Exponentiation of coefficient systems and exponential motives

TL;DR

This paper develops a new six-functor formalism called $C_{exp}$ from an existing one, enabling the study of cohomological invariants of varieties with potentials through exponential motives.

Contribution

It introduces a novel construction of exponential six-functor formalisms and defines associated motives, expanding the toolkit for studying varieties with potentials.

Findings

Constructed $C_{exp}$ from any six-functor formalism $C$.

Established the symmetric monoidal convolution product at the $mbda$-categorical level.

Analyzed properties of motives in $C_{exp}$ attached to varieties with potential.

Abstract

We construct new six-functor formalisms capturing cohomological invariants of varieties with potentials. Starting from any six-functor formalism $C$, encoded as a coefficient system, we associate a new six-functor formalism $C_{\text{exp}}$. This requires in particular constructing the convolution product symmetric monoidal structure at the $\infty$-categorical level. We study $C_{\text{exp}}$ and how it relates to $C$. We also define motives in $C_{\text{exp}}$ attached to varieties with potential and study their properties.