Let (X,mathfraka) be a log pair, and let |X| denote the set of closed points of X with the Zariski topology. Write operatornamemld_x(X,mathfraka) for the minimal log discrepancy at x. LSC conjecture. The function |X|→R_≥ 0cup-∞,qquad x mapstooperatornamemld_x(X,mathfraka) is lower semi-continuous. The conjecture is known when dim X≤ 3, when X is smooth, more generally when X is a normal local complete intersection variety, when X has only quotient singularities, and in certain positive-characteristic cases; the paper proves it for varieties with hyperquotient singularities and, more generally, quotients of complete intersections by finite linear group actions.

The LSC conjecture for minimal log discrepancies

About 7 years old · traced to 2019

Let $(X,\mathfrak{a})$ be a log pair, and let $|X|$ denote the set of closed points of $X$ with the Zariski topology. Write $\operatorname{mld}_x(X,\mathfrak{a})$ for the minimal log discrepancy at $x$ . LSC conjecture. The function

|X|\to\mathbb{R}_{\ge 0}\cup\{-\infty\},\qquad x\mapsto\operatorname{mld}_x(X,\mathfrak{a})

is lower semi-continuous. The conjecture is known when $\dim X\le 3$ , when $X$ is smooth, more generally when $X$ is a normal local complete intersection variety, when $X$ has only quotient singularities, and in certain positive-characteristic cases; the paper proves it for varieties with hyperquotient singularities and, more generally, quotients of complete intersections by finite linear group actions.

Equivalent formulations 1Other wordings

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

The LSC conjecture for minimal log discrepancies
Let $Y$ be a normal $\mathbb{Q}$ -Gorenstein variety and let $\mathfrak{a}$ be a non-zero $\mathbb{R}$ -ideal sheaf on $Y$ . LSC conjecture. The function
$|Y|_{\rm cl} \to \mathbb{R}_{\ge 0} \cup \{-\infty\}, \qquad y \mapsto \operatorname{mld}_y(Y,\mathfrak{a})$
is lower semi-continuous, where $|Y|_{\rm cl}$ denotes the set of closed points of $Y$ with the Zariski topology. This is the usual lower semi-continuity conjecture for minimal log discrepancies. The source explicitly states that it remains open; it is a special case of the family version above.
source: Yusuke Nakamura and Kohsuke Shibata, “A counterexample to the PIA conjecture for minimal log discrepancies”, arXiv:2404.06164 (2026).

References

Primary source

Yusuke Nakamura and Kohsuke Shibata, “Inversion of adjunction for quotient singularities”, arXiv:2011.07300 (2021).

Additional references

2 papers in this index state this conjecture (2019–2020). The statement above is taken from the most recent of them; the others are arXiv:1912.04665.

Posted 1 month ago

The LSC conjecture for minimal log discrepancies

The LSC conjecture for minimal log discrepancies

Primary source

Additional references

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