Boundedness of the space of stable logarithmic maps

TL;DR

This paper proves that the moduli space of stable logarithmic maps with fixed invariants is a proper algebraic stack, extending previous results to more general logarithmic structures of the target scheme.

Contribution

It establishes the properness of the moduli space of stable logarithmic maps without restrictions on the target's logarithmic structure.

Findings

The moduli space is a proper algebraic stack.

Properness holds for fixed numerical invariants.

Extension of previous results to broader logarithmic structures.

Abstract

We prove that the moduli space of stable logarithmic maps with fixed numerical invariants, from logarithmic curves to a fixed projective target logarithmic scheme with fine and saturated logarithmic structure, is a proper algebraic stack. This was previously known only with further restrictions on the logarithmic structure of the target.