On singularity properties of convolutions of algebraic morphisms - the general case

Let K be a field of characteristic zero, X and Y be smooth K-varieties, and let G be an algebraic K-group. Given two algebraic morphisms phi:X -> G and psi:Y -> G, we define their convolution phi*psi:XxY -> G by phi*psi(x,y)=phi(x)center dot psi(y). We then show that this operation yields morphisms with improved smoothness properties. More precisely, we show that for any morphism phi:X -> G which is dominant when restricted to each geometrically irreducible component of X, by convolving it with itself finitely many times, one obtains a flat morphism with reduced fibers of rational singularities. 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 {fQp:Qpn -> C}p is an element of primes is a collection of motivic functions, and fQp is L1 for any p large enough, then in fact there exists epsilon>0 such that fQp is L1+epsilon for any p large enough.

Automated summary

The text discusses the concept of defining algebraic morphism convolution on smooth K-varieties and proves that this operation yields morphisms with improved smoothness properties. By convolving with itself finitely many times, one can obtain morphisms with rational singularities and flat properties. In addition, uniform bounds on families of morphisms are provided.