X and a divisor Y′ ⊂ Y, such that X′ is in standard form with respect to Y′ and π^{−1}(Y) ⊂ Y′. Note that the divisor of π^{∗}(f) is contained in π^{−1}(Y) and thus in Y′.

Proposition 3.8 (and thus Theorem 1.1) now follows from Proposition 4.7, which asserts that the pair (X′, π^{∗}(f)) has property (*).

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Department of Mathematics, Oregon State University, Corvallis, OR 97331 Current mailing address: PMB 136, 333 South State St., Lake Oswego, OR

97034-3961, USA

E-mail address: zinovy@math.orst.edu

Department of Mathematics and Computer Science, University of the Negev, Be’er Sheva’, Israel

Current mailing address: Hashofar 26/3, Ma’ale Adumim, Israel

E-mail address: youssin@math.bgu.ac.il