[Paper Review] Remarks on the abundance conjecture

Kenta Hashizume|arXiv (Cornell University)|Sep 15, 2015

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper proves the relative abundance conjecture for log canonical $n$-folds with big boundary divisors, assuming the conjecture holds for $(n-1)$-folds. Using the log minimal model program with scaling and known termination results, it establishes the semi-ampleness of the log canonical divisor under nefness, leading to full termination of log MMP and finite generation of adjoint rings in dimension four.

ABSTRACT

We prove the abundance theorem for log canonical $n$-folds such that the boundary divisor is big assuming the abundance conjecture for log canonical $(n-1)$-folds. We also discuss the log minimal model program for log canonical $4$-folds.

Motivation & Objective

• To establish the relative abundance conjecture for log canonical $n$-folds when the boundary divisor is big, under the inductive assumption that the conjecture holds for $(n-1)$-folds.
• To extend the minimal model program (MMP) to log canonical 4-folds with big boundary divisors, proving termination and existence of good minimal models.
• To prove finite generation of adjoint rings for log canonical 4-folds with big boundary divisors, under the same inductive assumption.
• To provide a complete minimal model theory for log canonical 4-folds with big boundary divisors, relying on known existence of log flips in all dimensions.

Proposed method

• Uses induction on dimension, assuming the abundance conjecture holds for log canonical $(n-1)$-folds to prove it for $n$-folds.
• Applies the log minimal model program with scaling to reduce the problem to a finite sequence of flips and divisorial contractions.
• Relies on the known termination of log MMP with scaling for $\mathbb{Q}$-factorial dlt 4-folds, as established in prior work.
• Employs the fact that log flips exist for log canonical pairs in all dimensions, enabling the running of the MMP.
• Uses the equivalence between relative abundance and semi-ampleness under nefness to reduce the problem to proving semi-ampleness.
• Applies results from [B2] and [F4] on termination of log MMP with scaling to derive a contradiction in the infinite case, thus proving termination.

Experimental results

Research questions

• RQ1 Does the relative abundance conjecture hold for log canonical $n$-folds with big boundary divisors, assuming it holds for $(n-1)$-folds?
• RQ2 Can the log minimal model program with scaling be fully terminated for log canonical 4-folds with big boundary divisors?
• RQ3 Is the adjoint ring of a log canonical 4-fold with big boundary divisors finitely generated?
• RQ4 Does the relative abundance conjecture imply the existence of a good minimal model or Mori fiber space in dimension four under the big boundary condition?
• RQ5 Can the log MMP for log canonical 4-folds with big boundary divisors be shown to terminate under the pseudo-effectivity of the log canonical divisor?

Key findings

• The relative abundance conjecture holds for log canonical $n$-folds with big boundary divisors, provided it holds for $(n-1)$-folds.
• The log minimal model program with scaling terminates for log canonical 4-folds with big boundary divisors, yielding either a good minimal model or a Mori fiber space.
• Any log MMP for a log canonical 4-fold with a big boundary divisor terminates if the log canonical divisor is pseudo-effective over the base.
• The adjoint ring of $n$ big $\mathbb{Q}$-Cartier divisors on a log canonical 4-fold is finitely generated as an $\mathcal{O}_U$-algebra.
• The existence of log flips in all dimensions allows the full implementation of the log MMP for log canonical 4-folds with big boundary divisors.
• The proof relies on contradiction via infinite sequences of flips, which are ruled out by known termination results for $\mathbb{Q}$-factorial dlt 4-folds.