A Thom-Sebastiani Theorem in Characteristic p

TL;DR

This paper proves a Thom-Sebastiani type theorem in characteristic p, showing that the vanishing cycle of a sum of functions is the convolution of their individual vanishing cycles, using $ ext{l}$-adic Fourier transform and stationary phase methods.

Contribution

It establishes a new convolution formula for vanishing cycles in characteristic p, extending classical results to this setting with novel techniques.

Findings

Vanishing cycle at the sum point equals convolution of individual vanishing cycles.

Uses $ ext{l}$-adic Fourier transform and stationary phase principle.

Provides a new tool for studying singularities in characteristic p.

Abstract

Let $k$ be a perfect field of characteristic $p$, let $f_i:X_i\to\mathbb A_k^1$ $(i=1,2)$ be two $k$-morphism of finite type, and let $f:X_1\times_k X_2\to \mathbb A_k^1$ be the morphism defined by $f(z_1,z_2)=f_1(z_1)+f_2(z_2)$. For each $i\in\{1,2\}$, let $x_i$ be a $k$-rational point in the fiber $f_i^{-1}(0)$ such that $f_i$ is smooth on $X_i-\{x_i\}$. Using the $\ell$-adic Fourier transformation and the stationary phase principle of Laumon, we prove that the vanishing cycle of $f$ at $x=(x_1,x_2)$ is the convolution product of the vanishing cycles of $f_i$ at $x_i$ $(i=1,2)$.