Log smoothness and polystability over valuation rings

Karim Adiprasito, Gaku Liu, Igor Pak, Michael Temkin

Published 2018-06-24Version 1

Let $\mathcal{O}$ be a valuation ring of height one of residual characteristic exponent $p$ and with algebraically closed field of fractions. Our main result provides a best possible resolution of the monoidal structure $M_X$ of a log variety $X$ over $\mathcal{O}$: there exists a log modification $Y\to X$ such that the monoidal structure of $Y$ is polystable. In particular, if $X$ is log smooth over $\mathcal{O}$ then $Y$ is polystable. As a corollary we deduce that any log variety over $\mathcal{O}$ possesses a polystable alteration of degreee $p^n$. The core of our proof is a subdivision result for polyhedral complexes satisfying certain rationality conditions.