On singularity properties of convolutions of algebraic morphisms

Abstract

Let $K$ be a field of characteristic zero, $X$ and $Y$ be smooth $K$-varieties, and let $V$ be a finite dimensional $K$-vector space. For two algebraic morphisms $\phi :X\to V$ and $\psi :Y\to V$ we define a convolution operation, $\phi \ast \psi :X\times Y\to V$, by $\phi \ast \psi (x,y)=\phi (x)+\psi (y)$. We then study the singularity properties of the resulting morphism, and show that as in the case of convolution in analysis, it has improved smoothness properties. Explicitly, we show that for any morphism $\phi :X\to V$ which is dominant when restricted to each irreducible component of $X$, there exists $N\in \mathbb{N}$ such that for any $n>N$ the $n$-th convolution power ${\phi }^n:=\phi \ast \cdots \ast \phi$ is a flat morphism with reduced geometric fibers of rational singularities (this property is abbreviated (FRS)). By a theorem of Aizenbud and Avni, for $K=\mathbb{Q}$, this is equivalent to good asymptotic behavior of the size of the $\mathbb{Z}/p^k\mathbb{Z}$-fibers of ${\phi }^n$ when ranging over both $p$ and $k$. More generally, we show that given a family of morphisms $\{{\phi }_i:X_i\to V\}$ of complexity $D\in \mathbb{N}$ (i.e. that the number of variables and the degrees of the polynomials defining $X_i$ and ${\phi }_i$ are bounded by $D$), there exists $N(D)\in \mathbb{N}$ such that for any $n>N(D)$, the morphism ${\phi }_1\ast \cdots \ast {\phi }_n$ is (FRS).

Revised version following referee's suggestions. 36 pages, comments welcome

References in corpus (2)

• Counting points of schemes over finite rings and counting representations of arithmetic lattices
• Thom-Sebastiani theorems for filtered D-modules and for multiplier ideals

Cited by in corpus (4)

• On singularity properties of convolutions of algebraic morphisms -- the general case (with an appendix joint with Gady Kozma)
• On rational singularities and counting points of schemes over finite rings
• On the Fourier coefficients of word maps on unitary groups
• A number theoretic characterization of $E$-smooth and (FRS) morphisms: estimates on the number of $\mathbb{Z}/p^k\mathbb{Z}$-points