Deligne--Faltings Logarithmic Derived Geometry with a View towards Logarithmic GLSM

Many perfect obstruction theories and virtual cycles in enumerative geometry are shadows from derived geometry. In this paper, we give a general construction of reduced perfect obstruction theories arising from a geometric reduction of the derived stack $X$ with respect to a $(-1)$-shifted log function $w_{\alpha }$. Notably, this construction leads to a decomposition of the virtual cycle $[X{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}$ into the reduced virtual cycle $[X{]}^{\mathrm{r}\mathrm{e}\mathrm{d}}$ and a reduced boundary cycle [Δ_X]^\red. Our construction is motivated by future applications to the moduli stack of UMD logarithmic R-maps in log GLSM \cite{CJR21}. Along the way, we develop logarithmic analogs of several constructions in derived geometry, in particular proving a logarithmic Lagrangian intersection theorem.