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Yoshinori Hashimoto

Publications and source records attributed to Yoshinori Hashimoto.

At least 19 recordsLinked to original sources
arXiv2026

GIT stability and biquotients of $SU(3)$

We study double-sided actions of $(\mathbb{C}^*)^2$ on $SL(3,\mathbb{C})/U$ and the associated quotients, where $U$ is a maximal unipotent subgroup of $SL(3,\mathbb{C})$. The main results of this paper are a sufficient condition for the double-sided quotient to agree with the quotient in terms of the geometric invariant theory (GIT), and an explicit necessary and sufficient condition for $SL(3,\mathbb{C})/U$ to agree with the $χ$-stable locus in its affine closure. We apply this result to characterize certain complex structures on $SU(3)$ which are not left invariant by means of the GIT quotient.

math.AG↗
arXiv2026

Magnitude of metric measure spaces and integrals over geodesics

We propose a definition of magnitude for a length space with a Borel measure, which involves integrals over the set of geodesics. This quantity agrees with the magnitude of finite metric spaces, up to re-scaling the metric to ensure the convergence, when we use the counting measure on them. We also prove a version of the homogeneous magnitude theorem, by showing that the new definition agrees with the volume when we use the weight measure on a compact homogeneous Riemannian manifold. We compute various examples, which suggest that this quantity can capture information of non-uniqueness of geodesics, such as the injectivity radius, corresponding to the generating degrees of the magnitude homology.

math.DG↗
arXiv2026

Harmonic band theory: rigidity of non-zero degree harmonic maps from 2-torus to complex projective space

We prove the rigidity of isotropic harmonic maps from a 2-torus to a complex projective space, when they are constructed from holomorphic embeddings associated to complete linear systems. We also prove that this rigidity holds for any holomorphic embeddings without special hyperosculation points, with an extra assumption on the pullbacks of Fubini--Study symplectic forms. These results ensure the rigidity of towers of harmonic bands in condensed matter physics.

math-ph↗
arXiv2025

A Hilbert--Mumford criterion for nilsolitons

We give an algebraic criterion for a nilpotent real Lie algebra and prove that it provides a necessary and sufficient condition for the associated nilpotent Lie group to admit left-invariant Ricci solitons, called nilsolitons. As an application of this result, we generalise Nikolayevsky's criterion for the existence of nilsolitons to nilpotent Lie algebras without nice bases. We further prove a modified version of the Taketomi--Tamaru conjecture for nilpotent Lie groups which gives an obstruction to the existence of nilsolitons.

math.DG↗
arXiv2025

Mapping properties of the Hilbert and Fubini--Study maps in Kähler geometry

Suppose that we have a compact Kähler manifold $X$ with a very ample line bundle $\mathcal{L}$. We prove that any positive definite hermitian form on the space $H^0 (X,\mathcal{L})$ of holomorphic sections can be written as an $L^2$-inner product with respect to an appropriate hermitian metric on $\mathcal{L}$. We apply this result to show that the Fubini--Study map, which associates a hermitian metric on $\mathcal{L}$ to a hermitian form on $H^0 (X,\mathcal{L})$, is injective.

math.DG↗
arXiv2025

Quantitative injectivity of the Fubini--Study map

We prove a quantitative version of the injectivity of the Fubini--Study map that is polynomial in the exponent of the ample line bundle, and correct the arguments in the author's previous papers.

math.AG↗
arXiv2023

Balanced metrics for extremal Kähler metrics and Fano manifolds

The first three sections of this paper are a survey of the author's work on balanced metrics and stability notions in algebraic geometry. The last section is devoted to proving the well-known result that a geodesically convex function on a complete Riemannian manifold admits a critical point if and only if its asymptotic slope at infinity is positive, where we present a proof which relies only on the Hopf--Rinow theorem and extends to locally compact complete length metric spaces.

math.DG↗
arXiv2023

Ueda's lemma via uniform Hörmander estimates for flat line bundles

We establish Hörmander-type $L^2$-estimates for the $\overline{\partial}$-operators that hold uniformly for all nontrivial flat holomorphic line bundles on compact Kähler manifolds. Our result can be regarded as a $\overline{\partial}$-version of Ueda's lemma on the operator norm of Čech coboundaries for flat line bundles and indeed recovers the original version of Ueda's lemma for compact Kähler manifolds. A partial generalisation for $(p,0)$-forms on Ricci-flat manifolds is also given.

math.CV↗
arXiv2022

Anticanonically balanced metrics and the Hilbert-Mumford criterion for the $δ_m$-invariant of Fujita-Odaka

We prove that the stability condition for Fano manifolds defined by Saito-Takahashi, given in terms of the sum of the Ding invariant and the Chow weight, is equivalent to the existence of anticanonically balanced metrics. Combined with the result by Rubinstein-Tian-Zhang, we obtain the following algebro-geometric corollary: the $δ_m$-invariant of Fujita-Odaka satisfies $δ_m >1$ if and only if the Fano manifold is stable in the sense of Saito-Takahashi, establishing a Hilbert-Mumford type criterion for $δ_m >1$. We also extend this result to the Kähler-Ricci $g$-solitons and the coupled Kähler-Einstein metrics, and as a by-product we obtain a formula for the asymptotic slope of the coupled Ding functional in terms of multiple test configurations.

math.DG↗
arXiv2022

Expected centre of mass of the random Kodaira embedding

Let $X \subset \mathbb{P}^{N-1}$ be a smooth projective variety. To each $g \in SL (N , \mathbb{C})$ which induces the embedding $g \cdot X \subset \mathbb{P}^{N-1}$ given by the ambient linear action we can associate a matrix $\barμ_X (g)$ called the centre of mass, which depends nonlinearly on $g$. With respect to the probability measure on $SL (N , \mathbb{C})$ induced by the Haar measure and the Gaussian unitary ensemble, we prove that the expectation of the centre of mass is a constant multiple of the identity matrix for any smooth projective variety.

math.DG↗
arXiv2021

Quot-scheme limit of Fubini-Study metrics and Donaldson's functional for vector bundles

For a holomorphic vector bundle $E$ over a polarised Kähler manifold, we establish a direct link between the slope stability of $E$ and the asymptotic behaviour of Donaldson's functional, by defining the Quot-scheme limit of Fubini-Study metrics. In particular, we provide an explicit estimate which proves that Donaldson's functional is coercive on the set of Fubini-Study metrics if $E$ is slope stable, and give a new proof of Hermitian-Einstein metrics implying slope stability.

math.AG↗
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↗
arXiv2021

Quot-scheme limit of Fubini-Study metrics and its applications to balanced metrics

We present some results that complement our prequels [arXiv:1809.08425,arXiv:1907.05770] on holomorphic vector bundles. We apply the method of the Quot-scheme limit of Fubini-Study metrics developed therein to provide a generalisation to the singular case of the result originally obtained by X.W. Wang for the smooth case, which states that the existence of balanced metrics is equivalent to the Gieseker stability of the vector bundle. We also prove that the Bergman 1-parameter subgroups form subgeodesics in the space of hermitian metrics. This paper also contains a review of techniques developed in [arXiv:1809.08425,arXiv:1907.05770] and how they correspond to their counterparts developed in the study of the Yau-Tian-Donaldson conjecture.

math.AG↗
arXiv2019

A variational approach to the Hermitian-Einstein metrics and the Quot-scheme limit of Fubini-Study metrics

This is a sequel of our paper [arXiv:1809.08425] on the Quot-scheme limit and variational properties of Donaldson's functional, which established its coercivity for slope stable holomorphic vector bundles over smooth projective varieties. Assuming that the coercivity is uniform in a certain sense, we provide a new proof of the Donaldson-Uhlenbeck-Yau theorem, in such a way that the analysis involved in the proof is elementary except for the asymptotic expansion of the Bergman kernel.

math.AG↗
arXiv2017

Scalar curvature and Futaki invariant of Kähler metrics with cone singularities along a divisor

We study the scalar curvature of Kähler metrics that have cone singularities along a divisor, with a particular focus on certain specific classes of such metrics that enjoy some curvature estimates. Our main result is that, on the projective completion of a pluricanonical bundle over a product of Kähler--Einstein Fano manifolds with the second Betti number 1, momentum-constructed constant scalar curvature Kähler metrics with cone singularities along the $\infty$-section exist if and only if the log Futaki invariant vanishes on the fibrewise $\mathbb{C}^*$-action, giving a supporting evidence to the log version of the Yau--Tian--Donaldson conjecture for general polarisations. We also show that, for these classes of conically singular metrics, the scalar curvature can be defined on the whole manifold as a current, so that we can compute the log Futaki invariant with respect to them. Finally, we prove some partial invariance results for them.

math.DG↗