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Takahiro Aoi

Publications and source records attributed to Takahiro Aoi.

5 recordsLinked to original sources
arXiv2026

Skoda-Zeriahi type integrability and entropy compactness for some measure with $L^1$-density

In this paper, we prove the Skoda-Zeriahi type integrability theorem with respect to some measure with $L^1$-density. In addition, we introduce the log-log threshold in order to detect singularities of Kähler potentials. We prove the positivity of the integrability threshold for such a measure and Kähler potentials with uniform log-log threshold. As an application, we prove the entropy compactness theorem for a family of potential functions of Poincaré type Kähler metrics with uniform log-log threshold. The Ohsawa-Takegoshi $L^2$-extension theorem and Skoda-Zeriahi's integrability theorem play a very important role in this paper.

math.DG↗
arXiv2024

Microscopic stability thresholds and constant scalar curvature Kähler metrics

In this paper, we directly prove that if the limit of microscopic stability thresholds introduced by Berman for a polarized manifold satisfies some condition, then there exists a unique constant scalar curvature Kähler metric. This is an analogue of K.Zhang's result which is proved by the delta-invariant introduced by Fujita-Odaka. This work is motivated by Berman's result which shows that if a Fano manifold is uniformly Gibbs stable, then there exists a unique Kähler-Einstein metric, without uniform K-stability. We also give some sufficient conditions of the existence of a constant scalar curvature Kähler cone metric.

math.DG↗
arXiv2022

A conical approximation of constant scalar curvature Kähler metrics of Poincaré type

Let $(X,L_X)$ be a polarized manifold and $D$ be a smooth hypersurface such that $D \in | L_X |$. In this paper, we show that if there is no nontrivial holomorphic vector field on $D$ and ${\rm Aut}_0 ((X,L_X); D)$ is trivial, then constant scalar curvature Kähler metrics of Poincaré type on $X \setminus D$ can be approximated by constant scalar curvature Kähler metrics with cone singularities of sufficiently small angle along $D$. This result implies log K-semistability of $((X,L_X);D)$ with angle 0.

math.DG↗
arXiv2021

On uniform log $K$-stability for constant scalar curvature Kähler cone metrics

We prove that the existence of constant scalar curvature Kähler metrics with cone singularities along a divisor implies log $K$-polystability and $G$-uniform log $K$-stability, where $G$ is the automorphism group which preserves the divisor. We also show that a constant scalar curvature Kähler cone metric along an ample divisor of sufficiently large degree always exists. We further show several properties of the path of constant scalar curvature Kähler cone metrics and discuss uniform log $K$-stability of normal varieties.

math.DG↗