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

balanced metrics on non kahler calabi yau threefolds we construct balanced metrics on the family of non kahler calabi yau threefolds that are obtained by smoothing after contracting 1 1 rational curves on a kahler calabi yau threefold as an application we construct balanced metrics on complex manifolds diffeomorphic to the connected sum of k ge 2 copies of s 3 s 3

comparison results for quasilinear elliptic equations via a picone type identity part i quasilinear cases abstract using some identities of picone s types comparison results and uniqueness theorems are established for some general quasilinear elliptic equations

on a criterion for discrete or continuous spectrum of p laplace eigenvalue problems with singular sign changing weights abstract in this paper we study the existence of spectrum for one dimensional p laplace eigenvalue problems with singular weights subject to dirichlet boundary condition for certain class of singular sign changing weights we prove the existence of discrete spectrum proofs are based on the c 1 0 1 regularity of solutions and construction of the first eigenvalue in a variational set up we also present an example of a weight for which the spectrum is continuous and any corresponding eigenfunction does not belong to c 1 0 1

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

global bifurcation phenomena for singular one dimensional p laplacian in this paper we present global existence results for the following problem p p u t h t f u t 0 a e in 0 1 u 0 u 1 0 where p x x p 2x p 1 a positive parameter and h a nonnegative measurable function on 0 1 which may be singular at t 0 and or t 1 and f c r r with r 0 by applying the global bifurcation theorem and figuring the shape of unbounded subcontinua of solutions we obtain many different types of global existence results of positive solutions we also obtain existence results of sign changing solutions for p when f is an odd symmetric function

oscillation criteria for pde with p laplacian via the riccati technique abstract the conditions on the function c x r n r are derived which ensure that the pde div u p 2 u c x u p 2u 0 p 1 is oscillatory i e belongs to the closure of the set of zeros of every solution defined on the domain x r n x 1 the main tool is the riccati technique combined with suitable a priori bounds

dwork cohomology de rham cohomology and hypergeometric functions in the 1960s dwork developed a p adic cohomology theory of de rham type for varieties over finite fields based on a trace formula for the action of a frobenius operator on certain spaces of p analytic functions one can consider a purely algebraic analogue of dwork s theory for varieties over a field of characteristic zero and ask what is the connection between this theory and ordinary de rham cohomology katz showed that dwork cohomology coincides with the primitive part of de rham cohomology for smooth projective hypersurfaces but the exact relationship for varieties of higher codimension has been an open question in this article we settle the case of smooth affine complete intersections

existence of positive solutions for singular boundary value problems involving the one dimensional p laplacian this paper studies the existence of positive solutions for singular dirichlet boundary value problems these results are obtained by using the global continuation theorem fixed point index theory and approximate method

existence of the second positive radial solution for a p laplacian problem we investigate the existence nonexistence and multiplicity of positive radial solutions for the p laplacian problem with boundary parameters for proofs we mainly use a combination of a fixed point theorem the method of upper and lower solutions and fixed point index theory in the frame of the ordinary differential equation ode technique

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

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

growth conditions for oscillation of nonlinear differential equations with p laplacian abstract in this paper oscillation criteria are established for all solutions of second order nonlinear differential equations of the form p x 1 t p g x 0 t 0 here p y is the one dimensional p laplacian operator and g x satisfies the signum condition x g x 0 if x 0 but is not assumed to be monotone the equation naturally includes the famous euler differential equation and half linear differential equations the main purpose is to examine the influence of certain growth conditions of the nonlinear term g x on the oscillation of solutions the conditions are shown to be sharp some of differential inequalities play important roles to prove our results a simple example is included to illustrate the main result a conjecture on the inverse problem is also given finally elliptic equations with p laplacian operator are discussed as an application to our results

a picone identity for half linear elliptic equations and its applications to oscillation theory abstract a picone identity is established for a class of half linear elliptic operators and sturmian comparison and oscillation results are derived on the basis of the picone identity wirtinger inequalities and riccati inequalities are also obtained as corollaries of the picone identity and a new class of half linear elliptic equations is introduced

one dimensional p laplacian with a strong singular indefinite weight i eigenvalue abstract in this paper we prove the existence of eigenvalues for the problem p u t h t p u t 0 a e in 0 1 u 0 u 1 0 where p s s p 2 s p 1 is a real parameter and the indefinite weight h is a nonnegative measurable function on 0 1 which may be singular at 0 and or 1 and h 0 on any compact subinterval in 0 1 we derive similar properties of eigenvalues as known in linear case p 2 or continuous case h c 0 1 if h satisfies 0 1 t p 1 1 t p 1 h t d t when 1 p 2 and 0 1 2 p 1 s 1 2 h d d s 1 2 1 p 1 1 2 s h d d s when p 2 respectively for the result we establish the c 1 regularity of all solutions at the boundary for the above problem as well as the following problem p u t h t f u t 0 a e in 0 1 u 0 u 1 0 where f c r r s f s 0 for s 0 f is odd and f s p s is bounded above as s 0

oscillation of solutions of nonlinear parabolic equations via comparison method in the present paper sufficient conditions are found for oscillation of the solutions of nonlinear parabolic equations subject to boundary conditions of neumann or dirichlet type the results are obtained via the method of comparison of solutions of nonlinear parabolic equations by oscillatory solutions of certain ordinary differential equations

connecting moduli spaces of calabi yau threefolds we demonstrate that many families of calabi yau threefolds consist generically of small resolutions of nodal forms in other families and in fact that a large class of families is connected by this relation our result resonates with a conjecture of reid that calabi yau threefolds may have a universal moduli space even though they are of different homotopy types such ideas tie quite naturally to alluring prospects of unifying super string models

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

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

complete csck metrics on the local models of the conifold transition in this paper we construct complete constant scalar curvature kahler csck metrics on the complement of the zero section in the total space of mathcal o 1 oplus2 over mathbb p 1 which is biholomorphic to the smooth part of the cone c 0 in mathbb c 4 defined by equation sigma i 1 4 w i 2 0 on its small resolution and its deformation we also consider complete csck metrics and find that if the csck metrics are homogeneous then they must be ricci flat

integral averages and oscillation criteria for half linear partial differential equation the technique of weighted integral averages known in the oscillation theory of ordinary differential equations is extended to the half linear partial differential equation e div email protected email protected p email protected u c x u p 2u 0 p 1 this technique is used to obtain new oscillation criteria for e on the unbounded domains

oscillation of the half linear pde in general exterior domains the variational approach abstract in the paper a variational technique is used to show how to derive new oscillation criteria for the half linear partial differential equation via the oscillation criteria for the ordinary half linear differential equation in contrast to the results usually found in the literature general types of unbounded domains can be considered

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

existence results of sign changing solutions for singular one dimensional p laplacian problems abstract consider singular one dimensional p laplacian problems with dirichlet boundary condition p u t h t f u t 0 t 0 1 p u 0 0 u 1 d where p r r is defined by p x x p 2 x p 1 h a nonnegative measurable function on 0 1 which may be singular at t 0 and or t 1 and f c r r by applying the global bifurcation theorem and deriving the shape of the unbounded subcontinua of solutions we obtain the existence and multiplicity results of sign changing solutions for p d

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

ordinary differential equations in the oscillation theory of partial half linear differential equation abstract in the paper we study the damped half linear partial differential equation div a x u p 2 u b x u p 2 u c x u p 2 u 0 using radialization method we derive general oscillation results which allow to deduce new oscillation criteria for this equation from oscillation criteria for ordinary differential equations using careful radialization we improve several known oscillation criteria

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

oscillation of damped pde with p laplacian in unbounded domains abstract in this paper new oscillation theorems for the damped partial differential equation pde with p laplacian div a x u p 2 u b x u p 2 u c x u p 2 u 0 are established in unbounded domains our theorems as two special cases when a i or a i b x 0 improve and complement some existing results in the literature to illustrate our main results we give some corollaries and examples

conjugacy criteria for half linear ode in the theory of pde with generalized p laplacian and mixed powers abstract in this paper we use conjugacy criteria for a half linear ordinary differential equation to prove some nonexistence results for a partial differential equation with generalized p laplacian and mixed derivatives div a x y p 2 y b x y p 2 y c x y p 2 y i 1 m c i x y p i 2 y e x the results can be used also to study equations with strongly non radial potential c x since the usual dependence on the mean value of c x over a sphere can be replaced by a more general term

picone identities for half linear elliptic operators with p x laplacians and applications to sturmian comparison theory picone identities are established for a class of half linear elliptic operators with p x laplacians and sturmian comparison theorems are obtained on the basis of the picone identities specializations to half linear partial or ordinary differential inequalities with p x laplacians are shown

picone identity for quasilinear elliptic equations with p x laplacians and sturmian comparison theory picone identities for quasilinear elliptic equations with p x laplacians are established and sturmian comparison theory is developed on the basis of picone identities as specializations various elliptic equations such as linear elliptic equations half linear elliptic equations with p laplacians half linear elliptic equations with p x laplacians pseudo p laplacian with variable power are discussed generalizations are also investigated

sturmian comparison and oscillation theorems for a class of half linear elliptic equations abstract sturmian comparison theorems are established for a class of half linear elliptic equations via the picone identity and oscillation results are derived on the basis of comparison results obtained

bifurcation of sign changing solutions for one dimensional p laplacian with a strong singular weight p sublinear at abstract in this paper we consider one dimensional p laplacian problems with a singular weight which may not be in l 1 using the properties of eigenfunctions and the global bifurcation theory we prove existence uniqueness nonexistence and multiplicity results of positive solutions as well as sign changing solutions specially when the nonlinear term is p linear near 0 and p sublinear at