Precise inversion of adjunction for minimal log discrepancies

From papers

Let $P\in X$ be the germ of a normal variety, and let $S+B$ be an effective $\mathbf{R}$ -divisor on $X$ such that $S$ is a normal prime divisor not appearing in $B$ . Assume that $(X,S+B)$ is a pair, and write the adjunction formula

(K_X+S+B)|_S=K_S+B_S,

where $B_S$ is the different on $S$ of $B$ . Precise inversion of adjunction. One has

\operatorname{mld}_P(X,S+B)=\operatorname{mld}_P(S,B_S).

This compares the minimal log discrepancy of a pair with that of its restriction by adjunction and is used in the paper to transfer the required equality to a lower-dimensional restricted pair.

Progress summary

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Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Precise inversion of adjunction for minimal log discrepancies
Let $(X,G+\Delta)$ be a pair such that $G$ is a reduced divisor with no component in the support of the effective divisor $\Delta$ , and let $Z$ be a closed subset of $G$ . Let $\nu\colon G^\nu\to G$ be the normalisation and let $\Delta_{G^\nu}$ be the different on $G^\nu$ . Precise inversion of adjunction. Then
$\operatorname{mld}_Z(X,G+\Delta)=\operatorname{mld}_{\nu^{-1}(Z)}(G,\Delta_{G^\nu}).$
The source identifies this as the precise inversion of adjunction connected with the minimal-log-discrepancy conjecture; it does not provide a resolution status.
source: Masayuki Kawakita, “Ideal-adic semi-continuity problem for minimal log discrepancies”, arXiv:1012.0395 (2010).

Sources & referencesView supporting material

Primary source

Masayuki Kawakita, “On equivalent conjectures for minimal log discrepancies on smooth threefolds”, arXiv:1803.02539 (2018).

Additional references

3 papers in this index state this conjecture (2006–2018). The statement above is taken from the most recent of them; the others are arXiv:0903.0418, arXiv:math/0611859.

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