The log Grothendieck ring of varieties

TL;DR

This paper introduces a Grothendieck ring for log schemes, adding a new class, and addresses the challenges in defining log Hodge numbers, proposing a viable alternative.

Contribution

It extends the Grothendieck ring framework to log schemes and provides a new approach to defining log Hodge numbers.

Findings

Defined a Grothendieck ring for log schemes with an additional class P

Identified issues with naive log Hodge number definitions

Proposed an alternative definition for log Hodge numbers

Abstract

We define a Grothendieck ring of varieties for log schemes. It is generated by one additional class ``$P$'' over the usual Grothendieck ring. We show the na\"ive definition of log Hodge numbers does not make sense for all log schemes. We offer an alternative that does.