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On singularity properties of convolutions of algebraic morphisms - the general case
Journal of the London Mathematical Society ( IF 1.0 ) Pub Date : 2020-11-29 , DOI: 10.1112/jlms.12414
Itay Glazer 1 , Yotam I. Hendel 1
Affiliation  

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 φ : X G and ψ : Y G , we define their convolution φ ψ : X × Y G by φ ψ ( x , y ) = φ ( x ) · ψ ( y ) . We then show that this operation yields morphisms with improved smoothness properties. More precisely, we show that for any morphism φ : 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 { f Q p : Q p n C } p primes is a collection of motivic functions, and f Q p is L 1 for any p large enough, then in fact there exists ε > 0 such that f Q p is L 1 + ε for any p large enough.

中文翻译:

关于代数态射卷积的奇异性——一般情况

是特征为零的域, X 圆滑 - 品种,并让 G 成为代数 -团体。给定两个代数态射 φ X G ψ G ,我们定义它们的卷积 φ ψ X × G 经过 φ ψ ( X , ) = φ ( X ) · ψ ( ) . 然后我们证明这个操作产生了具有改进平滑特性的态射。更准确地说,我们证明对于任何态射 φ X G 当限制到每个几何上不可约的分量时,它占主导地位 X ,通过将它与自身有限多次卷积,可以得到一个具有减少的有理奇点纤维的平面态射。还给出了态射族的统一边界。此外,作为关键的分析步骤,我们还证明了动机整合的以下结果:如果 { F n C } 素数 是一个动机函数的集合,并且 F 1 对于任何 足够大,那么实际上存在 ε > 0 以至于 F 1 + ε 对于任何 足够大。
更新日期:2020-11-29
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