## Abstract

Let (Formula presented.) be a field of characteristic zero, (Formula presented.) and (Formula presented.) be smooth (Formula presented.) -varieties, and let (Formula presented.) be an algebraic (Formula presented.) -group. Given two algebraic morphisms (Formula presented.) and (Formula presented.), we define their convolution (Formula presented.) by (Formula presented.). We then show that this operation yields morphisms with improved smoothness properties. More precisely, we show that for any morphism (Formula presented.) which is dominant when restricted to each geometrically irreducible component of (Formula presented.), 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 (Formula presented.) is a collection of motivic functions, and (Formula presented.) is (Formula presented.) for any (Formula presented.) large enough, then in fact there exists (Formula presented.) such that (Formula presented.) is (Formula presented.) for any (Formula presented.) large enough.

Original language | American English |
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Pages (from-to) | 1453-1479 |

Number of pages | 27 |

Journal | Journal of the London Mathematical Society |

Volume | 103 |

Issue number | 4 |

DOIs | |

State | Published - 1 Jun 2021 |

Externally published | Yes |

## Keywords

- 03C98 (primary)
- 11G25
- 14B05
- 14E18
- 14G05
- 20G15 (secondary)

## All Science Journal Classification (ASJC) codes

- General Mathematics