A remark on conical K\"ahler-Einstein metrics

TL;DR

This paper investigates the limitations on conical K"ahler-Einstein metrics on Fano manifolds, revealing that the maximum cone angle is often less than the invariant R(M), and explores this through log K-stability.

Contribution

It provides new non-existence results for conical K"ahler-Einstein metrics and analyzes the discrepancy between cone angles and the invariant R(M) via log K-stability.

Findings

Maximum cone angle is generally smaller than R(M).

Non-existence results for certain conical K"ahler-Einstein metrics.

Discrepancy explained through log K-stability.

Abstract

We give some non-existence results for K\"ahler-Einstein metrics with conical singularities along a divisor on Fano manifolds. In particular we show that the maximal possible cone angle is in general smaller than the invariant R(M). We study this discrepancy from the point of view of log K-stability.