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Toshiki Mabuchi

Publications and source records attributed to Toshiki Mabuchi.

17 recordsLinked to original sources
arXiv2016

Asymptotic polybalanced kernels on extremal Kaehler manifolds

In this paper, improving a preceding work, we obtain asymptotic polybalanced kernels associated to extremal Kaehler metrics on polarized algebraic manifolds. As a corollary, we have a stronger asymptotic relative Chow-polystability for extremal Kaehler polarized algebraic manifolds. Finally, related to the Yau-Tian-Donaldson Conjecture for extremal Kaehler metrics, we shall discuss the difference between strong relative K-stability and relative K-stability.

math.DG↗
arXiv2013

Existence problem of extremal Kaehler metrics

In this paper, we shall give some affirmative answer to an extremal Kaehler version of the Yau-Tian-Donaldson Conjecture. For a polarized algebraic manifold $(X,L)$, we choose a maximal algebraic torus $T$ in the group of holomorphic automorphisms of $X$. Then the polarization class $c_1(L)$ will be shown to admit an extremal Kaehler metric if $(X,L)$ is strongly K-stable relative to $T$.

math.DG↗
arXiv2013

A remark on Li-Xu's pathology

For test configurations, the Donaldson-Futaki invariant F_1 is well-known. In this note, its refinement will be discussed. Then we see that Li-Xu's pathology doesn't occur, since their example of a non-normal test configuration, with trivial normalization, actually has non-vanishing F_1 in this refined sense.

math.DG↗
arXiv2013

The Donaldson-Futaki invariant for sequences of test configurations

In this note, given a polarized algebraic manifold $(X,L)$, we define the Donaldson-Futaki invariant for a sequence of test configurations for $(X,L)$ with exponents tending to infinity. This then allows us to define a strong version of K-stability or K-semistability for $(X,L)$. In particular, $(X,L)$ will be shown to be K-semistable in this strong sense if the polarization class $c_1(L)$ admits a constant scalar curvature Kaehler metric.

math.DG↗
arXiv2013

Strong K-stability and asymptotic Chow-stability

For a polarized algebraic manifold $(X,L)$, let $T$ be an algebraic torus in the group of all holomorphic automorphisms of $X$. Then strong relative K-stability will be shown to imply asymptotic relative Chow-stability. In particular, by taking $T$ to be trivial, we see that asymptotic Chow-stability follows from strong K-stability.

math.DG↗
arXiv2011

Asymptotics of polybalanced metrics under relative stability constraints

Under the assumption of asymptotic relative Chow-stability for polarized algebraic manifolds $(M, L)$, a series of weighted balanced metrics $ω_m$, $m \gg 1$, called polybalanced metrics, are obtained from complete linear systems $|L^m|$ on $M$. Then the asymptotic behavior of the weights as $m \to \infty$ will be studied.

math.DG↗
arXiv2011

New examples of Sasaki-Einstein manifolds

In this note, stimulated by the existence result of Futaki-Ono-Wang for toric Sasaki-Einstein metrics, we obtain new examples of Sasaki-Einstein metrics on S^1-bundles associated to canonical line bundles of P^1-bundles over Kähler-Einstein Fano manifolds, even though the Futaki's obstruction does not vanish. Here the method of Sakane and Koiso is used, and our examples include non-toric Sasaki-Einstein manifolds.

math.DG↗
arXiv2009

A stronger concept of K-stability

In this paper, by introducing a wider class of one-parameter group actions for test configurations, we have a stronger form of the definition of K-stability. This allows us to obtain some key step of my preceding work in proving that constant scalar curvature polarization implies K-stability for polarized algebraic manifolds.

math.DG↗
arXiv2008

K-stability of constant scalar curvature polarization

In this paper, we shall show that a polarized algebraic manifold is K-stable if the polarization class admits a Kaehler metric of constant scalar curvature. This generalizes the results of Chen-Tian, Donaldson and Stoppa. (Parts of the arguments are based on a forthcoming paper "A stronger concept of K-stability." )

math.DG↗
arXiv2007

An affine sphere equation associated to Einstein toric surfaces

As seen in the works of Calabi, Cheng-Yau and Loftin, affine sphere equations have a close relationship with Kaehler-Einstein metrics. The main purpose of this note is to show that an equation analogous to those of hyperbolic affine spheres arises naturally from Kaehler-Einstein metrics on Einstein toric surfaces. The case for the remaining toric surfaces with Kaehler-Ricci solitons will also be discussed.

math.DG↗
arXiv2006

Extremal metrics and stabilities on polarized manifolds

The Hitchin-Kobayashi correspondence for vector bundles, established by Donaldson, Kobayashi, Luebke, Uhlenbeck and Yau, states that an indecomposable holomorphic vector bundle over a compact Kaehler manifold is stable in the sense of Takemoto-Mumford if and only if the vector bundle admits a Hermitian-Einstein metric. Its manifold analogue known as Yau's conjecture, which originated from Calabi's conjecture, asks whether ``stability'' and ``existence of extremal metrics'' for polarized manifolds are equivalent. In this note, the recent progress of this subject, by Donaldson, Tian and our group, together with its relationship to algebraic geometry will be discussed.

math.DG↗
arXiv2004

An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds, II

Recently, Donaldson proved asymptotic stability for a polarized algebraic manifold $M$ with polarization class admitting a Kähler metric of constant scalar curvature, essentially when the linear algebraic part $H$ of $Aut^0(M)$ is semisimple. The purpose of this paper is to give a generalization of Donaldson's result to the case where the polarization class admits an extremal Kähler metric, even when $H$ is not semisimple.

math.DG↗
arXiv2004

An obstruction to asymptotic semistability and approximate critical metrics

In this paper, we consider an obstruction to asymptotic Chow-semistability of a polarized Kaehler algebraic manifold. Even when a linear algebraic group of positive dimension acts nontrivially and holomorphically on a polarized Kaehler algebraic manifold with constant scalar curvature, the vanishing of the obstruction allows us to generalize Donaldson's construction of approximate solutions for equations of balanced metrics.

math.DG↗
arXiv2004

Stability of extremal Kähler manifolds

In this paper, by generalizing the concept of balanced metrics, we shall show that Donaldson's asymptotic approximation of balanced metrics for constant scalar curvature cases can be generalized to extremal Kaehler cases.

math.DG↗