2023/01/20 by Yoshinori Gongyo, Gongyo, Yoshinori, Joaquín Moraga +1 · 1 citation
Mathematics · Computer Science · #Commutative Algebra and Its Applications #Algebraic Geometry and Number Theory #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2301.08395
In this article, we introduce the generalized complexity of a generalized Calabi--Yau pair (X,B,M). This invariant compares the dimension of X and Picard rank of X with the sum of the coefficients of B and M. It generalizes the complexity introduced by Shokurov. We show that a generalized log Calabi-Yau pair (X,B,M) of dimension 2 with generalized complexity 0 satisfies that X is toric. This generalizes a result due to Brown, McKernan, Svaldi, and Zhong in the case of surfaces. Furthermore, we show that a generalized klt log Calabi-Yau surface (X,M) with generalized complexity 0 satisfies that X≃ ℙ2 or X≃ ℙ1× ℙ1. Thus, this invariant interpolates between the characterization of toric varieties and the Kobayashi-Ochiai Theorem. As an application, we show that 3-fold singularities with generalized complexity 0 are toric. Furthermore, we show a local version of Kobayashi-Ochiai Theorem in dimension 3.