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

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

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

growth of balls of holomorphic sections and energy at equilibrium let l be a big line bundle on a compact complex manifold x given a non pluripolar compact subset k of x and a continuous hermitian metric e on l we define the energy at equilibrium of k as the monge ampere energy of the extremal psh weight associated to k we prove the differentiability of the energy at equilibrium with respect to and we show that this energy describes the asymptotic behaviour as k of the volume of the sup norm unit ball induced by k k on the space of global holomorphic sections h 0 x kl as a consequence of these results we recover and extend rumely s robin type formula for the transfinite diameter we also obtain an asymptotic description of the analytic torsion and extend yuan s equidistribution theorem for algebraic points of small height to the case of a big line bundle

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

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

from monge ampere equations to envelopes and geodesic rays in the zero temperature limit let x theta be a compact complex manifold x equipped with a smooth but not necessarily positive closed 1 1 form theta by a well known envelope construction this data determines in the case when the cohomology class theta is pseudoeffective a canonical theta psh function u theta when the class theta is kahler we introduce a family u beta of regularizations of u theta parametrized by a large positive number beta where u beta is defined as the unique smooth solution of a complex monge ampere equation of aubin yau type it is shown that as beta rightarrow infty the functions u beta converge to the envelope u theta uniformly on x in the holder space c 1 alpha x for any alpha in 0 1 which is optimal in terms of holder exponents a generalization of this result to the case of a nef and big cohomology class is also obtained and a weaker version of the result is obtained for big cohomology classes the proofs of the convergence results do not assume any a priori regularity of u theta applications to the regularization of omega psh functions and geodesic rays in the closure of the space of kahler metrics are given as briefly explained there is a statistical mechanical motivation for this regularization procedure where beta appears as the inverse temperature this point of view also leads to an interpretation of u beta as a transcendental bergman metric

bounded state solution of degenerate kirchhoff type problem with a critical exponent abstract in the present paper we investigate the following degenerate kirchhoff type problem b r n u 2 d x u v x u u 2 2 u in r n u d 1 2 r n where b is a positive constant v l n 2 r n is a given nonnegative function and 2 is the critical exponent quite a few papers have been published about the degenerate kirchhoff type problem with a critical exponent moreover this degenerate problem in r n n 5 has never been considered so far we obtain some sufficient conditions on the existence of bounded state solution for this degenerate problem as to the cases where n 5 it is the first time to consider the degenerate problem

a potential theory for the k curvature equation abstract in this paper we introduce a potential theory for the k curvature equation which can also be seen as a pde approach to curvature measures we assign a measure to a bounded upper semicontinuous function which is strictly subharmonic with respect to the k curvature operator and establish the weak continuity of the measure

fine topology silov boundary and ddc n on considere la topologie fine des fonctions plurisousharmoniques on etablit des theoremes de convergence on etudie les definitions de dd c n l operateur dd c n converge pour des limites monotones des fonctions plurisousharmoniques bornees

application of tensor analysis to the finite element method this paper presents the application of tensor analysis to the finite element fe method a fe formulation is presented where the suffix notation and the einstein s convention are used instead of the matrix algebra notation the usage of the proposed notation avoids the drawbacks of matrix algebra and allows the easy implementation of the fe method in a computer code the dirichlet boundary condition is included in the fe formulation through the use of suffix constraints afterwards the suffix notation is applied to the fe formulation of the poisson equation and it is implemented in a c language finally two numerical examples are solved to illustrate the advantages of using the proposed notation in the fe method

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

nonexistence for complete kahler einstein metrics on some noncompact manifolds let m be a compact kahler manifold and n be a subvariety with codimension greater than or equal to 2 we show that there are no complete kahler einstein metrics on m n as an application let e be an exceptional divisor of m then m e cannot admit any complete kahler einstein metric if blow down of e is a complex variety with only canonical or terminal singularities a similar result is shown for pairs

invariance of plurigenera in this paper we give a proof of the following long conjectured result on the invariance of the plurigenera main theorem let p x d be a smooth projective family of compact complex manifolds parametrized by the open unit 1 disk d assume that the bers xt py1 t t 2 d are of general type then for every positive integer m the plurigenus dimc c xt mkxt is independent of t 2 d where kxt is the canonical line bundle of xt

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

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

an inequality for mixed monge ampere measures we generalize an inequality for mixed monge ampere measures from kolodziej indiana univ math j 43 1321 1338 1994 we also give an example that shows that our assumptions are sharp the corresponding result in the setting of compact kahler manifold is also discussed

determinantal point processes and fermions on complex manifolds large deviations and bosonization we study determinantal random point processes on a compact complex manifold x associated to a hermitian metric on a line bundle over x and a probability measure on x physically this setup describes a gas of free fermions on x subject to a u 1 gauge field and when x is the riemann sphere it specializes to various random matrix ensembles our general setup will also include the setting of weighted orthogonal polynomials in mathbb c n as well as in mathbb r n it is shown that in the many particle limit the empirical random measures on x converge exponentially towards the deterministic pluripotential equilibrium measure defined in terms of the monge ampere operator of complex pluripotential theory more precisely a large deviation principle ldp is established with a good rate functional which coincides with the normalized pluricomplex energy of a measure recently introduced in berman et al publ math de l ihes 117 179 245 2013 we also express the ldp in terms of the ray singer analytic torsion this can be seen as an effective bosonization formula generalizing the previously known formula in the riemann surface case to higher dimensions and the paper is concluded with a heuristic quantum field theory interpretation of the resulting effective boson fermion correspondence