Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2021; 58(5): 1247-1260

Online first article April 8, 2021      Printed September 30, 2021

https://doi.org/10.4134/BKMS.b200928

Copyright © The Korean Mathematical Society.

On barycentric transformations of Fano polytopes

DongSeon Hwang, Yeonsu Kim

Ajou University; Ajou University

Abstract

We introduce the notion of barycentric transformation of Fano polytopes, from which we can assign a certain type to each Fano polytope. The type can be viewed as a measure of the extent to which the given Fano polytope is close to be K\"ahler-Einstein. In particular, we expect that every K\"ahler-Einstein Fano polytope is of type $B_\infty$. We verify this expectation for some low dimensional cases. We emphasize that for a Fano polytope $X$ of dimension $1$, $3$ or $5$, $X$ is K\"ahler-Einstein if and only if it is of type $B_\infty$.

Keywords: Barycentric transformation, K\"ahler-Einstein Fano polytope, symmetric Fano polytope

MSC numbers: Primary 52B20; Secondary 52B15, 14M25, 32Q20

Supported by: This research was supported by the Samsung Science and Technology Foundation under Project SSTF-BA1602-03.

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