https://maths-people.anu.edu.au/~wang/publications/2-AdvMath2004.pdf

Kahler-Ricci solitons on toric manifolds with positive first Chern class

In this paper we explain the complete biregular classification of all 4-dimensional smooth toric Fano varieties. The main result states that there exist exactly 123 different types of toric Fano 4-folds up to isomorphism.

On the Classification of Toric Fano 4-folds

This paper concerns the explicit construction of extremal Kaehler metrics on total spaces of projective bundles, which have been studied in many places. We present a unified approach, motivated by the theory of hamiltonian 2-forms (as introduced and studied in previous papers in the series) but this paper is largely independent of that theory. 
We obtain a characterization, on a large family of projective bundles, of those `admissible' Kaehler classes (i.e., the ones compatible with the bundle structure in a way we make precise) which contain an extremal Kaehler metric. In many cases, such as on geometrically ruled surfaces, every Kaehler class is admissible. In particular, our results complete the classification of extremal Kaehler metrics on geometrically ruled surfaces, answering several long-standing questions. 
We also find that our characterization agrees with a notion of K-stability for admissible Kaehler classes. Our examples and nonexistence results therefore provide a fertile testing ground for the rapidly developing theory of stability for projective varieties, and we discuss some of the ramifications. In particular we obtain examples of projective varieties which are destabilized by a non-algebraic degeneration.

Hamiltonian 2-forms in Kahler geometry, III Extremal metrics and stability

The biregular classification of smoothd-dimensional toric Fano varieties is equivalent to the classification of special simplicial polyhedraP in ℝ
 d
 , the so-called Fano polyhedra, up to an isomorphism of the standard lattice
 
% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqeeuuDJXwAKb% sr4rNCHbacfaGae8hjHi6aaWbaaSqabeaaieGacaGFKbaaaOGaeyOG% IWSae8xhHe6aaWbaaSqabeaacaGFKbaaaaaa!418C!

 
$$\mathbb{Z}^d \subset \mathbb{R}^d$$

 . In this paper, we explain the complete biregular classification of all 4-dimensional smooth toric Fano varieties. The main result states that there exist exactly 123 different types of toric Fano 4-folds somorphism.

On the classification of toric Fano 4-folds

Mabuchi introduced the Mabuchi functional in [Mb1], and it turns out that it is very useful for dealing with Kahler metrics with constant scalar curvatures on compact manifolds (see [BM], etc.). One can also expect that the existence of Kahler metric with constant scalar curvature is almost equivalent to the existence of a lower bound of the Mabuchi functional (see, e.g., [Ti]). But for the case of extremal metrics, the Mabuchi functional is not applicable. Therefore, we need a new (or a modified) functional for metrics which are invariant under a maximal compact connected subgroup K of Aut(M). We did not obtain this functional until the appearing of [FM] (while we were reviewing [FM] in 1995). Mabuchi also found this functional independently [Mb3] (see also [Sm]). A definition of this functional was given in [GC]. We shall give some results and applications in this paper. It turns out that our modified Mabuchi functional M(ω1, ω2) has the property that M(ω1, g∗ω2) = M(ω1, ω2), for any g ∈ CKC(K), where CKC(K) is the centralizer of K in the complexification K of K. Moreover, the extremal metrics are exactly the local minimal points of this functional. Therefore, we expect that the existence of an extremal metric is almost equivalent to the existence of a lower bound of this functional. Surprising enough that the first application of this functional is not the existence but the uniqueness of extremal metrics on smooth toric varieties, i.e., smooth Kahler manifolds with an open (C∗)n-orbit. Therefore, there is for example at most one extremal metric in any Kahler class of the manifold obtained by blowing up two points or three points of a two dimensional complex projective space. To have the uniqueness, we consider the Mabuchi moduli space of the Kahler metrics on the toric varieties (see [Mb2], which was rediscovered by Semmes [Se1] and Donaldson [Ch]). It turns out that the moduli space is flat in this situation (see also [Se1,2]). Moreover, for any two Kahler metrics there is a unique geodesic

/pdf/on-modified-mabuchi-functional-and-mabuchi-moduli-space-of-3m05wzonhw.pdf

On Modified Mabuchi Functional and Mabuchi Moduli Space of Kähler Metrics on Toric Bundles

Recently, we have succeeded in identifying the structure of the adsorption layer of ethanol on a silica surface in cyclohexane to be a hydrogen-bonded linear aggregate (polymer), which we call a surface molecular macrocluster. In this work, we studied the effect of the miscibility of liquids on the formation of the surface molecular macroclusters for confirming that this is a surface induced phenomenon. We investigated the interaction and the structure of methanol adsorbed on a silica surface in methanol−cyclohexane binary liquids by a combination of colloidal probe atomic force microscopy, adsorption excess isotherm measurement, and FTIR spectroscopy using the attenuated total reflection (ATR) mode, and compared the results with those of the ethanol−cyclohexane and 1-propanol−cyclohexane binary liquids. The former system is immiscible at methanol concentrations of ca. 8−90 mol %, and the latter two are miscible at any composition. At 0.03 mol % methanol, which is far from the critical concentration for t...

Surface induced hydrogen-bonded macrocluster formation of methanol on silica surfaces.

/pdf/an-isoperimetric-inequality-for-orbifolds-f3fyqdo1g8.pdf

An isoperimetric inequality for orbifolds

Bando-Calabi-Futaki characters of Kähler orbifolds

https://projecteuclid.org/download/pdf_1/euclid.tmj/1178225805

Classification of Einstein-Kähler toric Fano fourfolds

The Bando-Calabi-Futaki character of a compact Kahler manifold is an obstruction to the existence of Kahler metrics with constant scalar curvature, which is a generalization of the Futaki character of a Fano manifold In this paper, we study the Bando-Calabi-Futaki character of a compact toric manifold In particular, we shall prove that the Bando-Calabi-Futaki character of a compact toric manifold vanishes on the Lie algebra of the unipotent radical of the automorphism group

/pdf/bando-calabi-futaki-character-of-compact-toric-manifolds-220ky9rbwe.pdf

Yasuhiro Nakagawa

Papers

Surface induced hydrogen-bonded macrocluster formation of methanol on silica surfaces.

An isoperimetric inequality for orbifolds

Bando-Calabi-Futaki characters of Kähler orbifolds

Classification of Einstein-Kähler toric Fano fourfolds

Bando-Calabi-Futaki character of compact toric manifolds