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

bergman kernel from path integral we rederive the expansion of the bergman kernel on kahler manifolds developed by tian yau zelditch lu and catlin using path integral and perturbation theory and generalize it to supersymmetric quantum mechanics one physics interpretation of this result is as an expansion of the projector of wave functions on the lowest landau level in the special case that the magnetic field is proportional to the kahler form this is relevant for the quantum hall effect in curved space and for its higher dimensional generalizations other applications include the theory of coherent states the study of balanced metrics noncommutative field theory and a conjecture on metrics in black hole backgrounds discussed in 24 we give a short overview of these various topics from a conceptual point of view this expansion is noteworthy as it is a geometric expansion somewhat similar to the dewitt seeley gilkey et al short time expansion for the heat kernel but in this case describing the long time limit without depending on supersymmetry

sur le theoreme d atiyah singer on donne une version simplifiee de la demonstration probabiliste du theoreme de l indice pour l operateur de dirac donnee par j m bismut au lieu d utiliser la construction du mouvement brownien au moyen du fibre des reperes on utilise la construction de schwartz en plongeant la variete ambiante dans un espace vectoriel de dimension plus grande on evite aussi l utilisation de la decomposition de l espace de wiener en deux utilisee par j m bismut le calcul des variations stochastiques etant notre outil principal de ce fait on perd la relation avec la cohomologie de l espace des lacets

morse inequalities for fourier components of kohn rossi cohomology of cr manifolds with s 1 action let x be a compact connected cr manifold of dimension 2n 1 n ge 2 with a transversal cr s 1 action on x we study the fourier components of the kohn rossi cohomology with respect to the s 1 action by studying the szego kernel of the fourier components we establish the morse inequalities on x using the morse inequalities we have established on x we prove that there are abundant cr functions on x when x is weakly pseudoconvex and strongly pseudoconvex at a point

integral criteria for a classification of solutions of linear differential equations are positive continuous function on the interval 0 co by solution we always mean a continuously differentiable function such that r t x t has a continuous derivative satisfying e when the func tion r is continuous but does not have a continuous derivative e can be interpreted as the first order system for the vector x y x rx 381

homology computations for complex braid groups ii abstract we complete the computation of the integral homology of the generalized braid group b associated to an arbitrary irreducible complex reflection group w of exceptional type in order to do this we explicitly computed the recursively defined differential of a resolution of z as a z b module using parallel computing we also deduce from this general computation the rational homology of the milnor fiber of the singularity attached to most of these reflection groups

bergman kernels and local holomorphic morse inequalities let x be a hermitian manifold and let lk be a high power of a hermitian holomorphic line bundle over x local versions of demailly s holomorphic morse inequalities that give bounds on the dimension of the dolbeault cohomology groups associated to lk are presented after integration they give the usual holomorphic morse inequalities the local weak inequalities hold on any hermitian manifold x regardless of compactness and completeness the proofs which are elementary are based on a new approach to pointwise bergman kernel estimates where the kernels are estimated by a model kernel in open image in new window

generalized bergman kernels on symplectic manifolds we study the near diagonal asymptotic expansion of the generalized bergman kernel of the renormalized bochner laplacian on high tensor powers of a positive line bundle over a compact symplectic manifold we show how to compute the coefficients of the expansion by recurrence and give a closed formula for the first two of them as a consequence we calculate the density of states function of the bochner laplacian and establish a symplectic version of the convergence of the induced fubini study metric we also discuss generalizations of the asymptotic expansion for non compact or singular manifolds as well as their applications our approach is inspired by the analytic localization techniques of bismut and lebeau

on the asymptotic expansion of bergman kernel we study the asymptotic of the bergman kernel of the spin c dirac operator on high tensor powers of a line bundle

nonoscillation results for third order nonlinear differential equations abstract criteria are obtained in order that all nontrivial solutions of the third order nonlinear eqs a y q t y sgn y 0 and b y q t y sgn y 0 are nonoscillatory where 0 and q t is positive and continuous on 0 these results extend and complement previous results of kura and also may be considered as third order analogues of results for the second order nonlinear equations

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

the atiyah singer theorems a probabilistic approach ii the lefschetz fixed point formulas abstract in this paper the lefschetz fixed point formulas of m f atiyah and i m singer ann of math 87 1968 484 530 546 604 for classical elliptic complexes are proved by using probabilistic methods

braid pictures for artin groups we define the braid groups of a two dimensional orbifold and introduce conventions for drawing braid pictures we use these to realize the artin groups associated to the spherical coxeter diagrams a n b n c n and d n and the affine diagrams a n b n c n and d n as subgroups of the braid groups of various simple orbifolds the cases d n b n c n and d n are new in each case the artin group is a normal subgroup with abelian quotient in all cases except a n the quotient is finite we also illustrate the value of our braid calculus by giving a picture proof of the basic properties of the garside element of an artin group of type d n

on continuity and compactness of some nonlinear operators associated with differential equations in noncompact intervals soit x t f t x t t j x s ou jcr est un intervalle donne f j r n r n est une fonction continue et s un sous ensemble de c j r n l espace des fonctions continues de j dans r n on considere l equation operateur associee x t x ou t est un operateur non lineaire discontinu dans son domaine

szego kernel asymptotics and morse inequalities on cr manifolds let x be an abstract compact orientable cr manifold of dimension 2n 1 n 2 and let l k be the k th tensor power of a cr complex line bundle l over x we assume that condition y q holds at each point of x in this paper we obtain a scaling upper bound for the szego kernel on 0 q forms with values in l k for large k after integration this gives weak morse inequalities analogues of the holomorphic morse inequalities of demailly by a refined spectral analysis we obtain also strong morse inequalities we apply the strong morse inequalities to the embedding of some convex concave manifolds

the infinitesimal lefschetz formulas a heat equation proof abstract in this paper the lefschetz formulas of atiyah and bott ann of math 86 1967 374 407 88 1968 451 491 and atiyah and singer ann of math 87 1968 546 604 93 1971 119 138 are proved by heat equation methods in infinitesimal form i e without using the usual localization on the fixed point set the proof is obtained by a homotopic deformation of the dirac operator and uses probabilistic methods the main applications are potentially in noncompact situations where fixed point formulas do not always hold while their infinitesimal analogues are still valid

dimension n representations of the braid group on n strings abstract in 1996 e formanek classified all the irreducible complex representations of bn of dimension at most n 1 where bn is the artin braid group on n strings in this paper we extend this classification to the representations of dimension n for n 9 we prove that all such representations are equivalent to the tensor product of a one dimensional representation and a specialization of a certain one parameter family of n dimensional representations which was first discovered in 1996 by tong yang and ma in order to do this we classify all the irreducible complex representations of bn for which rank i 1 2 where the i are the standard generators

a kollar type vanishing theorem let f x rightarrow y be a smooth fibration between two complex manifolds x and y and let l be a pseudo effective line bundle on x we obtain a sufficient condition for r q f k x y otimes l to be reflexive and hence derive a kollar type vanishing theorem

proof of the k pi 1 conjecture for affine artin groups we prove the k pi 1 conjecture for affine artin groups the complexified complement of an affine reflection arrangement is a classifying space this is a long standing problem due to arnol d pham and thom our proof is based on recent advancements in the theory of dual coxeter and artin groups as well as on several new results and constructions in particular we show that all affine noncrossing partition posets are el shellable we use these posets to construct finite classifying spaces for dual affine artin groups we introduce new cw models for the orbit configuration spaces associated with arbitrary coxeter groups we construct finite classifying spaces for the braided crystallographic groups introduced by mccammond and sulway

on kneser solutions of nonlinear third order differential equations sufficient conditions are given under which the third order differential equation with the middle term y q t y r t f y 0 q positive has a kneser solution y the derivatives of which change their signs

big vector bundles and complex manifolds with semi positive tangent bundles we classify compact kahler manifolds with semi positive holomorphic bisectional and big tangent bundles we also classify compact complex surfaces with semi positive tangent bundles and compact complex 3 folds of the form mathbb p t x whose tangent bundles are nef moreover we show that if x is a fano manifold such that mathbb p t x has nef tangent bundle then x cong mathbb p n

morse inequalities a function space integral approach abstract in this paper we prove the morse inequalities in the non degenerate and degenerate cases like the approach of j m bismut ours is based on the idea suggested by witten in fact if anything our approach is closer to witten s original idea than bismut s

the witten complex and the degenerate morse inequalities on utilise le complexe introduit par witten pour demontrer des inegalites de morse dans le cas ou les points critiques de la fonction de morse sont isoles et aussi les inegalites de morse degenerees de bott

spectral theory for tensor products of hermitian holomorphic line bundles let l be a quasipositive line bundle over a compact complex hermitian manifold it has been conjectured that the smallest positive eigenvalue of the laplacian acting on the k th tensor power of l is uniformly bounded independent of k various special cases of the conjecture are proved but a counterexample is given to the general conjecture

a stochastic approach to the poincare hopf theorem in this paper the poincare hopf theorem hopf 8 is proved by a probabilistic method the proof of this theorem is reduced to an estimate of a supertrace of the fundamental solution we use the malliavian calculus to estimate the fundametal solution our proof can be applied to the case that the zero point set is the union of the submanifolds under some conditions

holomorphic morse inequalities for orbifolds we prove that demailly s holomorphic morse inequalities hold true for complex orbifolds by using a heat kernel method then we introduce the class of moishezon orbifolds and as an application of our inequalties we give a geometric criterion for a compact connected orbifold to be a moishezon orbifolds thus generalizing siu s and demailly s answers to the grauert riemenschneider conjecture to the orbifold case

the atiyah singer theorems a probabilistic approach i the index theorem abstract the atiyah singer index theorem for classical elliptic complexes is proved by using probabilistic methods the general idea is to use a probabilistic construction of the heat equation kernel which permits the direct derivation of the index formulas without using the theory of invariants of gilkey stochastic calculus on the exterior algebra is then used to find the classical local formula for the index theorem the lefschetz fixed point formulas of atiyah singer will be proved in part ii

oscillation criteria of a certain class of third order nonlinear delay differential equations with damping abstract in this paper we are concerned with the oscillation of third order nonlinear delay differential equations of the form r 2 t r 1 t y p t y q t f y g t 0 by using a generalized riccati transformation and integral averaging technique we establish some new sufficient conditions which insure that any solution of this equation oscillates or converges to zero in particular several examples are given to illustrate the importance of our results

morse type inequalities for dynamical systems and the witten laplacian we provide an analytic proof of morse type inequalities for vector flelds determining a morse decomposition with normally hyperbolic dynamics in the demonstration we reduce the problem to the gradient case using a morse bott lyapunov function and

morse homotopy and chern simons perturbation theory we define and invariant of a three manifold equipped with a flat bundle with vanishing homology the construction is based on morse theory using several morse functions simultaneously and is regarded as a higher loop analogue of various product operations in algebraic topology there is a heuristic argument that this invariant is related to perturbative chern simons gauge theory by axelrod singer etc there is also a theorem which gives a relation of the construction to open string theory on the cotangent bundle

demailly s asymptotic morse inequalities a heat equation proof on donne une demonstration qui s appuie sur l equation de la chaleur pour les inegalites asymptotiques de demailly pour le complexe

on the qualitative behaviors of solutions of third order nonlinear differential equations abstract we present some new results about oscillation and asymptotic behavior of solutions of third order nonlinear differential equations of the form r 2 t r 1 t y p t y q t f y g t 0 our work is significantly different from the previous works and the results of this paper improve extend and complement a number of existing results in the literature several examples are also given to illustrate the importance of our results

kirwan novikov inequalities on a manifold with boundary we extend the novikov morse type inequalities for closed 1 forms in 2 directions first we consider manifolds with boundary second we allow a very degenerate structure of the critical set of the form assuming only that the form is non degenerated in the sense of kirwan in particular we obtain a generalization of a result of floer about the usual morse inequalities on a manifold with boundary we also obtain an equivariant version of our inequalities our proof is based on an application of the witten deformation technique the main novelty here is that we consider the neighborhood of the critical set as a manifold with a cylindrical end this leads to a considerable simplification of the local analysis in particular we obtain a new analytic proof of the morse bott inequalities on a closed manifold

semiclassical determinants of schrodinger operators on line bundles heat kernel methods are used to compute asymptotic expressions for the zeta regularized determinant and local heat invariants of a schrodinger operator acting on sections of the kth tensor power of a line bundle as k the result is an infinite series representation involving todd polynomials of the curvature

lp estimates and existence theorems for the operator on complete kahler manifolds abstract in this paper we establish some l p estimates and existence theorems for the operator on a holomorphic hermitian vector bundle over a complete kahler manifold as applications we prove some vanishing theorems of the l p dolbeault cohomology on complete kahler manifolds with suitable geometric conditions

cohomology of affine artin groups and applications the result of this paper is the determination of the cohomology of artin groups of type an b n and a n with non trivial local coefficients the main result is an explicit computation of the cohomology of the artin group of type b n with coefficients over the module q q 1 t 1 here the first n 1 standard generators of the group act by q multiplication while the last one acts by t multiplication the proof uses some technical results from previous papers plus computations over a suitable spectral sequence the remaining cases follow from an application of shapiro s lemma by considering some well known inclusions we obtain the rational cohomology of the artin group of affine type an as well as the cohomology of the classical braid group br n with coefficients in the n dimensional representation presented in tong yang and ma 1996 the topological counterpart is the explicit construction of finite cw complexes endowed with a free action of the artin groups which are known to be k 1 spaces in some cases including finite type groups particularly simple formulas for the euler characteristic of these orbit spaces are derived

oscillation of third order nonlinear damped delay differential equations this paper is concerned with the oscillation of certain third order nonlinear delay differential equations with damping we give new characterizations of oscillation of the third order equation in terms of oscillation of a related well studied second order linear differential equation without damping we also establish new oscillation results for the third order equation by using the integral averaging technique due to philos numerous examples are given throughout

on monotone solution of the third order differential equation the existence and uniqueness of monotone solution of linear differential equation of the third order are proved using ky fan fixed point theorem the existence of such solution of nonlinear equation of the third order is obtained

a transcendental approach to kollar s injectivity theorem ii we treat kollar s injectivity theorem from the analytic or differential geometric viewpoint more precisely we give a curvature condition which implies kollar type cohomology injectivity theorems our main theorem is formulated for a compact kahler manifold but the proof uses the space of harmonic forms on a zariski open set with a suitable complete kahler metric we need neither covering tricks desingularizations nor leray s spectral sequence

oscillation results for emden fowler type differential equations abstract the third order nonlinear differential equation x a t x b t f x 0 is considered we present oscillation and nonoscillation criteria which extend and improve previous results existing in the literature in particular some results recently stated by m gregus and m gregus jr j math anal appl 181 1994 575 585 in addition contributions to the classification of solutions are given the techniques used are based on a transformation which reduces to a suitable disconjugate form to this aim auxiliary results on the asymptotic behavior of solutions of a second order linear differential equation associated to are stated they are presented in an independent form because they may be applied also to simplify and improve other qualitative problems concerning differential equations with quasiderivatives

heat equation derivative formulas for vector bundles we use martingale methods to give bismut type derivative formulas for differentials and co differentials of heat semigroups on forms and more generally for sections of vector bundles the formulas are mainly in terms of weitzenbock curvature terms in most cases derivatives of the curvature are not involved in particular our results improve b k driver s formula in 1997 j math pures appl 9 76 703 737 for logarithmic derivatives of the heat kernel measure on a riemannian manifold our formulas also include the formulas of k d elworthy and x m li 1998 c r acad sci paris ser i math 327 87 92

nonoscillation criteria for second order nonlinear differential equations abstract a number of known nonoscillation criteria for the second order nonlinear differential equation y q x y 0 0 where q is positive and locally of bounded variation are improved by energy function techniques

integral criteria for oscillation of third order nonlinear differential equations abstract in this paper we are concerned with the oscillation of third order nonlinear differential equations of the form r 2 t r 1 t y p t y q t f y 0 t t 0 where t 0 0 is a fixed real number r 1 r 2 p q i t 0 r are nonnegative continuous functions with r 1 0 r 2 0 f r r is a continuous function by making use of a generalized riccati transformation we establish some new sufficient integral conditions under which the equation has at least one oscillatory solution examples are given to illustrate the importance of our results