Finite generation of relative log canonical algebras of semi-dlt pairs

Abstract

We prove that the relative log canonical algebra of a semi-dlt pair $(X,{\Delta }_X)$ projective over a scheme $T$ is finitely generated, equivalently, that $(X,{\Delta }_X)$ admits a relative stable model, provided that the normalization admits a log canonical model over $T$, that $(X,{\Delta }_X)$ admits a stable model over an open dense subscheme $T^0\subset T$ whose exceptional locus does not contain any stratum of the conductor, and that the log centers contained in the conductor have images meeting $T^0$. Our main contribution is a direct proof that avoids Kollár's gluing theory entirely. As a consequence, we recover demi-normal versions of results of Hacon--Xu and Birkar. Furthermore, we derive the existence of certain MMP steps for slc pairs, recovering results of Ambro and Kollár. Finally, these results lay the groundwork for a streamlined proof of the properness of the Kollár--Shepherd-Barron--Alexeev moduli space of stable pairs, to be completed in forthcoming work.

arXiv comment34 pages, comments welcome!