Towards boundedness of minimal log discrepancies by the Riemann-Roch theorem

Abstract

We introduce an approach via the Riemann-Roch theorem to the boundedness problem of minimal log discrepancies in fixed dimension. After reducing it to the case of a Gorenstein terminal singularity, firstly we prove that the minimal log discrepancy is bounded if either multiplicity or embedding dimension is bounded. Secondly we recover the characterization of a Gorenstein terminal three-fold singularity by Reid, and the sharp bound for its minimal log discrepancy by Markushevich, without explicit classification. Finally we provide the sharp bound for a special four-fold singularity, whose general hyperplane section has a terminal piece.