# On minimal log discrepancies on varieties with fixed Gorenstein index

@article{Nakamura2015OnML,
title={On minimal log discrepancies on varieties with fixed Gorenstein index},
author={Yusuke Nakamura},
journal={arXiv: Algebraic Geometry},
year={2015}
}
• Yusuke Nakamura
• Published 26 January 2015
• Mathematics
• arXiv: Algebraic Geometry
We generalize the rationality theorem of the accumulation points of log canonical thresholds which was proved by Hacon, M\textsuperscript{c}Kernan, and Xu. Further, we apply the rationality to the ACC problem on the minimal log discrepancies. We study the set of log discrepancies on varieties with fixed Gorenstein index. As a corollary, we prove that the minimal log discrepancies of three-dimensional canonical pairs with fixed coefficients satisfy the ACC.
15 Citations
Small quotient minimal log discrepancies
We prove that for each positive integer $n$ there exists a positive number $\epsilon_n$ so that $n$-dimensional toric quotient singularities satisfy the ACC for mld's on the interval
ACC for minimal log discrepancies of exceptional singularities.
• Mathematics
• 2019
We prove the existence of $n$-complements for pairs with DCC coefficients and the ACC for minimal log discrepancies of exceptional singularities. In order to prove these results, we develop the
Divisors computing minimal log discrepancies on lc surfaces
• Mathematics
• 2021
Let (X ∋ x,B) be an lc surface germ. If X ∋ x is klt, we show that there exists a divisor computing the minimal log discrepancy of (X ∋ x,B) that is a Kollár component of X ∋ x. If B 6= 0 or X ∋ x is
On generalized minimal log discrepancy
• Mathematics
• 2021
We discuss the ACC conjecture and the LSC conjecture for minimal log discrepancies of generalized pairs. We prove that some known results on these two conjectures for usual pairs are still valid for
ACC for log canonical threshold polytopes
• Mathematics
• 2017
We show that the log canonical threshold polytopes of varieties with log canonical singularities satisfy the ascending chain condition.
ACC for local volumes and boundedness of singularities
• Mathematics
• 2020
The ACC conjecture for local volumes predicts that the set of local volumes of klt singularities $x\in (X,\Delta)$ satisfies the ACC if the coefficients of $\Delta$ belong to a DCC set. In this
Inversion of adjunction for quotient singularities
• Mathematics
Algebraic Geometry
• 2022
We prove the precise inversion of adjunction formula for quotient singularities and klt Cartier divisors. As an application, we prove the semi-continuity of minimal log discrepancies for klt
Rational nef and anti-nef polytopes are not uniform
We give two examples which show that rational nef and anti-nef polytopes are not uniform even for klt surface pairs, answering a question of Chen-Han. We also show that rational nef polytopes are
A variant of the effective adjunction conjecture with applications
We propose a variant of the effective adjunction conjecture for lc-trivial fibrations. This variant is suitable for inductions and can be used to treat real coefficients.
Moishezon morphisms
We try to understand which morphisms of complex analytic spaces come from algebraic geometry. We start with a series of conjectures, and then give some partial solutions.

## References

SHOWING 1-10 OF 22 REFERENCES
On minimal log discrepancies
An explanation to the boundness of minimal log discrepancies conjectured by V.V. Shokurov would be that the minimal log discrepancies of a variety in its closed points define a lower semi-continuous
Towards boundedness of minimal log discrepancies by the Riemann-Roch theorem
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,
Two two-dimensional terminations
Varieties with log terminal and log canonical singularities are considered in the Minimal Model Program, see \cite{...} for introduction. In \cite{shokurov:hyp} it was conjectured that many of the
A connectedness theorem over the spectrum of a formal power series ring
We study the connectedness of the non-klt locus over the spectrum of a formal power series ring. In dimension 3, we prove the existence and normality of the smallest lc center, and apply it to the
Discreteness of log discrepancies over log canonical triples on a fixed pair
For a fixed pair and fixed exponents, we prove the discreteness of log discrepancies over all log canonical triples formed by attaching a product of ideals with given exponents.
Limits of log canonical thresholds
• Mathematics
• 2007
Let Tn denote the set of log canonical thresholds of pairs (X, Y ), with X a nonsingular variety of dimension n, and Y a nonempty closed subscheme of X. Using non-standard methods, we show that every
Minimal Discrepancy for a Terminal cDV Singularity Is 1
An answer to a question raised by Shokurov on the minimal discrepancy of a terminal singularity of index 1 is given. It is proved that the minimal discrepancy is 1 (it is 2 for a non-singular point
On the birational automorphisms of varieties of general type
• Mathematics
• 2010
We show that the number of birational automorphisms of a variety of general type X is bounded by c vol(X;KX), where c is a constant that only depends on the dimension of X.
The set of toric minimal log discrepancies
We describe the set of minimal log discrepancies of toric log varieties, and study its accumulation points.