McKernan's singularity-boundedness conjecture for Fano contractions

Let $f:X\rightarrow Z$ be a Fano contraction, meaning a contraction with $-K_X$ ample over $Z$ . The singularities of $Z$ should be bounded in terms of the singularities of $X$ .

McKernan's conjecture. Fix a positive integer $r$ and a real number $0<\epsilon\leq 1$ . There exists $\delta>0$ depending only on $r,\epsilon$ such that, whenever $f:X\rightarrow Z$ is a contraction of relative dimension $r$ , $X$ is $\epsilon$ -lc, $-K_X$ is ample over $Z$ , and $Z$ is $\mathbb Q$ -Gorenstein, then $Z$ is $\delta$ -lc.

This conjecture asserts a uniform control of the singularities on the base of a Fano contraction. The source states that it was proved by Birkar as a consequence of the more general Shokurov conjecture.

The conjecture predicts that the singularities of the base of a Fano contraction are uniformly controlled by the relative dimension and the singularities of the total space. It is reported as a consequence of Birkar’s proof of the more general Shokurov conjecture.

Known results

Han, Jiang, and Luo proved the optimal bound $\delta=1/2$ when the relative dimension is $1$ and $\epsilon=1$ .
Chen obtained $\delta=\epsilon^2/2$ in relative dimension $1$ .
Chen’s toric result gives an explicit bound for relative dimension $r$ , including $\delta=\frac{\epsilon^{2^r}}{2^{2^r}\prod_{i=1}^{r-1}i^{2^i}}$ .

December 2024 confirmation

On December 4, 2024, Bingyi Chen’s version 2 reported that Birkar had confirmed McKernan’s conjecture through the general Shokurov theorem, whose conclusion gives a $\delta>0$ depending only on the dimension and $\epsilon$ . The claim is treated here as unverified.

Current status (as of September 2026): Birkar’s proof is reported to settle the conjecture, while independent verification of the proof is not established in this scan.

McKernan's singularity-boundedness conjecture for Fano contractions

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Progress summary

Solutions 0