Termination of flips

Work over an algebraically closed field of characteristic $0$. A pair $(X,B)$ consists of a normal variety $X$ and an effective $\mathbb R$-divisor $B$ such that $K_X+B$ is $\mathbb R$-Cartier. For a log resolution $f\colon Y\to X$ write

$$K_Y+B_Y=f^*(K_X+B),$$

where $B_Y$ is the sum of the strict transform of $B$ and an exceptional divisor. The pair is klt (Kawamata log terminal) if every coefficient of $B_Y$ is $<1$ for every such $f$. It is lc (log canonical) if every coefficient is $\le1$.

Let $(X,B)$ be a klt pair with $X$ $\mathbb Q$-factorial, projective over a variety $S$. A $(K_X+B)$-flip over $S$ is a diagram $X\xrightarrow{\varphi}Z\xleftarrow{\varphi^+}X^+$ over $S$ with the following properties:

• $\varphi$ is the contraction of a $(K_X+B)$-negative extremal ray of $\overline{NE}(X/S)$;
• $\varphi$ and $\varphi^+$ are small (isomorphisms in codimension one) projective birational morphisms;
• $K_{X^+}+B^+$ is $\varphi^+$-ample, where $B^+$ is the strict transform of $B$.

The pair $(X^+,B^+)$ is again $\mathbb Q$-factorial and klt.

Conjecture (termination of flips). For every $n$, there is no infinite sequence

$$(X_0,B_0)\dashrightarrow(X_1,B_1)\dashrightarrow(X_2,B_2)\dashrightarrow\cdots$$

of $(K_{X_i}+B_i)$-flips over $S$ starting from a $\mathbb Q$-factorial klt pair of dimension $n$.

The same conjecture is made for dlt and lc pairs, and Shokurov's formulation covers all of them. It is a theorem for $n\le3$ and is partly known for $n=4$. It is open for arbitrary sequences in every dimension $n\ge5$.

Background

The minimal model program (MMP) seeks, for a given variety or pair, a birational model on which $K_X+B$ is nef or which carries a Mori fibre space structure. It runs by repeatedly contracting a $(K_X+B)$-negative extremal ray. Divisorial contractions decrease the Picard number, so only finitely many occur. A small contraction cannot be followed by a contraction in the same category, and is replaced by its flip. The MMP terminates exactly when every sequence of flips it produces is finite.

Existence of flips for klt pairs in all dimensions was proved by Birkar, Cascini, Hacon and McKernan, together with Hacon–McKernan [5]. Termination is therefore the main missing step in the MMP for klt pairs. Shokurov showed that termination for lc pairs in dimension $n$ follows from two conjectures on minimal log discrepancies in dimension $n$: the ascending chain condition and lower semicontinuity. Birkar showed that the ascending chain condition for log canonical thresholds, together with existence of flips and the MMP in lower dimension, implies termination for lc pairs with $K_X+B$ effective [6]. Kollár and Mori's book gives the standard account [9].