A motivic local Cauchy-Crofton formula
- PDF / 284,063 Bytes
- 11 Pages / 439.37 x 666.142 pts Page_size
- 63 Downloads / 277 Views
© The Author(s) 2020
Arthur Forey
A motivic local Cauchy-Crofton formula Received: 22 March 2019 / Accepted: 30 October 2020 Abstract. 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.
1. Introduction The classical Cauchy-Crofton formula is a geometric measure theory result stating that the volume of some 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 [11]. It has been used by Lion in [13] to show the existence of the local density of semi-Pfaffian sets. In [7] and [8], Comte has established a local version of the formula for sets X ⊆ Rn 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 [8]. This result was generalized by Valette in [14] who shows that the continuity also holds along Whitney’s strata. 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 ⊆ Rn . It is shown by Comte and Merle in [10] that one can recover one set of invariants by linear combination of the other, see also [9]. A notion of local density for definable sets in Henselian valued fields of characteristic zero has been developed by the author in [12]. The aim of this note is to establish a motivic analogue of the local Cauchy-Crofton formula. Our formula is a new step toward developing a theory of higher local curvature invariants in non-Archimedean geometry. We now describe briefly our formula in a particular case, see the next section for precise definitions. Fix an algebraically closed field k of characteristic zero and K = k((t)). Fix X a semi-algebraic subset of K n of dimension d (or more A. Forey (B): D-MATH, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland. e-mail: [email protected] Mathematics Subject Classification: Primary 03C98 · 12J10 · 14B05 · 32Sxx · Secondary 03C68 · 11S80
https://doi.org/10.1007/s00229-020-01258-3
2
A. Forey
generaly a definable set in the Denef-Pas language). We use the motivic integral of Cluckers and Loeser [5], which takes values in the localised Grothendieck ring of varieties K(Var k )loc . Denote by d (X, x) the motivic local density of X at x. Let G(n, d) be the Grassmannian variety of d-dimensional vector subspaces of K n . There is a volume form ωn,d invariant under G L n (O K )-transformations such that 1 = G(n,d) ωn,d . For V ∈ G(n, n − d), let pV : K n → K n /V be the canonical projection. There is a dense definable subset ⊆ G(n, n − d) such that for V ∈ , pV is finite-to-on
Data Loading...
