A NOTE FOR THE CHANGE OF VARIABLE FORMULA IN ARC SPACES

(1)

CHIN-LUNG WANG

Let φ : Y → X be a proper birational morphism between two n-dimensional complex smooth varieties. Let Sk ⊂ L(Y ) be the subset of formal arcs such that ordtJ (φ) = k. What is the structure of φ_∗: Sk → L(X)? We will see later that in the set level φ_∗ is almost an one to one map. However, the important observation which leads to the change of variable formula in motivic integration is that this map is indeed a piece-wise trivial C^k fibration onto its image when one considers it in formal arcs of certain finite level.

More precisely, instead of working on the infinite dimensional spaces L(Y ) and L(X), one may consider the following diagram about truncations to discuss only algebraic varieties L_m(Y ) and L_m(X):

L(Y )

φ

πm // Lm(Y )

φ_m

L(X) _π

m // Lm(X)

The map φ₀ = φ is the original map. The map φ₁ is the tangent map T_Y → TX

since the first order arc is nothing but the tangent space.

Denef and Loeser showed that

Theorem 0.1. For each k ∈ N, there exists m^k∈ N such that for m ≥ mk, πmSk

is a union of fibers of φm. Moreover, φm|π_mS_k is a piece-wise trivial C^k fibration onto its image.

This was also proved by them for singular variety X, with suitable modifications on the set Sk. Here we concern only the smooth case. In the smooth case πm are all surjective (every finite arc can be lifted), so we may also define Sk directly on certain large enough level m.

The purpose of this note is to demonstrate a key step in the proof of Denef and Loeser’s result. We will first do it for a simple blowing-up at one smooth point in §1, then in §2 we give the proof for the general case using the inverse function theorem (or Hensel’s lemma). §2 is independent of §1, however §1 gives a down-to-the-earth treatment which I feel also helpful in understanding the real content of change of variable formula.

1. A Simple Blowing-Up Example

Consider a blowing-up φ : Y := ˜Cⁿ → X := Cⁿ at one smooth point 0 ∈ Cⁿ. For the affine open set U1of Y with coordinates (y1, · · · , yn), the map φ takes the

Date: October 2002.

1

(2)

2 CHIN-LUNG WANG

form

x1= y1, x2= y1y2, · · · xn= y1yn,

where (x1, · · · , xn) are coordinates of X. The Jacobian J := J (φ) = (y₁ⁿ⁻¹) which corresponds to the divisor (n − 1)E which appears in the holomorphic change of variable formula K_Y = φ^∗K_X+ (n − 1)E, where E = φ⁻¹(0) is the exceptional divisor. In this affine chart it is defined by y₁= 0.

A formal arc γ ∈ L(Y ) is represented by γ(t) = (y₁(t), · · · y_n(t)). Then ord_tJ (γ) = ord_t(y₁(t)ⁿ⁻¹) = s(n − 1) if y₁(t) = a_1st^s+ · · · with a_s6= 0. That is, S_k 6= ∅ only for k being of the form k = s(n − 1). The first (and trivial) case is k = 0 hence s = 0 too (i.e. a106= 0). In this case, (yi) is unique solvable by (xi). By putting together all n affine open sets U0, . . . , Unof the standard covering of Y we simply get S0= Y \E ∼= X\0, a trivial C⁰ fibration. In this case m0can be any non-negative integer.

The next case is k = n − 1 and s = 1. That is, y1(t) = a11t¹+ · · · with a116= 0.

From

x1(t) = y1(t), x2(t) = y1(t)y2(t), · · · xn(t) = y1(t)yn(t),

we see that t|xi(t) for all i. Moreover, this is the only condition needs to be satisfied for x_i(t), 2 ≤ i ≤ n. The image of φ_∗consists of all x(t) = tv(t) = t(v₁(t), · · · , v_n(t)) with t 6 |v₁(t). By gluing together over the standard affine covering, we see the image consists of all x(t) = tv(t) with t 6 |v_i(t) for some i (i.e. v(0) 6= 0). Also from the formula we see that y(t) ∈ L(Y ) is uniquely solvable for any such x(t). This shows that φ_∗ : S_n−1 → L(X) is one to one. The similar argument also shows that φ_∗: S_k → L(X) is one to one for any k = s(n − 1).

In order to look at finite truncations, we first claim

Lemma 1.1. Let φ_∗y(t) = x(t) and y(t) ∈ S_k. Then for ˜x(t) = x(t) + t^`v with

` ≥ k and t 6 |v, there exists an unique ˜y(t) = y(t) + t^`−ku such that φ_∗y(t) = ˜˜ x(t).

Proof. Since v(0) 6= 0, by reordering the variables we may assume that v₁(t) 6= 0.

Then we will perform the computations in the chart U₁on Y .

For ˜x₁(t) = ˜y₁(t), we need to solve x₁(t) + t^`v₁= y₁(t) + t^`−ku₁. We may simply set u1= t^kv1.

For 2 ≤ i ≤ n, to solve ˜xi(t) = ˜y1(t)˜yi(t) we need to solve x_i(t) + t^`v_i = (y₁(t) + t^`v₁)(y_i(t) + t^`−ku_i)

= y1(t)yi(t) + t^`v1yi(t) + t^`−k(y1(t) + t^`v1)ui. This is equivalent to solve u_i from the equation

t^`−k(y1(t) + t^`v1)ui= t^`(vi− v1yi(t)).

The condition y(t) ∈ S_k means that ord_ty₁(t)ⁿ⁻¹ = k. That is, ord_ty₁(t) = k/(n − 1). In particular, the order in t in the LHS is ` − k + k/(n − 1) ≤ `. Since the order in the RHS is at least `, we see that u_i can be uniquely solved. Proof of the theorem in our special case. Suppose that we are given the equation φ_∗(y(t)) = x(t) with y(t) ∈ Sk, k = s(n − 1). Notice that in order for the solution

˜

y(t) in the above lemma to be in Sk, we need to require that ` ≥ k/(n − 1) + 1.

It would be sufficient to require that ` ≥ k + 1. (Indeed, for general birational morphisms φ, the number n − 1 appears as the codimension (of the blowing-up center) minus one, so in general it can take the value 1.) We will show that m_k:=

k + 1 will be enough for the theorem to be true.

(3)

A NOTE FOR THE CHANGE OF VARIABLE FORMULA IN ARC SPACES 3

Let m ≥ mk. By applying the lemma to the case ` = m + 1, we see that in order to solve φm(˜y(t) mod t^m+1) = x(t) mod t^m+1, we may assume that ˜y(t) = y(t) + t^m−1+ku. The solutions will be the residue classes ¯u = u mod t^k. This space is C^kn, but we will show that only a k-dimensional subspace will give rise to solutions.

From x₁(t) = y₁(t) + t^m+1−ku₁ mod t^m+1we see that t^k|u₁hence that ¯u₁= 0.

For other i with 2 ≤ i ≤ n, the equation

x_i(t) = y₁(t)(y_i(t) + t^m+1−ku_i) mod t^m+1

implies that y₁(t)t^m+1−ku_i= 0 mod t^m+1. That is, t^k|y1(t)u_i. Since ord_ty₁(t) = s, we get ord_tu_i≥ k − s. Equivalently the solutions of u_ihas dimension k − (k − s) = s by counting the number of coefficients of u_i(t) mod t^k. Since there are n − 1 such i’s, we see the total solutions have dimension s(n − 1) = k. Remark 1.2. By adding a few extra coordinates, the same argument also proves the theorem in the case that φ : Y → X is the blowing-up of a smooth variety X along a smooth center.

2. The General Case for Smooth Varieties

Now for a birational morphism φ : Y → X with Y and X smooth, φ naturally induces a map φ∗ : L(Y ) → L(X). If KY = φ^∗KX+ E with E a normal crossing divisor, the change of variable formula of Denef and Loeser states that

Z

S

L^−fdµ_X= Z

φ⁻¹(S)

L^{−f ◦φ}^∗^−ord^t^{J φ}dµ_Y.

Here J φ := OY(−E) is the ideal sheaf generated by the holomorphic Jacobian factor, ord_tI : L(X) → N ∪ {0} for any ideal sheaf I is the function of minimal degree in t. Namely for γ ∈ L(X), ord_tI(γ) := ming∈Ideg_tg ◦ γ(t).

We do not define the motivic integration here. We only remark that this formula follows from Theorem 0.1 once we know the definition of integration. Also we will only prove that the fiber of φ_m|_S_k is C^k and ignore the piece-wise-trivial- fibration statement since it involves other tools in doing so (Boolean algebra and semi-algebraic geometry).

The proof is indeed an application of the inverse function theorem over power series rings which traces carefully the orders in t. Let φ : Y → X be the birational morphism with Ered ⊂ Y and Z ⊂ X be the exceptional loci in Y and X respectively.

For each k ∈ N ∪ {0} let S^k ⊂ L(Y ) be the subset γ ∈ L(Y ) such that ordtJ φ(γ) = k. By the inverse function theorem, the map φ∗ : L(Y ) → L(X) is a bijection between L(Y )^× := L(Y )\L(Ered) and L(X)^× := L(X)\L(Z), thus there is no interesting geometry on the map φ∗|S_k : Sk → L(X)^×. However, the important observation by Denef and Loeser is that when one takes finite truncations in

L(Y )

φ

πm // Lm(Y )

φ_m

L(X) _π

m // Lm(X)

(4)

4 CHIN-LUNG WANG

for all large enough m the induced map φm|π_m(S_k): πm(Sk) → Lm(X) is indeed a piece-wise trivial C^k fibration over its image. Together with the fact that L(Z) is of measure zero in L(X), this will imply the change of variable formula.

To investigate the fibration structure near one arc γ ∈ L(Y ), it is enough to restrict the map to formal neighborhoods φ : ˆCⁿ(0) → ˆCⁿ(0). Or equivalently to represent φ by an algebraic map (still called φ) on power series φ : C[[t]]ⁿ→ C[[t]]ⁿ with φ(0) = 0. Let φ(y(t)) = x(t) with y(t) ∈ Skand let ` ≥ 2k + 1. We first notice that for each v ∈ C[[t]]ⁿ, there is a unique solution u ∈ C[[t]]ⁿ of the equation

φ(y(t) + t^`−ku) = x(t) + t^`v.

Indeed by Taylor’s expansion

φ(y(t) + t^`−ku) = φ(y(t)) + Dφ(y(t))t^`−ku + t^2(`−k)R(t, u).

Let A = Dφ(y(t)). The equation becomes Au + R(t, u)t^`−k= t^kv. That is, u = (det A)⁻¹t^kA^∗(v − R(t, u)t^`−2k).

Here A^∗ is the adjoint matrix of A. Since ordtdet A = ordtJ φ(y(t)) = k, the term (det A)⁻¹t^k has order zero. Also since ` − 2k ≥ 1, by repeated substitutions this relation solves u as a vector in formal power series.

Now let m ≥ 2k and let ` = m + 1. The above discussion shows that in order to find all solutions of φ(˜y(t) mod t^m+1) = x(t) mod t^m+1, we may assume that

˜

y(t) = y(t) + t^m+1−ku. Notice that the residue classes ¯u = u mod t^k form a linear space isomorphic to C^nk. By Hensel’s lemma, in order to count the solutions we may simply consider the equation At^m+1−ku = 0 mod t¯ ^m+1. That is, A¯u = 0 mod t^k. Since ordtdet(A^∗) = (n − 1)k, the solution space of ¯u has dimension nk − (n − 1)k = k as expected.

This verifies that φ⁻¹_mx(t) ∼¯ = C^k. The piece-wise triviality needs other tools to prove it, which will not be reported here. For the complete details the readers are referred to the original paper.

Remark 2.1. For S = L(X) and E =Pn

i=1eiEi a normal crossing, the change of variable formula gives

[X] = Z

L(X)

L⁰dµX =X

I⊂{1,...,n}[E_I^◦]Y

i∈I

L − 1 L^eⁱ⁺¹− 1.

Since L^e+1− 1 = (L − 1)[P^e], this lives in the localization of K0(Var_C) in projective spaces.