[Paper Review] On minimal model theory for log abundant lc pairs

Kenta Hashizume, Zhengyu Hu|arXiv (Cornell University)|Jun 3, 2019

Algebraic Geometry and Number Theory36 references4 citations

TL;DR

This paper establishes the minimal model theory for log abundant log canonical (lc) pairs under the assumption of the existence of good minimal models or Mori fiber spaces for projective klt pairs of dimension $n$. By leveraging inductive techniques and numerical dimension constraints on lc centers, the authors prove the existence of good minimal models or Mori fiber spaces for such pairs, and further establish termination of log MMP with scaling. The key contribution is a foundational step toward the abundance conjecture in the lc setting via inductive reduction on dimension.

ABSTRACT

Under the assumption of the minimal model theory for projective klt pairs of dimension $n$, we establish the minimal model theory for lc pairs $(X/Z,Δ)$ such that the log canonical divisor is relatively log abundant and its restriction to any lc center has relative numerical dimension at most $n$. We also give another detailed proof of results by the second author, and study termination of log MMP with scaling.

Motivation & Objective

• To establish the minimal model program (MMP) for log abundant lc pairs under dimensionally inductive assumptions.
• To prove termination of the log MMP with scaling for such pairs.
• To provide a new proof of results on lc pairs with big boundaries, strengthening prior work by the second author.
• To extend applications to Calabi–Yau pairs and Fano-type varieties under dimension constraints on lc centers.

Proposed method

• Uses induction on dimension, assuming the existence of good minimal models or Mori fiber spaces for klt pairs of dimension $n$.
• Applies the concept of relative numerical dimension $\kappa_\sigma(S^\nu/Z, (K_X+\Delta)|_{S^\nu})$ to constrain behavior on lc centers.
• Employs the notion of $\pi$-log abundance, requiring $K_X+\Delta$ and its pullbacks to lc centers to be abundant over $Z$.
• Introduces a scaling technique in the log MMP, tracking decreasing scaling coefficients $\lambda_i$ to ensure termination.
• Relies on the contradiction argument via [5, Theorem 4.1 (iii)] to prove termination of the MMP when $K_X+B+\lambda A$ is pseudo-effective but not nef.
• Uses birational transforms and extremal ray theory to show that each step of the MMP preserves the scaling property and leads to a minimal model or Mori fiber space.

Experimental results

Research questions

• RQ1 Under what conditions does a log abundant lc pair admit a good minimal model or Mori fiber space over a base variety?
• RQ2 Can the log MMP with scaling terminate for log abundant lc pairs under dimension constraints on lc centers?
• RQ3 How does the relative numerical dimension of the log canonical divisor on lc centers affect the minimal model program?
• RQ4 To what extent can results on lc pairs with big boundaries be reproven using inductive arguments on log abundance?
• RQ5 What implications does the existence of good minimal models for klt pairs have for the abundance conjecture in the lc setting?

Key findings

• Theorem 1.1 establishes the existence of a good minimal model or Mori fiber space for an lc pair $(X,\Delta)$ over $Z$ if $K_X+\Delta$ is $\pi$-log abundant and $\kappa_\sigma(S^\nu/Z, (K_X+\Delta)|_{S^\nu}) \leq n$ for all lc centers $S$.
• Theorem 1.2 generalizes this to pairs where $K_X+\Delta$ is abundant or $\Delta$ is big over $Z$, under the same dimension constraint on lc centers.
• Theorem 1.3 proves finite generation of the graded algebra $\mathcal{R}(X,D)$ for Calabi–Yau pairs with $-K_X$ big and lc centers of dimension at most 3.
• Theorem 1.4 shows that if $\dim X - \dim Z \leq n$ and all fibers have constant dimension, then $(X,\Delta)$ admits a good minimal model or Mori fiber space over $Z$, linking absolute and relative minimal model theory.
• Theorem 1.5 provides an alternative proof of the existence of good minimal models for lc pairs with big boundaries via a $\pi$-ample divisor $A$, under the same inductive assumption.
• The log MMP with scaling terminates for $(K_X+B)$-MMP over $Z$ with scaling of $A$, as shown by contradiction using pseudo-effectivity and the non-nefness of $K_X+B+\lambda A$.