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

k polystability of q fano varieties admitting kahler einstein metrics it is shown that any possibly singular fano variety x admitting a kahler einstein metric is k polystable thus confirming one direction of the yau tian donaldson conjecture in the setting of mathbb q fano varieties equipped with their anti canonical polarization the proof is based on a new formula expressing the donaldson futaki invariants in terms of the slope of the ding functional along a geodesic ray in the space of all bounded positively curved metrics on the anti canonical line bundle of x one consequence is that a toric fano variety x is k polystable iff it is k polystable along toric degenerations iff 0 is the barycenter of the canonical weight polytope p associated to x the results also extend to the logarithmic setting and in particular to the setting of kahler einstein metrics with edge cone singularities applications to geodesic stability bounds on the ricci potential and perelman s lambda entropy functional on k unstable fano manifolds are also given

uniqueness and short time regularity of the weak kahler ricci flow let x be a compact kahler manifold we prove that the kahler ricci flow starting from arbitrary closed positive 1 1 currents is smooth outside some analytic subset this regularity result is optimal meaning that the flow has positive lelong numbers for short time if the initial current has we also prove that the flow is unique when starting from currents with zero lelong numbers

bessel functions heat kernel and the conical kahler ricci flow abstract following donaldson s openness theorem on deforming a conical kahler einstein metric we prove a parabolic schauder type estimate with respect to conical metrics as a corollary we show that the conical kahler ricci flow exists for short time the key is to establish the relevant heat kernel estimates where we use the weber formula on bessel function of the second kind and carslaw s heat kernel representation in 8

ricci measure for some singular riemannian metrics we define the ricci curvature as a measure for certain singular torsion free connections on the tangent bundle of a manifold the definition uses an integral formula and vector valued half densities we give relevant examples in which the ricci measure can be computed in the time dependent setting we give a weak notion of a ricci flow solution on a manifold

kahler einstein metrics and the kahler ricci flow on log fano varieties we prove the existence and uniqueness of kahler einstein metrics on q fano varieties with log terminal singularities and more generally on log fano pairs whose mabuchi functional is proper we study analogues of the works of perelman on the convergence of the normalized kahler ricci flow and of keller rubinstein on its discrete version ricci iteration in the special case of non singular fano manifolds our results on ricci iteration yield smooth convergence without any additional condition improving on previous results our result for the kahler ricci flow provides weak convergence independently of perelman s celebrated estimates

a modified kahler ricci flow in this note we study a kahler ricci flow modified from the classic version in the non degenerate case strong convergence at infinite time is achieved the main focus should be on degenerate case where some partial results are presented

monge ampere equations in big cohomology classes we define non pluripolar products of an arbitrary number of closed positive 1 1 currents on a compact kahler manifold x given a big 1 1 cohomology class on x i e a class that can be represented by a strictly positive current and a positive measure on x of total mass equal to the volume of and putting no mass on pluripolar sets we show that can be written in a unique way as the top degree self intersection in the non pluripolar sense of a closed positive current in we then extend kolodziedj s approach to sup norm estimates to show that the solution has minimal singularities in the sense of demailly if has l 1 e density with respect to lebesgue measure if is smooth and positive everywhere we prove that t is smooth on the ample locus of provided is nef using a fixed point theorem we finally explain how to construct singular kahler einstein volume forms with minimal singularities on varieties of general type

a numerical criterion for very ample line bundles let x be a projective algebraic manifold of dimension n and let l be an ample line bundle over x we give a numerical criterion ensuring that the adjoint bundle kx l is very ample the sufficient conditions are expressed in terms of lower bounds for the intersection numbers l y over subvarieties y of x in the case of surfaces our criterion gives universal bounds and is only slightly weaker than i reider s criterion when dimx 3 and codimy 2 the lower bounds for l y involve a numerical constant which depends on the geometry of x by means of an iteration process it is finally shown that 2kx ml is very ample form 12n our approach is mostly analytic and based on a combination of hormander s l estimates for the operator lelong number theory and the aubin calabi yau theorem

cohomology of q convex spaces in top degrees it is shown that every strongly q complete subvariety of a complex analytic space has a fundamental system of strongly q complete neighborhoods as a consequence we find a simple proof of ohsawa s result that every non compact irreducible n dimensional analytic space is strongly n complete finally it is shown that l cohomology theory readily implies ohsawa s hodge decomposition and lefschetz isomorphism theorems for absolutely q convex manifolds

convergence of calabi yau manifolds abstract in this paper we study the convergence of calabi yau manifolds under kahler degeneration to orbifold singularities and complex degeneration to canonical singularities including the conifold singularities and the collapsing of a family of calabi yau manifolds

recognition of object classes from range data abstract we develop techniques for recognizing instances of 3d object classes which may consist of multiple and or repeated sub parts with internal degrees of freedom linked by parameterized transformations from sets of 3d feature observations recognition of a class instance is structured as a search of an interpretation tree in which geometric constraints on pairs of sensed features not only prune the tree but are used to determine upper and lower bounds on the model parameter values of the instance a real valued constraint propagation network unifies the representations of the model parameters model constraints and feature constraints and provides a simple and effective mechanism for accessing and updating parameter values recognition of objects with multiple internal degrees of freedom including non uniform scaling and stretching articulations and sub part repetitions is demonstrated and analysed for two different types of real range data 3d edge fragments from a stereo vision system and position surface normal data derived from planar patches extracted from a range image

numerical characterization of the kahler cone of a compact kahler manifold the goal of this work is to give a precise numerical description of the kahler cone of a compact kahler manifold our main result states that the kahler cone depends only on the intersection form of the cohomology ring the hodge structure and the homology classes of analytic cycles if x is a compact kahler manifold the kahler cone k of x is one of the connected components of the set p of real 1 1 cohomology classes which are numerically positive on analytic cycles i e y p 0 for every irreducible analytic set y in x p dim y this result is new even in the case of projective manifolds where it can be seen as a generalization of the well known nakai moishezon criterion and it also extends previous results by campana peternell and eyssidieux the principal technical step is to show that every nef class which has positive highest self intersection numberx n 0 contains a kahler current this is done by using the calabi yau theorem and a mass concentration technique for monge ampere equations the main result admits a number of variants and corollaries including a description of the cone of numerically effective 1 1 classes and their dual cone another important consequence is the fact that for an arbitrary deformation x s of compact kahler manifolds the kahler cone of a very general fibre xt is independent of t i e invariant by parallel transport under the 1 1 component of the gauss manin connection

uniqueness in e x ω abstract we prove uniqueness for the dirichlet problem for the complex monge ampere equation on compact kahler manifolds in the case of probability measures vanishing on pluripolar sets the proof uses the mass concentration technique due to kolodziej coupled with inequalities for mixed monge ampere measures and the comparison principle

the weighted monge ampere energy of quasiplurisubharmonic functions abstract we study degenerate complex monge ampere equations on a compact kahler manifold x we show that the complex monge ampere operator d d c n is well defined on the class e x of plurisubharmonic functions with finite weighted monge ampere energy the class e x is the largest class of psh functions on which the monge ampere operator is well defined and the comparison principle is valid it contains several functions whose gradient is not square integrable we give a complete description of the range of the operator d d c n on e x as well as on some of its subclasses we also study uniqueness properties extending calabi s result to this unbounded and degenerate situation and we give applications to complex dynamics and to the existence of singular kahler einstein metrics

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

the complex monge ampere type equation on compact hermitian manifolds and applications abstract we prove the existence and uniqueness of continuous solutions to the complex monge ampere type equation with the right hand side in l p p 1 on compact hermitian manifolds next we generalise results of eyssidieux guedj and zeriahi 17 18 to compact hermitian manifolds which a priori are not in the fujiki class these generalisations lead to a number of applications we obtain partial results on a conjecture of tosatti and weinkove 40 and on a weak form of a conjecture of demailly and paun 11

the kahler ricci flow through singularities we prove the existence and uniqueness of the weak kahler ricci flow on projective varieties with log terminal singularities we also show that the weak kahler ricci flow can be uniquely continued through divisorial contractions and flips if they exist finally we propose an analytic version of the minimal model program with ricci flow

a viscosity approach to the dirichlet problem for complex monge ampere equations we present a viscosity approach to the dirichlet problem for the complex monge ampere equation det u bar j k f x u our approach differs from previous viscosity approaches to this equation in several ways it is based on contact set techniques the alexandrov bakelman pucci estimate on extensive applications of sup inf convolutions and on a relation between real and complex hessians more specifically this paper includes a notion of viscosity solutions a comparison principle and a solvability theorem the equivalence between viscosity and pluripotential solutions an estimate of the modulus of continuity of a solution in terms of that of a given subsolution and of the right hand side f and an alexandrov bakelman pucci type of l estimate

the kahler ricci flow on surfaces of positive kodaira dimension 4 estimates 9 4 1 the zeroth order and volume estimates 9 4 2 a partial second order estimate 13 4 3 gradient estimates 14 4 4 the second order estimates 18

kahler ricci flow with degenerate initial class in the earlier joint work 3 we introduced the weak kahler ricci flow for various geometric motivations in this current work we take further consideration on setting up the weak flow by allowing the initial class to be not necessary kahler it s shown that the construction is compatible with the earlier construction in kahler case we also discuss the convergence as t 0 which is of great interest in this topic and provide related motivation 1 motivation and set up of weak flow the kahler ricci flow the complex version of ricci flow has been under intensive study over the last twenty some years in 18 and more recently 17 g tian proposed the intriguing program of constructing globally existing weak kahlerricci flow with canonical singular limit at time infinity and applying it to the study of algebraic manifolds and even kahler manifolds in general it can be viewed as the analytic version of the famous minimal model program in algebraic geometry in general people should expect the classic smooth kahler ricci flow to encounter singularity at some finite time which is completely decided by cohomology information according to the optimal existence result in 20 just as what people have been doing and had successes in cases for ricci flow surgeries for the underlying manifold should be expected for the kahler ricci flow we naturally expect the surgery to have flavours from algebraic geometry a simple example would be surfaces of general type one only needs the blowingdown of 1 curves when applying the general construction at the end of 3 to push the flow through the finite time of singularities where the measure restriction is actually not so involved as explained later in example 4 4 the degenerate class at the singularity time would become kahler for the new manifold because those 1 curves causing the cohomology degeneration would eventually been crushed to points things can get significantly more complicated for higher dimension for example a standard procedure called flip is introduced in the algebraic geometry context which is of great importance for the business about the algebraic minimal model program simply speaking one needs to blow up the manifold and then perform a different blowing down process naturally we should expect the transformation of the degenerate class is still not kahler in this note we confirm that this is not a problem under the assumption that formally the kahler ricci flow is instantly date received on january 6 2012 and accepted on august 22 2012 2010 mathematics subject classification primary 53c44 secondary 14e30 58j35

on the convergence of a modified kahler ricci flow we study the convergence of a modified kahler ricci flow defined by zhou zhang we show that the modified kahler ricci flow converges to a singular metric when the limit class is degenerate this proves a conjecture of zhang

the moduli space of fano manifolds with kahler ricci solitons abstract we construct a canonical hausdorff complex analytic moduli space of fano manifolds with kahler ricci solitons this enlarges the moduli space of fano manifolds with kahler einstein metrics we discover a moment map picture for kahler ricci solitons and give complex analytic charts on the topological space consisting of kahler ricci solitons by studying differential geometric aspects of this moment map some stacky words and arguments on gromov hausdorff convergence help to glue them together in the holomorphic manner

weak solutions to degenerate complex monge ampere flows ii abstract studying the long term behavior of the kahler ricci flow on mildly singular varieties one is naturally led to study weak solutions of degenerate parabolic complex monge ampere equations the purpose of this article the second of a series on this subject is to develop a viscosity theory for degenerate complex monge ampere flows on compact kahler manifolds our general theory allows in particular to define and study the normalized kahler ricci flow on varieties with canonical singularities generalizing results of song and tian

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

the complex monge ampere equation on compact kahler manifolds we consider the complex monge ampere equation on a compact kahler manifold m g when the right hand side f has rather weak regularity in particular we prove that estimate of triangle phi and the gradient estimate hold when f is in w 1 p 0 for any p 0 2n as an application we show that there exists a classical solution in w 3 p 0 for the complex monge ampere equation when f is in w 1 p 0