The log tangent space of the log jet space

TL;DR

This paper introduces new concepts of log jet spaces in log geometry, demonstrating their properties, computing their differentials, and contrasting them with classical jet spaces, highlighting their behavior on mildly singular spaces.

Contribution

It defines and analyzes new notions of log jet spaces, computes their Kähler differentials, and compares them with traditional jet spaces, advancing understanding of log geometric structures.

Findings

Log jet spaces behave like jet spaces of smooth varieties on mildly singular spaces.

Computed log Kähler differentials of log jet and arc spaces.

Contrasted log jet spaces with classical jet spaces, highlighting differences.

Abstract

We introduce new notions of log jet spaces. Mildly singular spaces are ``smooth'' in log geometry, so their log jet spaces behave like the jet spaces of smooth varieties. Myriad examples contrast log jet spaces with the usual jet spaces of schemes. We then compute the log K\"ahler differentials of the log jet and arc spaces after de Fernex and Docampo. We obtain some of their applications to the structure of the log jet space. We conclude with comparison remarks with other log jet spaces.