Shokurov's Gorensteinness conjecture for quotient singularities

Let XX be an nn-dimensional Q\mathbb{Q}-Gorenstein variety, let pXp\in X be a closed point, and let mldp(X)\operatorname{mld}_p(X) denote the minimal log discrepancy at pp.

Shokurov's Gorensteinness conjecture. The following assertions hold:

  1. mldp(X)n\operatorname{mld}_p(X)\leq n.
  2. If mldp(X)>n1\operatorname{mld}_p(X)>n-1, then XX is smooth at pp.
  3. If mldp(X)=n1\operatorname{mld}_p(X)=n-1, then XX is Gorenstein at pp.

The source states that this conjecture is proved for quotient singularities, including the assertion that one can take r(n,n1)=1r(n,n-1)=1 in the index conjecture.

Sources & referencesView supporting material

Primary source

Yusuke Nakamura and Kohsuke Shibata, “Shokurov's index conjecture for quotient singularities”, arXiv:2209.04845 (2024).

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