On singularity properties of convolutions of algebraic morphisms -- the general case (with an appendix joint with Gady Kozma)

Abstract

Let $K$ be a field of characteristic zero, $X$ and $Y$ be smooth $K$-varieties, and let $G$ be a algebraic $K$-group. Given two algebraic morphisms $\phi :X\to G$ and $\psi :Y\to G$, we define their convolution $\phi \ast \psi :X\times Y\to G$ by $\phi \ast \psi (x,y)=\phi (x)\cdot \psi (y)$. We then show that this operation yields morphisms with improved smoothness properties. More precisely, we show that for any morphism $\phi :X\to G$ which is dominant when restricted to each absolutely irreducible component of $X$, by convolving it with itself finitely many times, one can obtain a flat morphism with reduced fibers of rational singularities, generalizing the main result of our previous paper. Uniform bounds on families of morphisms are given as well. Moreover, as a key analytic step, we also prove the following result in motivic integration; if $\{f_{{\mathbb{Q}}_p}:{\mathbb{Q}}_p^n\to \mathbb{C}{\}}_{p\in \mathrm{primes}}$ is a collection of functions which is motivic in the sense of Denef-Pas, and $f_{{\mathbb{Q}}_p}$ is $L^1$ for any $p$ large enough, then in fact there exists $\epsilon >0$ such that $f_{{\mathbb{Q}}_p}$ is $L^{1+\epsilon }$ for any $p$ large enough.

Revised version following referee's suggestions. 26 pages, with an appendix joint with Gady Kozma, comments welcome