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Operations on constructible functions and generalized valuations

Andreas Bernig, Vadim Lebovici

Abstract

Alesker's theory of generalized valuations unifies smooth measures and constructible functions on real analytic manifolds, extending classical operations on functions and measures. Alesker showed that these operations agree with the sheaf-theoretic ones on constructible functions under restrictive assumptions, leaving key aspects conjectural. In this article, we close this gap by proving that the two approaches indeed coincide on constructible functions under mild transversality assumptions. Our proof is based on a comparison with the corresponding operations on characteristic cycles. As applications, we extend additive kinematic formulas from convex bodies to compact subanalytic sets in Euclidean spaces and derive new kinematic formulas on the 3-sphere.

Operations on constructible functions and generalized valuations

Abstract

Alesker's theory of generalized valuations unifies smooth measures and constructible functions on real analytic manifolds, extending classical operations on functions and measures. Alesker showed that these operations agree with the sheaf-theoretic ones on constructible functions under restrictive assumptions, leaving key aspects conjectural. In this article, we close this gap by proving that the two approaches indeed coincide on constructible functions under mild transversality assumptions. Our proof is based on a comparison with the corresponding operations on characteristic cycles. As applications, we extend additive kinematic formulas from convex bodies to compact subanalytic sets in Euclidean spaces and derive new kinematic formulas on the 3-sphere.
Paper Structure (29 sections, 34 theorems, 134 equations)

This paper contains 29 sections, 34 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Context
  3. Contributions
  4. Plan of the paper.
  5. Preliminaries
  6. Notations
  7. Orientations.
  8. Currents
  9. Borel-Moore homology
  10. Subanalytic currents and Borel-Moore homology.
  11. Constructible functions
  12. Characteristic cycles
  13. Generalized valuations
  14. Constructible functions as generalized valuations
  15. Wave front set of subanalytic currents
  16. ...and 14 more sections

Key Result

Theorem 1

Let $f:X\to Y$ be a real analytic map between real analytic manifolds and let $\psi\in\mathrm{CF}_{}(Y)$. If then the pullback $f^*\!\left[\psi\right]\in\mathcal{V}^{-\infty}(X)$ is well-defined and $f^*\!\left[\psi\right] = \left[f^*\psi\right]$.

Theorems & Definitions (55)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Corollary 4
  • Theorem 5
  • Theorem 6
  • Proposition 2.1: Hor03
  • Lemma 2.2: Fu90
  • Lemma 2.3
  • Lemma 2.4
  • ...and 45 more