A motivic local Cauchy-Crofton formula

In this note, we establish a version of the local Cauchy-Crofton formula for definable sets in Henselian discretely valued fields of characteristic zero. It allows to compute the motivic local density of a set from the densities of its projections integrated over the Grassmannian.


Introduction
The aim of this note is to establish a motivic analogue of the local Cauchy-Crofton formula. The classical Cauchy-Crofton formula is a geometric measure theory result stating that the volume of a set X of dimension d can be recovered by integrating over the Grassmannian the number of points of intersection of X with affine spaces of codimension d, see for example [9]. It has been used by Lion [11] to show the existence of the local density of semi-Pfaffian sets. Comte [5], [6] has established a local version of the formula for sets X ⊆ R n definable in an o-minimal structure. The formula states that the local density of such a set X can be recovered by integrating over a Grassmannian the density of the projection of X on subspaces. This allows him to show the continuity of the real local density along Verdier's strata in [6].
The local Cauchy-Crofton formula appears as a first step toward comparing the local Lipschitz-Killing curvature invariants and the polar invariants of a germ of a definable set X ⊆ R n . It is shown by Comte and Merle in [8] that one can recover one set of invariants by linear combination of the other, see also [7].
A notion of local density for definable sets in Henselian valued field of characteristic zero has been develloped by the autor in [10]. Our formula is a new step toward developing a theory of higher local curvature invariants in non-archimedean geometry.
A p-adic analogue has been developed by Cluckers, Comte and Loeser in [2, Section 6]. We will follow closely their approach. Our precise result appears as Theorem 4.1 at the end of Section 4.
Acknoledgements. Many thanks to Georges Comte for encouraging me to work on this project and useful discussions. I also thanks Michel Raibaut for interesting comments.

Motivic integration and local density
We assume the reader is familiar with the notion of motivic local density developed by the autor in [10] and in particular with Cluckers and Loeser's theory of motivic integration [3], [4]. See [10, Section 2.7] for a short summary of the theory.
Key words and phrases. Motivic integration, Henselian valued fields, local density, metric invariants.
We adopt the notations and conventions of [10]. In particular, we fix T , a tame or mixed-tame theory of valued fields in the sense of [4]. Such a T always admits a Henselian discretely valued field of characteristic zero as a model. Definable means definable without parameters in T and K is (the underlining valued field of) a model of T with discrete value group and residue field k enough saturated.
For example, we can take for T the theory of a discretely valued field of characteristic zero in the Denef-Pas language ; if the residue field is of characteristic p > 0, one need to add the higher angular components.
For each definable set X, Cluckers and Loeser define a ring of constructible motivic functions C(X), which include the characteristic functions of any definable set Y ⊆ X.
For ϕ ∈ C(X) of support of dimension d and that is integrable, they define If d = dim(X), we drop the d from the notation. If ϕ is the characteristic function of some definable set Y ⊆ X, we denote the integral µ d (Y ). If the residue field k is algebraically closed, the target ring of motivic integration is equal to where B(x, n) is the ball of center x and valuative radius n.
There is some e ∈ N * such that for each i, the subsequence (θ ke+i ) k∈N converges to some d i ∈ C({ * }). Here C({ * }) has a topology induced by the degree in L. We define the motivic local density of X at x to be It is shown in [10] that one can compute the motivic local density on the tangent cone. Fix some Λ ∈ D, where D = {Λ n,m | n, m ∈ N * }, The Λ-tangent cone with multiplicities is a definable function CM Λ x (X) ∈ C(K m ), of support C Λ x (X), well defined up to a set of dimension < d. For example, if X ⊆ K m is of dimension m, there is no muliplicity to take into account and CM Λ x (X) is the characteristic function of C Λ x (X). Theorem 3.25 and 5.12 from [10] state that there is a Λ such that for all Λ ′ ⊆ Λ,

Local constructible functions
Consider a definable function π : X → Y between definable sets X and Y of dimension n. Recall form [4] the notation π ! (ϕ) ∈ C(Y ) for any motivic constructible function ϕ ∈ C(X).
If X is a definable subset of K n and x ∈ K n , define the ring of germs of constructible motivic functions at x by C(X) x : Consider now a linear projection π : It does not depend on the large enough r chosen. Indeed, as π : being the projection on Y . Hence (up to taking a finite definable partition of X), for some r 0 , for any r ≥ r 0 ,

Grassmannians
Fix a point x ∈ K n and view K n as a K-vector space with origin 0. Then denote by G(n, d) K the Grassmannian of dimension d subvector spaces of K n . The canonical volume form on G(n, d) K invariant under GL n (O K )-transformations induce a constuctible function ω n,d on G(n, d) invariant under GL n (O K ) transformations, see [3, Section 15] for details. Since G(n, d) k is smooth and proper, the motivic volume of G(n, d) K is equal to the class [G(n, d) k ] of G(n, d) k is the (localized) Grothendieck group of varieties over the residue field k. Denoting F q the finite field with q elements, it is known, see for example [1], that |G(n, d)(F q )| = (q n − 1)(q n − q) · · · (q n − q r−1 ) (q r − 1)(q r − q) · · · (q r − q r−1 ) .
The proof rely on the fact that Since the analog of this formula holds in K(Var k ), the same proof shows that Note that even if the right hand side can be written without denominator, hence as an element of K(Var k ), one has to work in K(Var k ) loc to show the equality with this method. In particular, this shows that [G(n, d) k ] is invertible in K(Var k ) loc . The motivic volume of G(n, r) K is then invertible. Hence we can normalize ω n,r such that 1 = V ∈G(n,r) ω n,r (V ).
For V ∈ G(n, n − d), define p V : K n → K n /V the canonical projection. We identify K n /V to K d as follows. There is some g ∈ GL n (O K ) such that g(K n−d × {0} d ) = V . We identify K n /V to g({0} n−d × K n−d ). The particular choice of g does not matter thanks to the change of variable formula. If X is a dimension d definable subset of K n then there is an dense definable subset Ω = Ω(X, x) of G(n, n−d) such that for every V ∈ Ω, π V satisfies condition ( * ). Indeed the tangent K × -cone of X is of dimension at most d and it suffices to set which is indeed dense in G(n, n − d) In particular, for any V ∈ Ω, p !,x (ϕ) is well defined for any ϕ ∈ C(X) x . With these notations, we can now state our motivic local Cauchy-Crofton formula. Recall that Θ d (X, x) ∈ C( * ) ⊗ Q is the motivic local density of X at x, Theorem 4.1 (local Cauchy-Crofton). Let X ⊆ K n a definable set of dimension d and x ∈ K n . Then By [10, Proposition 3.8], we may assume X = X. We can also assume x = 0 and 0 ∈ X. Indeed, if 0 / ∈ X, then both sides of the formula are 0.

Tangential Crofton formula
We start by proving the theorem in the particular case where X is a Λ-cone.
Lemma 5.1. Let X be a definable Λ-cone with origin 0 contained in some Π ∈ G(n, d). Then Proof. Assume Λ = Λ e,r . Fix some V ∈ G(n, n − d) such that π V : Π → K d is bijective. As X is a Λ-cone and π V is linear, π V (X) is also a Λ-cone. From the definition of local density, see also [10,Remark 3.11], we have Then since X is a Λ-cone and π V is bijective, we have a disjoint union The set C j is indeed a disjoint union since it is the image of X ∩ S(0, i) by the application Indeed the function ϕ V restricted to X ∩ S(0, i) is a definable bijection of image D V i since π V is linear and bijective on Π.
By the change of variable formula, we have L −v(Jac(ϕV (x))) .
Since ω n,n−d is invariant under GL n (O K )-transformations and ϕ V (x) = ϕ gV (x ′ ), by the change of variable formula we get that C i (x) is equal to V ∈G(n,n−d) Moreover, it is independent of i by linearity of π V , hence we denote it by C and we have We also have Combining Equations 2, 6 and 4, we get V ∈G(n,n−d) This last expression is equal to = CΘ d (X, 0). We find C = 1 by computing both sides of the previous equality with X = Π. This following lemma is the motivic analogous of the classical spherical Crofton formula, see for example [9,Theorem 3.2.48]. See also [2, Remark 6.2.4] for a reformulation in the p-adic case.
Lemma 5.2. Let X be a definable Λ-cone with origin 0. Then Proof. We only have to modify slightly the proof of Lemma 5.1. Indeed, we use now the function restricted to the smooth part of X. It is now longer injective on X ∩S(0, i), however the motivic volume of the fibers is taken into account in p V !,0 (X). Hence we get similarly V ∈G(n,n−d) which is equal to CΘ d (X, 0). Once again, we find C = 1 by computing both sides for X a vector space of dimension d.

General case
Before proving Theorem 4.1, we need a technical lemma. Lemma 6.1. Let X ⊆ K n be a definable set of dimension d and V ∈ G(n, n − d) such that the projection p V : C Λ 0 (X) → K d is finite-to-one. Then there is a definable k-partition of X such that for each k-part X ξ , there is a ξ-definable set C ξ of dimension less than d such that p V is injective on C Λ 0 (X ξ )\C ξ . Proof. We can assume Λ = K × . As the projection p V : C Λ 0 (X) → K d is finiteto-one, by finite b-minimality one can find a k-partition of C Λ 0 (X) such that p V is injective on each k-part of C Λ 0 (X). For a k-part C Λ 0 (X) ξ , define B ξ to be the ξ-definable subset of K n defined as the union of lines ℓ passing through 0 such that the distance between ℓ ∩ S(0, 0) and C Λ 0 (X) ξ ∩ S(0, 0) is strictly smaller than the distance between ℓ∩S(0, 0) and C Λ 0 (X) ξ ′ ∩S(0, 0) for every ξ ′ = ξ. Set X ξ = X ∩B ξ . Then setting Y = X\ ∪ ξ X ξ , we have C Λ 0 (Y ) empty and C Λ 0 (X ξ ) ⊆ C Λ 0 (X) ξ . Hence we set C ξ = C Λ 0 (X) ξ \C Λ 0 (X) ξ and we have X ξ and C ξ as required. Proof of Theorem 4.1. From [10, Theorem 3.25], there is a Λ ∈ D such that Θ d (X, 0) = Θ d (CM Λ 0 (X), 0). As in the proof of [10, Theorem 3.25], by [10, Proposition 2.14 and Lemma 3.9], we can assume X is the graph of a 1-Lipschitz function defined on some definable set U ⊂ K d . In this case, CM Λ 0 (X) = 1 C Λ 0 (X) . From Lemma 5.2, we have Θ d (C Λ 0 (X), 0) = V ∈G(n,n−d) Θ d (p V !,0 (C Λ 0 (X)), 0)ω n,n−d (V ).
Hence we need to show that for every V in a dense subset of G(n, n − d), Θ d (p V !,0 (C Λ 0 (X)), 0) = Θ d (p V !,0 (X), 0). We can find a k-partition of X such that p V is injective on the k-parts. Replace X by one of the k-parts and suppose then that p V is injective on X.