On K-stability of reductive varieties

G. Tian and S.K. Donaldson formulated a conjecture relating GIT stability of a polarized algebraic variety to the existence of a Kähler metric of constant scalar curvature. In [D3] Donaldson partially confirmed it in the case of projective toric varieties. In this paper we extend Donaldson’s results and computations to a new case, that of reductive varieties.Received: November 2003 Revision: January 2004 Accepted: January 2004     

Extremal metrics and K-stability

We propose an algebraic geometric stability criterion for apolarised variety to admit an extremal Kähler metric. Thisgeneralises conjectures by Yau, Tian and Donaldson, which relateto the case of Kähler–Einstein and constant scalarcurvature metrics. We give a result in geometric invariant theorythat motivates this conjecture, and an example computation thatsupports it.     

Uniform K-stability for extremal metrics on toric varieties

In this paper we prove that for toric varieties the uniform K-stability is a necessary condition for the existence of extremal metrics.     

Relative K-stability and modified K-energy on toric manifolds

In this paper, we give a sufficient condition for both the relative K-stability and the properness of modified K-energy associated to Calabi's extremal metrics on toric manifolds. In addition, several examples of toric manifolds which satisfy the sufficient condition are presented.     

Toric surfaces,K-stability and Calabi flow

Let $X$ be a toric surface and $u$ be a normalized symplectic potential on the corresponding polygon $P$ . Suppose that the Riemannian curvature is bounded by a constant $C_1$ and $ \int _{\partial P} u d \sigma < C_2, $ then there exists a constant $C_3$ depending only on $C_1, C_2$ and $P$ such that the diameter of $X$ is bounded by $C_3$ . Moreoever, we can show that there is a constant $M > 0$ depending only on $C_1, C_2$ and $P$ such that Donaldson’s $M$ -condition holds for $u$ . As an application, we show that if $(X,P)$ is (analytic) relative $K$ -stable, then the modified Calabi flow converges to an extremal metric exponentially fast by assuming that the Calabi flow exists for all time and the Riemannian curvature is uniformly bounded along the Calabi flow.     

环流形上的极值度量——存在性和K-稳定性

本文主要研究环流形上的极值度量的存在性和K-稳定性.本文将Donaldson关于环流形上有关常数量曲率度量的稳定性概念的约化推广到一般的极值度量的情形.通过这个约化,本文证明环流形上极值度量的存在性可以推出流形对于环形变的相对K-稳定性.在不知道是否存在极值度量的情形下,本文还给出环流形相对K-稳定的一个充分性条件.对环曲面的情形,基于Arrezo-Pacard-Singer的工作,本文证明任意一个环曲面上存在含有极值度量的Ka¨hler类,并给出一些环曲面上有不存在极值度量的K¨ahler类的例子.关于一般的环流形上的极值度量的存在性,本文用变分方法研究其弱解,证明在能量泛函逆紧性假设下,存在弱极小化子.     

K-stability of constant scalar curvature Kähler manifolds

We show that a polarised manifold with a constant scalar curvature Kähler metric and discrete automorphisms is K-stable. This refines the K-semistability proved by S.K. Donaldson.     

Lifting Bailey pairs to WP-Bailey pairs

A pair of sequences such that and

     

Existence of best proximity pairs and equilibrium pairs

In this paper, using the fixed point theorem for Kakutani factorizable multifunctions, we shall prove new existence theorems of best proximity pairs and equilibrium pairs for free abstract economies, which include the previous fixed point theorems and equilibrium existence theorems.     

Eulerian pairs on Fibonacci words

Recently, Sagan and Savage introduced the notion of Eulerian pairs. In this note, we find Eulerian pairs on Fibonacci words based on Foata’s first transformation or Han’s bijection and a map in the spirit of a bijection of Steingrímsson.     

Surgery on pairs of closed manifolds

To apply surgery theory to the problem of classifying pairs of closed manifolds, it is necessary to know the subgroup of the group LP* generated by those elements which are realized by normal maps to a pair of closed manifolds. This closely relates to the surgery problem for a closed manifold and to the computation of the assembly map. In this paper we completely determine such subgroups for many cases of Browder-Livesay pairs of closed manifolds. Moreover, very explicit results are obtained in the case of an elementary fundamental group. Then we generalize them, and obtain several further results about the realization of elements in the Browder-Quinn surgery obstruction groups by means of normal maps to a closed manifold filtered by closed submanifolds. Partially supported by the RFFI Grant No. 05-01-00993, by the GNSAGA of the National Research Council of Italy, by the MIUR (Ministero della Istruzione, Università e Ricerca) of Italy within the project ‘Proprietà Geometriche delle Varietà Reali e Complesse’, and by a Research Grant of the University of Modena and Reggio Emilia. The second author would like to thank the MPI (Bonn), the ICTP (Trieste), and the University of Modena and Reggio Emilia for their kind hospitality and support in different periods during the year 2007, and the Commission on Development and Exchange of IMU for the travel support.     

Closed geodesics on pairs of pants

We study the non-simple closed geodesics of the Riemann surfaces of signature (0, 3). In the aim of classifying them, we define one parameter: the number of strings. We show that for a given number of strings, a minimal geodesic exists; we then give its representation and its length which depends on the boundary geodesics.     

Note on models

The first part of this note deals with the importance of the polytechnical tradition for mathematical models.¹ In the second part I discuss the question of the “semantic shift” in the mathematical usage of the word “model” (M. Epple),² that means, the passage from concrete material objects to models in the modern sense of the word – abstract entities satisfying a certain system of axioms. My aim is to provide some new information on the theme and to open a new and in my opinion essential aspect – the polytechnical tradition – to look at it. The two parts are only in a loose connection.     

Note on non-D-rings

Abstract

Recall that an integral domain R is said to be a non-D-ring if there exists a non-constant polynomial f (X) in R[X] (called a uv-polynomial) such that f (a) is a unit of R for every a in R. In this note we generalize this notion to commutative rings (that are not necessarily integral domains) as follows: for a positive integer n, we say that R is an n-non-D-ring if there exists a polynomial f of degree n in R[X] such that f (a) is a unit of R for every a in R. We then investigate the properties of this notion in di?erent contexts of commutative rings.