Thickening of the diagonal, interleaving distance and Fourier-Sato transform

Francois Petit, Pierre Schapira

Published 2020-06-23Version 1

Given a "good" metric space $X$ we construct an interleaving distance $\mathrm{dist}_X$ on the bounded derived category of abelian sheaves on $X$. Our main tool is the family of kernels associated with thickenings of the diagonal. Complete Riemannian manifolds having a strictly positive convexity radius are examples of good metric spaces. We prove a kind of (proper and non proper) stability theorem in this framework and also define the notion of a Lipschitz kernel on $Y\times X$ which will be proved to define a Lipschitz map for the interleaving distances. Finally, we show that the Fourier-Sato transform of sheaves on Euclidian spheres may be obtained as a thickening of the diagonal and thus defines an isometry. We also obtain a similar result for the Radon transform of sheaves on projective spaces.