ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 5, Pages 136–145 (November 17, 1999) S 1079-6762(99)00072-4

PARUSIŃSKI’S “KEY LEMMA” VIA ALGEBRAIC GEOMETRY

Z. REICHSTEIN AND B. YOUSSIN

(Communicated by David Kazhdan)

Abstract. The following “Key Lemma” plays an important role in the work by Parusiński on the existence of Lipschitz stratifications in the class of semianalytic sets: For any positive integer n, there is a finite set of homogeneous symmetric polynomials W_{1}, …, W_{N} in Z[x_{1}, …, x_{n}] and a constant M > 0 such that

|dx_{i}/x_{i}| ≤ M max_{j=1,...,N} |dW_{j}/W_{j}| ,

as densely defined functions on the tangent bundle of ℂ^{n}. We give a new algebro-geometric proof of this result.

1. Introduction

Parusiński’s fundamental work on the existence of Lipschitz stratifications in the class of semianalytic sets relies on the following result.

Theorem 1.1 (Parusiński [P, 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}.

Parusiński refers to Theorem 1.1 as the “Key Lemma”; the proof of this result in [P, Section 6] is quite difficult, being apparently the hardest part of [P]. 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 group actions on algebraic varieties; these results, in turn, rely on canonical resolution of singularities.

We remark that Parusiński proves the inequality (1.1) under the additional assumption that dV (p, v) = 0 if V (p) = 0 for every V belonging to a 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

\begin{matrix}c \\ ◯ \end{matrix}1999 American Mathematical Society

Received by the editors October 16, 1999.

1991 Mathematics Subject Classification. Primary 14E15, 14F10, 14L30; Secondary 16S35, 32B10, 58A40.

Z. Reichstein was partially supported by NSF grant DMS-9801675 and (during his stay at MSRI) by NSF grant DMS-9701755.

that the statement of the Key Lemma in [P] only asserts the existence of polynomials W_{1}, …, W_{N} with real coefficients; however, the construction of W_{1}, …, W_{N} given there, produces polynomials over ℤ. Thus, while Theorem 1.1 appears to be stronger than the “Key Lemma” in [P], the two are, in fact, equivalent.

We are grateful to R. Guralnick and S. Montgomery for informative conversations on the subject of group algebras, and to D. Kazhdan for helpful comments on an earlier draft of this paper.

2. Preliminaries

Notational conventions. All algebraic varieties considered in this paper, are assumed to be irreducible and defined 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 we shall always understand a closed point. Given an embedding k ⊂ ℂ and a rational function f on X, we shall denote the corresponding rational function on X_{ℂ} = X ⊗_{k} 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 G-varieties we shall mean a G-equivariant morphism (respectively, rational map, birational isomorphism).

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; cf. [MFK, Definition 0.4]. The subgroups NStab(x) and Stab(x) of G 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}; Stab(x) fixes each of these points while NStab(x) permutes them. As an 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. Here Stab(x) = {1} for any x ∈ 𝔸_{ℚ}^{1} − {0}. 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.

Semilinear representations and skew group rings. Let X be a G-variety 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 “semilinear” in the following sense.

Definition 2.1. Suppose a finite group H acts by automorphisms on a field K. A semilinear 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 the skew group algebra K ∗ H is defined as the set of formal sums ∑_{h∈H} a_{h}h (where a_{h} ∈ K), with componentwise addition and with multiplication given, distributively, by (a_{1}h_{1})(a_{2}h_{2}) = a_{1}h_{1}(a_{2})h_{1}h_{2}. A semilinear 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 Wedderburn’s Theorem every semisimple ring R is a direct product of simple rings, called the simple components of R; see [B, 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., [Mo, 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. It is easy to see directly that dim_{K^{H}} (Z(K ∗ H)) ≤ |H′|. Hence, m ≤ |H′|, as claimed.

The following proposition describes the particular kind of skew group rings we shall 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-dimensional semilinear 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 semilinear 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 semilinear 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}).

To construct V_{χ}, write χ ∈ (H′)^{∗} as χ = χ_{1}^{l_{1}} … χ_{m}^{l_{m}} for some nonnegative integers l_{1}, …, l_{m}, and 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. As dim_{K}(V_{χ}) = 1, V_{χ} is a 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 [B, 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

We begin by restating (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}. Since ℙ_{ℂ}^{n−1} is compact in the metric topology, this will allow us to pass from local to global estimates.

Proposition 3.1. 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 3.2. Proposition 3.1 ⟹ Theorem 1.1.

Indeed, apply Proposition 3.1 with k = ℚ, X = ℙ_{ℚ}^{n−1}, G = S_{n} and f = x_{1}/s_{1}; here S_{n} acts on ℙ_{ℚ}^{n−1} by permuting the homogeneous coordinates x_{1}, …, x_{n}, and s_{1} = x_{1} + ⋯ + x_{n}. Write each β_{j} as a quotient of two homogeneous polynomials (of the same degree, with integer coefficients) in x_{1}, …, x_{n}:

β_{1} = W_{1}/W_{2} , …, β_{m} = 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. By symmetry, (1.1) holds for all i.

Definition 3.3. 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 3.4. Proposition 3.1 holds, assuming the pair (X, f) that appears there, has property (*).

Indeed, the Zariski open cover ⋃_{i} U_{i} of X, as in Definition 3.3, gives rise to a Zariski open cover ⋃_{i} U_{i,ℂ} of X_{ℂ}. The functions γ_{ij} are continuous on U_{i,ℂ} with respect to the metric topology. Thus any point x ∈ X has an open 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 is 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 3.1 holds.

Reduction 3.5. Suppose X and X′ are birationally isomorphic G-varieties over k ⊂ ℂ. If Proposition 3.1 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 it holds 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 the 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 3.1 holds for X′ as claimed.

We have thus shown that Theorem 1.1 is a consequence of the following, purely algebraic statement (see Reductions 3.2, 3.4 and 3.5).

Proposition 3.6. Let G be a finite group, X a projective G-variety, and f ∈ k(X). Then there exists a birational morphism π: X′ ⟶ X of G-varieties such that the pair (X′, π^{∗}(f)) has property (*) (see Definition 3.3).

A proof of Proposition 3.6 (and thus of Theorem 1.1) will be given in the next section. The idea is to construct π: X′ ⟶ X by resolving the G-action on X to “standard form” with respect to a divisor containing the divisor of f; see below. The simplest (affine) example of such X′ is X′ = 𝔸^{1} = Spec k[t], where k contains a primitive mth root 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}; the equality df/f = \frac{1}{m}dβ/β shows that (X′, f) has property (*).

4. Conclusion of the proof

G-varieties in standard form.

Definition 4.1 ([RY, Definition 3.1]). We say that a generically free G-variety 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_{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 through 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) on 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 a primitive eth root 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) semilinearly; 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′)^{∗}. Combining this observation with Theorem 4.3(c), we conclude 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” the finite set Gx; see [Sh, Theorem III.1.1]. 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 one-dimen-sional 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, we can write

𝔪_{x}/𝔪_{x}^{2} = ⟨\overline{u}_{1}⟩ ⊕ ⋯ ⊕ ⟨\overline{u}_{m}⟩ ⊕ V_{1} ⊕ ⋯ ⊕ V_{l},

where each V_{i} is a simple (k(x)∗H)-module and dim_{k(x)}(V_{i}) = 1. Choose \overline{v}_{1}, …, \overline{v}_{l} ∈ 𝔪_{x}/𝔪_{x}^{2} so that \overline{v}_{i} generates V_{i} as a k(x)-vector space. 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} ≥ 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 H-invariant 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

(4.3) \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}.

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},

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

for every g ∈ G, and every i = 1, …, m. Consequently, u_{1}^{e_{j1}} ⋯ u_{m}^{e_{jm}}I_{Gx} is a Ginvariant 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),

(a_{j})_{y} ∈ (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}.

Note that since \overline{h}_{j} is H-invariant, (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}.

The homomorphism 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 revisit property (*) of Definition 3.3.

Lemma 4.6. Suppose X is 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

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′.

We claim that M = M′. It is clear that M ⊂ M′; to prove the opposite inclusion, we shall show that dw_{1}/w_{1}, …, dw_{m}/w_{m}, dh_{1}/h_{1}, …, dh_{l}/h_{l} generate M′ as an O_{x,X}-module.

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′ as a k(x)-vector space. Consequently, by Nakayama’s lemma these elements generate M′ as an O_{x,X}-module. Thus M′ = M, as claimed.

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; hence,

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

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 3.3.

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}} .

The last assertion is immediate from Lemma 4.6.

Proof of Theorem 1.1. As we showed in Section 3, it is enough to prove Proposition 3.6.

Assume X is 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 hence in Y′. Proposition 3.6 (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

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

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

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

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