Logarithmic motives with compact support

TL;DR

This paper develops a theory of motives with compact support for logarithmic schemes, establishing key properties like Gysin sequences, K{"u}nneth formula, and duality, advancing the understanding of log schemes in algebraic geometry.

Contribution

It introduces a novel theory of motives with compact support for log schemes, including new invariance, duality, and cancellation theorems, underpinned by the development of related homology and cohomology theories.

Findings

Established a Gysin sequence for log motives.

Proved a K{"u}nneth formula assuming resolution of singularities.

Demonstrated $ar{ullet}$-invariance and duality for log schemes.

Abstract

We develop a theory of motives with compact support for logarithmic schemes over a field. Starting from the notion of finite logarithmic correspondences with compact support, we define the logarithmic motive with compact support analogous to the classical case for schemes. We then establish an analog of a Gysin sequence and, assuming resolution of singularities, a K{\"u}nneth formula. This implies that our theory is $\overline{\square}$-invariant, which presents a critical feature that is absent in the classical case. Further assuming resolution of singularities, we prove a duality theorem for log schemes which we use to establish a cancellation theorem for log schemes whose underlying scheme is proper. Moreover, we discuss new homology and cohomology theories for log smooth fs logarithmic schemes based on our results.