singular kahler einstein metrics we study degenerate complex monge ampere equations of the form omega dd c f n e t f mu where omega is a big semi kahler form on a compact kahler manifold x of dimension n t in r and mu f omega n is a positive measure with density f in l p x omega n p 1 we prove the existence and unicity of continuous o plurisubharmonic solutions in case x is projective and omega psi omega where psi x to v is a proper birational morphism to a normal projective variety omega in ns r v is an ample class and mu has only algebraic singularities we prove that the solution is smooth in the regular locus of the equation we use these results to construct singular kahler einstein metrics of non positive curvature on projective klt pairs in particular on canonical models of algebraic varieties of general type

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

profinite groups in which the probabilistic zeta function coincides with the subgroup zeta function abstract we examine finitely generated profinite groups in which two formal dirichlet series the subgroup zeta function and the probabilistic zeta function coincide we call these groups reversible in the class of prosolvable groups of finite rank we show some sufficient conditions for this property to hold and we produce a structural characterisation of torsion free prosolvable groups of rank two which are reversible for pro p groups several results support the conjecture that reversibility is equivalent to the property that every open subgroup has the same minimal number of generators of the group itself in particular this holds for powerful pro p groups

the space of kahler metrics this paper the second of a series deals with the function space mathcal h of all smooth kahler metrics in any given n dimensional closed complex manifold v these metrics being restricted to a given fixed real cohomology class called a polarization of v this function space is equipped with a pre hilbert metric structure introduced by t mabuchi 10 who also showed that formally this metric has nonpositive curvature in the first paper of this series 4 the second author showed that the same space is a path length space he also proved that mathcal h is geodesically convex in the sense that for any two points of mathcal h there is a unique geodesic path joining them which is always length minimizing and of class c1 1 this partially verifies two conjectures of donaldson 8 on the subject in the present paper we show first of all that the space is as expected a path length space of nonpositive curvature in the sense of a d aleksandrov a second result is related to the theory of extremal kahler metrics namely that the gradient flow in mathcal h of the k energy of v has the property that it strictly decreases the length of all paths in mathcal h except those induced by one parameter families of holomorphic automorphisms of m

scalar curvature and stability of toric varieties we define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a kahler metric of constant scalar curvature the main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces we prove that under the stability hypothesis the mabuchi functional is bounded below on invariant metrics and that minimising sequences have a certain convergence property in the reverse direction we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature the proofs use a general framework developed by guillemin and abreu in which invariant kahler metrics correspond to convex functions on the moment polytope of a toric variety this paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a kahler metric of constant scalar curvature the pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of stability in the sense of geometric invariant theory this expectation is probably now an item of folklore going back to suggestions put forward by yau in the case of kahlereinstein metrics and the many results of tian and others in this case reinforced by a detailed formal picture which makes clear the analogy with the well established relation between the stability of vector bundles and yang mills connections 5 here we begin the investigation of

hyperdiscriminant polytopes chow polytopes and mabuchi energy asymptotics letx n p n be a smooth linearly normal algebraic variety it is shown that the mabuchi energy of x fsjx restricted to the bergman metrics is completely determined by the x hyperdiscriminant of format n 1 and the chow form of x as a corollary it is shown that the mabuchi energy is bounded from below for all degenerations in g if and only if the hyperdiscriminant polytope dominates the chow polytope for all maximal algebraic tori h of g

smooth approximations of the conical kahler ricci flows in this note we show that the conical kahler ricci flows introduced in chen and wang bessel functions heat kernel and the conical kahler ricci flow j funct anal 269 2 2013 exist for all time t in 0 infty in the weak sense as in definition 1 2 as a key ingredient of the proof we show that a conical kahler ricci flow is actually the limit of a sequence of smooth kahler ricci flows

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

obstructions to the existence of sasaki einstein metrics we describe two simple obstructions to the existence of ricci flat kahler cone metrics on isolated gorenstein singularities or equivalently to the existence of sasaki einstein metrics on the links of these singularities in particular this also leads to new obstructions for kahler einstein metrics on fano orbifolds we present several families of hypersurface singularities that are obstructed including 3 fold and 4 fold singularities of ade type that have been studied previously in the physics literature we show that the ads cft dual of one obstruction is that the r charge of a gauge invariant chiral primary operator violates the unitarity bound

symmetries quotients and kahler einstein metrics we consider fano manifolds m that admit a collection of finite automorphism groups g1 gk such that the quotients m gi are smooth

weighted projective embeddings stability of orbifolds and constant scalar curvature kahler metrics we embed polarised orbifolds with cyclic stabiliser groups into weighted projective space via a weighted form of kodaira embedding dividing by the non reductive automorphisms of weighted projective space then formally gives a moduli space of orbifolds we show how to express this as a reductive quotient and so a git problem thus defining a notion of stability for orbifolds we then prove an orbifold version of donaldson s theorem the existence of an orbifold kahler metric of constant scalar curvature implies k semistability by extending the notion of slope stability to orbifolds we therefore get an explicit obstruction to the existence of constant scalar curvature orbifold kahler metrics we describe the manifold applications of this orbifold result and show how many previously known results troyanov ghigi kollar rollin singer the adscft sasaki einstein obstructions of gauntlett martelli sparks yau fit into this framework

a thermodynamical formalism for monge ampere equations moser trudinger inequalities and kahler einstein metrics we develop a variational calculus for a certain free energy functional on the space of all probability measures on a kahler manifold x this functional can be seen as a generalization of mabuchi s k energy functional and its twisted versions to more singular situations applications to monge ampere equations of mean field type twisted kahler einstein metrics and moser trudinger type inequalities on miller manifolds are given tian s alpha invariant is generalized to singular measures allowing in particular a proof of the existence of kahler einstein metrics with positive ricci curvature that are singular along a given anti canonical divisor which combined with very recent developments concerning miller metrics with conical singularities confirms a recent conjecture of donaldson as another application we show that if the calabi flow in the anti canonical class exists for all times then it converges to a kahler einstein metric when a unique one exists which is in line with a well known conjecture c 2013 elsevier inc all rights reserved

kahler ricci solitons on toric fano orbifolds we prove the existence of kahler ricci solitons on toric fano orbifolds hence extend the theorem of wang and zhu adv math 188 87 103 2004 to the orbifold case

kahler ricci solitons on toric manifolds with positive first chern class abstract in this paper we prove there exists a kahler ricci soliton unique up to holomorphic automorphisms on any toric kahler manifold with positive first chern class and the kahler ricci soliton is a kahler einstein metric if and only if the futaki invariant vanishes

geometric analysis of chow mumford stability abstract in this paper we relate the chow mumford stability of a projective manifold x to the properness of a certain energy functional on the subspace of bergman metrics in the kahler cone of x we use this relationship to rederive the stability of smooth curves over c this is accomplished by analyzing the asymptotics of some rational integrals

kahler einstein metrics and volume minimization abstract we prove that if a polarized q fano variety admits a kahler einstein metric then the normalized volume functional over the associated affine cone is globally minimized at the canonical divisorial valuation obtained by blowing up the vertex this is also generalized to the logarithmic and the orbifold settings a refinement of this result together with the result in 41 gives an equivalent characterization of k semistability for any smooth fano manifold we also prove that the valuation associated to the reeb vector field of a smooth sasaki einstein metric minimizes the normalized volume over the corresponding kahler cone these results strengthen and generalize the minimization result of martelli sparks yau

metric contraction of the cone divisor by the conical kahler ricci flow we use the momentum construction of calabi to study the conical kahler ricci flow on hirzebruch surfaces with cone angle along the exceptional curve and show that either the flow gromov hausdorff converges to the riemann sphere or a single point in finite time or the flow contracts only the cone divisor to a single point and gromov hausdorff converges to a two dimensional projective orbifold the limiting behaviour depends only on the cone angle numerical properties of the initial kahler class and the degree of the hirzebruch surface this gives the first example of the conical kahler ricci flow contracting the cone divisor to a single point and shows that the conical flow may contract curves of self intersection less than 1 as opposed to the smooth kahler ricci flow at the end we introduce a conjectural picture of the geometry of finite time non collapsing singularities of the flow on kahler surfaces in general

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

curvature of vector bundles associated to holomorphic fibrations let l be a semi positive line bundle over a kahler manifold x fibered over a complex manifold y assuming the fibers are compact and nonsingular we prove that the hermitian vector bundle e over y whose fibers over points y are the spaces of global sections over xy to l kx y endowed with the l2 metric is semi positive in the sense of nakano we also discuss various applications among them a partial result on a conjecture of griffiths on the positivity of ample bundles

greatest lower bounds on ricci curvature for toric fano manifolds abstract in this short note based on the work of wang and zhu 2004 8 we determine the greatest lower bounds on ricci curvature for all toric 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

convergence of kahler ricci flow in this paper we prove a theorem on convergence of kahler ricci flow on a compact kahler manifold which admits a kahler ricci soliton

transport exponents of sturmian hamiltonians 2015 elsevier inc we consider discrete schrodinger operators with sturmian potentials and study the transport exponents associated with them under suitable assumptions on the frequency we establish upper and lower bounds for the upper transport exponents as an application of these bounds we identify the large coupling asymptotics of the upper transport exponents for frequencies of constant type we also bound the large coupling asymptotics uniformly from above for lebesgue typical frequency a particular consequence of these results is that for most frequencies of constant type transport is faster than for lebesgue almost every frequency we also show quasi ballistic transport for all coupling constants generic frequencies and suitable phases

bergman kernels and the pseudoeffectivity of relative canonical bundles a criterion for the pseudoeffectivity of twisted relative canonical bundles of surjective projective maps

gromov hausdorff limits of kahler manifolds and algebraic geometry we prove a general result about the geometry of holomorphic line bundles over kahler manifolds

minimizing normalized volumes of valuations for any mathbb q gorenstein klt singularity x o we introduce a normalized volume function widehat mathrm vol that is defined on the space of real valuations centered at o and consider the problem of minimizing widehat mathrm vol we prove that the normalized volume has a uniform positive lower bound by proving an izumi type estimate for any mathbb q gorenstein klt singularity furthermore by proving a properness estimate we show that the set of real valuations with uniformly bounded normalized volumes is compact and hence reduce the existence of minimizers for widehat mathrm vol to a conjectural lower semicontinuity property for a smooth point we use an inequality of de fernex ein musta to show that the divisorial valuation associated to the exceptional divisor of the standard blow up is a minimizer of widehat mathrm vol finally the relation to fujita s work on divisorial stability is pointed out

asymptotically log fano varieties abstract motivated by the study of fano type varieties we define a new class of log pairs that we call asymptotically log fano varieties and strongly asymptotically log fano varieties we study their properties in dimension two under an additional assumption of log smoothness and give a complete classification of two dimensional strongly asymptotically log smooth log fano varieties based on this classification we formulate an asymptotic logarithmic version of calabi s conjecture for del pezzo surfaces for the existence of kahler einstein edge metrics in this regime we make some initial progress towards its proof by demonstrating some existence and non existence results among them a generalization of matsushima s result on the reductivity of the automorphism group of the pair and results on log canonical thresholds of pairs one by product of this study is a new conjectural picture for the small angle regime and limit which reveals a rich structure in the asymptotic regime of which a folklore conjecture concerning the case of a fano manifold with an anticanonical divisor is a special case

twisted and conical kahler ricci solitons on fano manifolds in this paper we first study the relationship between the existence of twisted kahler ricci solitons and the properness of modified twisted k energy approximating by a sequence of smooth twisted kahler ricci solitons we obtain an existence result of conical kahler ricci solitons

special test configuration and k stability of fano varieties for any at projective family x l c such that the generic bre x is a klt q fano variety and ljx q kx we use the techniques from the minimal model program mmp to modify the total family the end product is a family such that every ber is a klt q fano variety moreover we can prove that the donaldson futaki invariants of the appearing models decrease when the family is a test conguration of a xed fano variety x kx this implies tian s conjecture given x a fano manifold to test its k semi poly stability we only need to test on the special test congurations

kahler einstein metrics with positive scalar curvature in this paper we prove that the existence of kahler einstein metrics implies the stability of the underlying kahler manifold in a suitable sense in particular this disproves a long standing conjecture that a compact kahler manifold admits kahler einstein metrics if it has positive first chern class and no nontrivial holomorphic vector fields we will also establish an analytic criterion for the existence of kahler einstein metrics our arguments also yield that the analytic criterion is satisfied on stable kahler manifolds provided that the partial c 0 estimate posed in t6 is true

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

prescribing curvature on compact surfaces with conical singularities we study the berger nirenberg problem on surfaces with conical singularities i e we discuss conditions under which a function on a riemann surface is the gaussian curvature of some conformal metric with a prescribed set of singularities of conical types

cohomogeneity one einstein sasaki 5 manifolds we consider hypersurfaces in einstein sasaki 5 manifolds which are tangent to the characteristic vector field we introduce evolution equations that can be used to reconstruct the 5 dimensional metric from such a hypersurface analogous to the nearly hypo and half flat evolution equations in higher dimensions we use these equations to classify einstein sasaki 5 manifolds of cohomogeneity one

the dirichlet problem for nonlinear second order elliptic equations ii complex monge ampere and uniformaly elliptic equations on considere le probleme de dirichlet pour des equations elliptiques non lineaires d ordre 2 pour une fonction reelle dans un domaine borne de r n a frontiere lisse

note on k stability of pairs we prove that a pair x d with x fano and d an anti canonical divisor is k unstable for negative angles and is k semistable for zero angle