Log smoothness and polystability over valuation rings

TL;DR

This paper proves that log varieties over certain valuation rings can be modified to achieve polystability, providing a resolution of their monoidal structure, with applications to alterations of varieties.

Contribution

It establishes a best possible resolution of the monoidal structure of log varieties over valuation rings, introducing a subdivision technique for polyhedral complexes to achieve polystability.

Findings

Existence of a log modification making the monoidal structure polystable.

Any variety over the valuation ring admits a polystable alteration of degree p^n.

The subdivision result for polyhedral complexes is key to the proof.

Abstract

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 $\calO$ with a vertical log structure: 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 with a smooth generic fiber. As a corollary we deduce that any 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.