arXiv subjects

Yoshinori Hashimoto

Publications and source records attributed to Yoshinori Hashimoto.

23 records · Page 2Linked to original sources

arXiv2017

About J-flow, J-balanced metrics, uniform J-stability and K-stability

From the work of Dervan-Keller, there exists a quantization of the critical equation for the J-flow. This leads to the notion of J-balanced metrics. We prove that the existence of J-balanced metrics has a purely algebro-geometric characterization in terms of Chow stability, complementing the result of Dervan-Keller. We also obtain various criteria that imply uniform J-stability and uniform K-stability. Eventually, we discuss the case of Kähler classes that may not be integral over a compact manifold.

math.AG↗

arXiv2017

Quantisation of extremal Kähler metrics

Suppose that a polarised Kähler manifold $(X,L)$ admits an extremal metric $ω$. We prove that there exists a sequence of Kähler metrics $\{ ω_k \}_k$, converging to $ω$ as $k \to \infty$, each of which satisfies the equation $\bar{\partial} \text{grad}^{1,0}_{ω_k} ρ_k (ω_k)=0$; the $(1,0)$-part of the gradient of the Bergman function is a holomorphic vector field.

math.DG↗

arXiv2017

Relative stability associated to quantised extremal Kähler metrics

We study algebro-geometric consequences of the quantised extremal Kähler metrics, introduced in the previous work of the author. We prove that the existence of quantised extremal metrics implies weak relative Chow polystability. As a consequence, we obtain asymptotic weak relative Chow polystability and $K$-semistability of extremal manifolds by using quantised extremal metrics; this gives an alternative proof of the results of Mabuchi and Stoppa--Székelyhidi. In proving them, we further provide an explicit local density formula for the equivariant Riemann--Roch theorem.

math.AG↗

arXiv2015

Stability and canonical metrics on projective spaces blown up along a line

Let $\text{Bl}_{\mathbb{P}^1} \mathbb{P}^n$ be a Kähler manifold obtained by blowing up a complex projective space $\mathbb{P}^n$ along a line $\mathbb{P}^1$. We prove that $\text{Bl}_{\mathbb{P}^1} \mathbb{P}^n$ does not admit constant scalar curvature Kähler metrics in any rational Kähler class, but admits extremal metrics, with an explicit formula in action-angle coordinates, in Kähler classes that are close to the pullback of the Fubini--Study class.

math.DG↗

arXiv2015

Existence of twisted constant scalar curvature Kähler metrics with a large twist

Suppose that there exist two Kähler metrics $ω$ and $α$ such that the metric contraction of $α$ with respect to $ω$ is constant, i.e. $Λ_ω α= \text{const}$. We prove that for all large enough $R>0$ there exists a twisted constant scalar curvature Kähler metric $ω'$ in the cohomology class $[ ω]$, satisfying $S(ω' ) - R Λ_{ω' } α= \text{const}$. We discuss its implication to $K$-stability and the continuity method recently proposed by X.X. Chen.

math.DG↗