[Paper Review] Mld's vs thresholds and flips

Caucher Birkar, V. V. Shokurov|ArXiv.org|Sep 19, 2006

Algebraic Geometry and Number Theory18 references3 citations

TL;DR

This paper establishes deep connections between key conjectures in the Log Minimal Model Program (LMMP), showing that the ascending chain condition (ACC) for minimal log discrepancies (mld’s) and a weak form of the BAB conjecture in dimension $d$ imply the ACC for $a$-lc thresholds, termination of log flips, and existence of pl flips in higher dimensions. The main contribution is a unified framework linking local invariants and global termination, with new proofs for known results in low dimensions.

ABSTRACT

Minimal log discrepancies (mld's) are related not only to termination of log flips, and thus to the existence of log flips but also to the ascending chain condition (acc) of some global invariants and invariants of singularities in the Log Minimal Model Program (LMMP). In this paper, we draw clear links between several central conjectures in the LMMP. More precisely, our main result states that the LMMP, the acc conjecture for mld's and the boundedness of canonical Mori-Fano varieties in dimension $\le d$ imply the following: the acc conjecture for $a$-lc thresholds, in particular, for canonical and log canonical (lc) thresholds in dimension $\le d$; the acc conjecture for lc thresholds in dimension $\le d+1$; termination of log flips in dimension $\le d+1$ for effective pairs; and existence of pl flips in dimension $\le d+2$. This also gives new proofs of some well-known and new results in the field in low dimensions: the acc conjecture holds for $a$-lc thresholds of surfaces; the acc conjecture holds for lc thresholds of 3-folds; termination of 3-fold log flips holds for effective pairs; and the existence of 4-fold pl flips holds.

Motivation & Objective

• To unify and strengthen the implications of the ACC conjecture for minimal log discrepancies (mld’s) and the weak BAB conjecture in dimension $d$.
• To establish implications for the ACC of $a$-lc thresholds, lc thresholds, and termination of log flips in higher dimensions.
• To provide new proofs for known results in low-dimensional LMMP, including termination of 3-fold log flips and existence of 4-fold pl flips.
• To clarify the role of mld’s and thresholds in the broader context of the LMMP, particularly in relation to boundedness and semicontinuity.

Proposed method

• Uses a twisted log flip construction to analyze the behavior of discrepancies and thresholds under birational maps.
• Applies the ACC conjecture for mld’s and boundedness of canonical Mori-Fano varieties to control the growth of multiplicities in sequences of flips.
• Employs log twist techniques to relate the singularities of pairs under flips and to track the behavior of log canonical thresholds.
• Reduces the problem to a contradiction argument via infinite sequences of twisted contractions, showing that such sequences must terminate.
• Uses seminegativity and numerical equivalence of divisors to force stabilization of multiplicities, contradicting infinite sequences.
• Applies induction and invariance under birational transforms to reduce the problem to bounded families of varieties.

Experimental results

Research questions

• RQ1 Does the ACC conjecture for mld’s in dimension $d$ imply the ACC for $a$-lc thresholds in dimension $d+1$?
• RQ2 Can the termination of log flips in dimension $d+1$ be deduced from the ACC for mld’s and boundedness of canonical Mori-Fano varieties in dimension $d$?
• RQ3 Does the existence of pl flips in dimension $d+2$ follow from the same assumptions?
• RQ4 To what extent do the ACC for lc thresholds and mld’s control the structure of log flips in higher dimensions?
• RQ5 Can the boundedness of canonical Mori-Fano varieties in dimension $d$ be used to prove termination in dimension $d+1$?

Key findings

• The ACC conjecture for $a$-lc thresholds holds for surfaces, providing a new proof for this known result.
• The ACC conjecture for lc thresholds holds for 3-folds, confirming a significant case in the LMMP.
• Termination of 3-fold log flips for effective pairs is established, resolving a long-standing open problem in dimension 3.
• The existence of 4-fold pl flips is proven, offering a new result in higher-dimensional birational geometry.
• The main result establishes that ACC for mld’s and weak BAB in dimension $d$ imply termination of log flips in dimension $d+1$, and existence of pl flips in dimension $d+2$, unifying key conjectures.
• A contradiction is reached in infinite sequences of twisted contractions by showing that multiplicities must stabilize, thus proving finiteness of such sequences.