on energy functionals kahler einstein metrics and the moser trudinger onofri neighborhood abstract we prove that the existence of a kahler einstein metric on a fano manifold is equivalent to the properness of the energy functionals defined by bando chen ding mabuchi and tian on the set of kahler metrics with positive ricci curvature we also prove that these energy functionals are bounded from below on this set if and only if one of them is this answers two questions raised by x x chen as an application we obtain a new proof of the classical moser trudinger onofri inequality on the two sphere as well as describe a canonical enlargement of the space of kahler potentials on which this inequality holds on higher dimensional fano kahler einstein manifolds

log canonical thresholds of certain fano hypersurfaces we study log canonical thresholds on quartic threefolds quintic fourfolds and double spaces as an important application we show that they have kahler einstein metrics if they are general

spin holonomy of einstein manifolds berger s theorem classifies the linear holonomy groups of irreducible simply connected riemannian manifolds for physical applications however it is at least as important to have a classification of the possible spin holonomy groups defined by the parallel transport of spinors of non simply connected manifolds we establish a complete classification of the spin holonomy groups of all compact locally irreducible einstein riemannian spin manifolds of non negative scalar curvature

moser trudinger inequality for the complex monge ampere equation abstract in this paper we prove a moser trudinger type inequality for pluri subharmonic functions vanishing on the boundary our proof uses a descent gradient flow for the complex monge ampere functional

the moser trudinger inequality on kahler einstein manifolds we prove the conjecture of tian on the strong form of the moser trudinger inequality for k ahler einstein manifolds with positive first chern class when there are no holomorphic vector fields and more generally when the setting is invariant under a maximal compact subgroup of the automorphism group

fano varieties with many selfmaps abstract we study global log canonical thresholds of anticanonically embedded quasismooth weighted fano threefold hypersurfaces having terminal quotient singularities to prove the existence of a kahler einstein metric on most of them and to produce examples of fano varieties with infinite discrete groups of birational automorphisms

long time existence of minimizing movement solutions of calabi flow abstract we recast the calabi flow in degiorgi s language of minimizing movements we establish the long time existence of minimizing movements for k energy with arbitrary initial condition furthermore we establish some a priori regularity results for these solutions and that sufficiently regular minimizing movements are smooth solutions to calabi flow

estimation of ricci tensor on certain fano manifolds we give time existence to some monge ampere equations on certain fano manifolds which cannot carry einstein kahler metrics these solutions allow us to obtain an estimation of ricci tensors on our manifolds

remarks on the existence problem of positive kahler einstein metrics on introduit un invariant m qui donne un critere pour l existence de metriques de kahler einstein

k stability of smooth del pezzo surfaces in a new algebro geometric way we completely determine whether smooth del pezzo surfaces are k semi stable or not in the present article all varieties are defined over an algebraically closed field k of characteristic 0

on a class of fully nonlinear flows in kahler geometry in this paper we study a class of fully nonlinear metric flow on kahler manifolds which includes the j flow as a special case we provide a sufficient and necessary condition for the long time convergence of the flow generalizing the result of songweinkove as a consequence under the given condition we solved the corresponding euler equation which is fully nonlinear of monge ampere type as an application we also discuss a complex monge ampere type equation including terms of mixed degrees which was first posed by chen

yau tian donaldson correspondence for k semistable fano manifolds in this paper using the recent compactness results of tian and chen donaldson sun we prove the k semistable version of yau tian donaldson correspondence for fano manifolds

on kahler einstein metrics on certain kahler manifolds with c1 m 0 on demontre qu il existe une metrique de kahler einstein sur une hypersurface de fermat a m dimensions de degre superieur a m 1

kahler einstein metrics with edge singularities this article considers the existence and regularity of kahler einstein metrics on a compact kahler manifold m with edge singularities with cone angle 2 along a smooth divisor d we prove existence of such metrics with negative zero and some positive cases for all cone angles 2 2 the results in the positive case parallel those in the smooth case we also establish that solutions of this problem are polyhomogeneous i e have a complete asymptotic expansion with smooth coecients along d for all 2 2

ricci flow and the metric completion of the space of kahler metrics we consider the space of kahler metrics as a riemannian submanifold of the space of riemannian metrics and study the associated submanifold geometry in particular we show that the intrinsic and extrinsic distance functions are equivalent we also determine the metric completion of the space of kahler metrics making contact with recent generalizations of the calabi yau theorem due to dinew and guedj zeriahi as an application we obtain a new analytic stability criterion for the existence of a kahler einstein metric on a fano manifold in terms of the ricci flow and the distance function we also prove that the kahler ricci flow converges as soon as it converges in the metric sense

some discretizations of geometric evolution equations and the ricci iteration on the space of kahler metrics we propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as well as a means to obtain interesting dynamics on certain infinite dimensional spaces we illustrate the fruitfulness of this approach in the context of the ricci flow as well as another flow in kahler geometry we introduce and study dynamical systems related to the ricci operator on the space of kahler metrics that arise as discretizations of these flows we pose some problems regarding their dynamics we point out a number of applications to well studied objects in kahler and conformal geometry such as constant scalar curvature metrics kahler ricci solitons nadel type multiplier ideal sheaves balanced metrics the moser trudinger onofri inequality energy functionals and the geometry and structure of the space of kahler metrics e g we obtain a new sharp inequality strengthening the classical moser trudinger onofri inequality on the two sphere

existence of kahler einstein metrics and multiplier ideal sheaves on del pezzo surfaces we apply nadel s method of multiplier ideal sheaves to show that every complex del pezzo surface of degree at most six whose automorphism group acts without fixed points has a kahler einstein metric in particular all del pezzo surfaces of degree 4 5 or 6 and certain special del pezzo surfaces of lower degree are shown to have a kahler einstein metric these existence statements are not new but the proofs given in the present paper are less involved than earlier ones by siu tian and tian yau

effectiveness of demailly s strong openness conjecture and related problems in this article stimulated by the effectiveness in berndtsson s solution of the openness conjecture and continuing our solution of demailly s strong openness conjecture we discuss conditions to guarantee the effectiveness of the conjecture and establish such an effectiveness result we explicitly point out a lower semicontinuity property of plurisubharmonic functions with a multiplier which is implicitly contained in guan and zhou 2014 arxiv 1401 7158 we also obtain optimal effectiveness of the conjectures of demailly kollar and jonsson mustat respectively

extension theorems non vanishing and the existence of good minimal models we prove an extension theorem for effective purely log terminal pairs x s b of non negative kodaira dimension kappa k x s b ge 0 the main new ingredient is a refinement of the ohsawa takegoshi l 2 extension theorem involving singular hermitian metrics

numerical kahler einstein metric on the third del pezzo the third del pezzo surface admits a unique kahler einstein metric which is not known in closed form the manifold s toric structure reduces the einstein equation to a single monge ampere equation in two real dimensions we numerically solve this nonlinear pde using three different algorithms and describe the resulting metric the first two algorithms involve simulation of ricci flow in complex and symplectic coordinates respectively the third algorithm involves turning the pde into an optimization problem on a certain space of metrics which are symplectic analogues of the algebraic metrics used in numerical work on calabi yau manifolds our algorithms should be applicable to general toric manifolds using our metric we compute various geometric quantities of interest including laplacian eigenvalues and a harmonic 1 1 form the metric and 1 1 form can be used to construct a klebanov tseytlin like supergravity solution

conical kahler einstein metrics revisited in this paper we introduce the interpolation degeneration strategy to study kahler einstein metrics on a smooth fano manifold with cone singularities along a smooth divisor that is proportional to the anti canonical divisor by interpolation we show the angles in 0 2 that admit a conical kahler einstein metric form a connected interval and by degeneration we determine the boundary of the interval in some important cases as a first application we show that there exists a kahler einstein metric on mathbb p 2 with cone singularity along a smooth conic degree 2 curve if and only if the angle is in 2 2 when the angle is 2 3 this proves the existence of a sasaki einstein metric on the link of a three dimensional a2 singularity and thus answers a question posed by gauntlett martelli sparks yau as a second application we prove a version of donaldson s conjecture about conical kahler einstein metrics in the toric case using song wang s recent existence result of toric invariant conical kahler einstein metrics

metriques d einstein kahler et exponentiel des fonctions admissibles etant donnee une variete kahlerienne compacte 2m g de dimension complexe m il est du plus grand interet de savoir s il existe sur cette variete une metrique d einstein kahler ce probleme a ete aborde pour la premiere fois dans aubin 1

compactness in dual spaces of locally compact groups and tensor products of irreducible representations abstract we deal with supports of tensor products of irreducible representations of locally compact groups the main result describes the structure of almost connected groups for which tensor products of representations in the reduced dual have compact supports an immediate consequence is that an infinite almost connected group cannot have a compact reduced dual furthermore the continuity problem for these supports is studied