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The LSC conjecture for minimal log discrepancies

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

∣X∣→R≥0∪{−∞},x↦mld⁡x(X,a)|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≤3\dim X\le 3, when XX is smooth, more generally when XX is a normal local complete intersection variety, when XX 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 1

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

  1. The LSC conjecture for minimal log discrepancies

    Let YY be a normal Q\mathbb{Q}-Gorenstein variety and let a\mathfrak{a} be a non-zero R\mathbb{R}-ideal sheaf on YY. LSC conjecture. The function

    ∣Y∣cl→R≥0∪{−∞},y↦mld⁡y(Y,a)|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∣cl|Y|_{\rm cl} denotes the set of closed points of YY 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).

Sources & referencesView supporting material

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.

birational-geometrylower-semicontinuityminimal-log-discrepanciessingularities

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