Moduli of morphisms of logarithmic schemes

Abstract

We show that there is a logarithmic algebraic space parameterizing logarithmic morphisms between fixed logarithmic schemes when those logarithmic schemes satisfy natural hypotheses. As a corollary, we obtain the representability of the stack of stable logarithmic maps from logarithmic curves to a fixed target without restriction on the logarithmic structure of the target.

An intermediate step requires a left adjoint to pullback of étale sheaves, whose construction appears to be new in the generality considered here, and which may be of independent interest.