The log homotopy exact sequence

TL;DR

This paper proves the exactness of the homotopy sequence for the logarithmic fundamental group in specific log scheme morphisms, generalizing previous results and introducing a log Stein factorization in certain cases.

Contribution

It extends the exactness results of the homotopy sequence to broader log scheme morphisms and constructs a new log Stein factorization in particular cases.

Findings

Exactness of the homotopy sequence for log smooth, proper, saturated morphisms.

Generalization of Hoshi's results from log regular to broader cases.

Construction of a log Stein factorization in specific scenarios.

Abstract

We show exactness of the homotopy sequence for the logarithmic fundamental group in the case of log smooth, finitely presented, proper and saturated morphisms of fs log schemes over a field. This generalizes earlier results of Hoshi in the log regular case. In passing, we also construct a "log Stein factorization" in some particular cases.