[Paper Review] Existence of log canonical closures

Christopher D. Hacon, Chenyang Xu|arXiv (Cornell University)|May 5, 2011

Algebraic Geometry and Number Theory28 references7 citations

TL;DR

This paper establishes the existence of good minimal models for dlt pairs over a base variety under mild conditions, proving that if a dlt pair has a good minimal model over a dense open subset of the base and all non-klt centers meet this subset, then it admits a good minimal model over the whole base. The key contribution is a foundational result enabling log canonical compactifications and verifying the valuative criterion for properness of the moduli functor of stable schemes.

ABSTRACT

Let $f:X o U$ be a projective morphism of normal varieties and $(X,Δ)$ a dlt pair. We prove that if there is an open set $U^0\subset U$, such that $(X,Δ) imes_U U^0$ has a good minimal model over $U^0$ and the images of all the non-klt centers intersect $U^0$, then $(X,Δ)$ has a good minimal model over $U$. As consequences we show the existence of log canonical compactifications for open log canonical pairs, and the fact that the moduli functor of stable schemes satisfies the valuative criterion for properness.

Motivation & Objective

• To establish the existence of good minimal models for dlt pairs over a base variety under conditions ensuring good behavior on a dense open subset.
• To prove the existence of log canonical compactifications for open log canonical pairs, extending such pairs to projective log canonical pairs over a larger base.
• To verify the valuative criterion for properness of the moduli functor of stable schemes using the minimal model program.
• To confirm conjectures on base change compactifications for log canonical morphisms and flips in the log canonical setting.
• To provide a foundational tool for studying $ε$-lc centers and the existence of flips in the log canonical category.

Proposed method

• Use induction on dimension, reducing the problem to the non-klt locus $S = \lfloor \Delta \rfloor$, which is treated as an sdlt pair.
• Apply Kollár’s gluing theory to extend minimal model results from the components of $S$ to the full non-klt locus.
• Leverage Fujino’s generalization of Kawamata’s base point free theorem to show that minimal models are automatically good models under the given conditions.
• Apply the canonical bundle formula to analyze the Iitaka fibration and control the behavior of log canonical divisors under base change.
• Use the existence of good minimal models over $U^0$ and the intersection condition on non-klt centers to propagate the good model structure to the whole base $U$.
• Apply gluing theory to extend log canonical models over a curve germ and construct the compactified family via involution extension and quotient construction.

Experimental results

Research questions

• RQ1 Under what conditions does a dlt pair over a base variety admit a good minimal model if it has a good minimal model over a dense open subset?
• RQ2 Can every open log canonical pair be compactified to a log canonical pair over a projective base?
• RQ3 Does the moduli functor of stable schemes satisfy the valuative criterion for properness?
• RQ4 Do flips exist for log canonical pairs under the assumption of a good minimal model over a dense open set?
• RQ5 Can the minimal model program for log canonical pairs be extended via base change and gluing techniques?

Key findings

• If a dlt pair $(X, \Delta)$ over a base $U$ has a good minimal model over a dense open subset $U^0 \subset U$ and all non-klt centers meet $X^0 = X \times_U U^0$, then $(X, \Delta)$ admits a good minimal model over $U$.
• Every open log canonical pair $(X^0, \Delta^0) \to U^0$ admits a log canonical compactification to a projective log canonical pair $(X, \Delta) \to U$ over a normal quasi-projective variety $U$.
• For any affine finite type lc morphism $f^0: X^0 \to U$ with $U$ a smooth curve, there exists a finite dominating base change $\widetilde{U} \to U$ and a projective lc morphism $f: X \to \widetilde{U}$ extending $f^0$.
• The moduli functor of semi-log canonical models satisfies the valuative criterion for properness, as shown via base change and gluing of log canonical models.
• The $(K_X + \Delta'')$-MMP with scaling over $U$ terminates with a good minimal model or a Mori fibration when $K_X + \Delta' + \Delta'' \sim_{\mathbb{Q}, U} 0$ and $(X, \Delta'')$ is dlt.
• Flips exist for log canonical pairs, as a consequence of the termination of the MMP with scaling under the stated conditions.