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The concept of a charge density wave (CDW) permeates much of condensed matter physics and chemistry. CDWs have their origin rooted in the instability of a one-dimensional system described by Peierls. The extension of this concept to reduced dimensional systems has led to the concept of Fermi surface nesting (FSN), which dictates the wave vector (q CDW ) of the CDW and the corresponding lattice distortion. The idea is that segments of the Fermi contours are connected byq CDW , resulting in the effective screening of phonons inducing Kohn anomalies in their dispersion at q CDW , driving a lattice restructuring at low temperatures. There is growing theoretical and experimental evidence that this picture fails in many real systems and in fact it is the momentum dependence of the electron-phonon coupling (EPC) matrix element that determines the characteristic of the CDW phase. Based on the published results for the prototypical CDW system 2H-NbSe 2 , we show how well theq-dependent EPC matrix element, but not the FSN, can describe the origin of the CDW. We further demonstrate a procedure of combing electronic band and phonon measurements to extract the EPC matrix element, allowing the electronic states involved in the EPC to be identified. Thus, we show that a large EPC does not necessarily induce the CDW phase, with Bi 2 Sr 2 CaCu 2 O 8+δ as the example, and the charge-ordered phenomena observed in various cuprates are not driven by FSN or EPC. To experimentally resolve the microscopic picture of EPC will lead to a fundamental change in the way we think about, write about, and classify charge density waves.charge density wave | phonon | nesting T he phrase charge density wave (CDW) was first used by Fröhlich (1, 2) but originates from Peierls' description of a fundamental instability in a one-dimensional (1D) chain of atoms equally spaced by a lattice constant a (2). Fig. 1A shows the free electron band of such a 1D chain with one electron per atomic site. The Fermi points are at k F = ±π/2a and are connected by the nesting vector q = 2k F . In 1930 Peierls asserted that this system is unstable, showing an electronic disturbance with the wave vector 2k F , changing the periodicity of the chain, and opening up a gap at the zone boundary (k = π/2a) of the new unit cell containing two atoms (1, 2). The conjecture was that the gain in electronic energy would always overwhelm the cost of restructuring the atoms (1). Consequently in the Peierls model there would be a transition from the metallic high-temperature state to the insulating-dimerized ground state at a critical temperature T CDW . Kohn (3) pointed out that there is an image of the Fermi surface in the vibrational spectrum, because the zero energy electronic excitations at 2k F will effectively screen any lattice motion with this wave vector. Fig. 1B shows the phonon dispersion for this 1D chain at different temperatures (1). Below T CDW the phonon energy at q = 2k F becomes imaginary, meaning there is a new lattice structure. Above T CDW there is a sharp dip (K...

Abstract. In this paper, a new holomorphic invariant is defined on a compact Kähler manifold with positive first Chern class and nontrivial holomorphic vector fields. This invariant generalizes the Futaki invariant. We prove that this invariant is an obstruction to the existence of Kähler-Ricci solitons. In particular, using this invariant together with the main result in [TZ1], we solve completely the uniqueness problem of Kähler-Ricci solitons. Two functionals associated to the new holomorphic invariant are also discussed. The main result here was announced in [TZ2]. Mathematics Subject Classification (2000). Primary 53C25; Secondary 32J15, 53C55, 58E11.Keywords. New holomorphic invariant, Kähler-Ricci soliton, holomorphic vector field. IntroductionThe purpose of this paper is to introduce a new holomorphic invariant and apply it to studying the uniqueness of Kähler-Ricci solitons on compact Kähler manifolds.Let M be an n-dimensional compact complex manifold with positive first Chernbe a Kähler metric on M with its Kähler formthere is a smooth function h g such that 2π ∂θ X (g)

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GANG TIAN AND XIAOHUA ZHU IntroductionIn this paper, we prove a theorem on convergence of Kähler-Ricci flow on a compact Kähler manifold M which admits a Kähler-Ricci soliton. A Kähler metric h is called a Kähler-Ricci soliton if its Kähler form ω h satisfies equationwhere Ric(ω h ) is the Ricci form of h and L X ω h denotes the Lie derivative of ω h along a holomorphic vector field X on M . As usual, we denote a Kähler-Ricci soliton by a pair (g KS , X). According to [TZ1], X should lie in the center of a reductive Lie subalgebra η r (M ) of η(M ), which consists of all holomorphic vector fields on2π ∂∂θ for some real-valued smooth function θ. It follows that the first Chern class c 1 (M ) is positive and it is represented by ω h .The Ricci flow was first introduced by R. Hamilton in [Ha]. If the underlying manifold M is Kähler with positive first Chern class, it is more natural to study the following Kähler-Ricci flow (normalized):where g 0 is a given metric with its Kähler class representing c 1 (M ). It can be shown that (0.1) preserves the Kähler class.Let Aut r (M ) be the connected Lie subgroup of the automorphism group of M corresponding to η r (M ). Let K be a maximal compact subgroup of Aut r (M ). According to [TZ1], we may assume that a Kähler-Ricci soliton (g KS , X) is Kinvariant and the imaginary part Im(X) of X generates a one-parameter subgroup K X of K. The following is our main result. Main

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.

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