asymptotic expansion of complex integrals via mellin transform soit x une variete complexe connexe a n 1 dimensions et f une fonction holomorphe non constante sur x avec 0 comme seule valeur critique possible on demontre un theoreme de developpement asymptotique d integrales complexes par la transformation de mellin

regularizations of residue currents under assumptions about complete intersection we prove that coleff herrera type currents satisfy a robust calculus in the sense that natural regularizations of such currents can be multiplied to yield regularizations of the coleff herrera product of the currents

regularizations of products of residue and principal value currents let f 1 and f 2 be two functions on some complex n manifold and let be a test form of bidegree n n 2 assume that f 1 f 2 defines a complete intersection the integral of f 1 f 2 on f 1 2 e lunate 1 f 2 2 e lunate 2 is the residue integral i f1 f 2 e lunate 1 e lunate 2 it is in general discontinuous at the origin let 1 and 2 be smooth functions on 0 such that j 0 0 and j 1 we prove that the regularized residue integral defined as the integral of over 1 over 2 f 1 f 2 where j j f j 2 e lunate j is holder continuous on the closed first quarter and that the value at zero is the coleff herrera residue current acting on in fact we prove that if is a test form of bidegree n n 1 then the integral of 1 over 2 f 1 f 2 is holder continuous and tends to the over potential 1 f 1 over 1 f 2 of the coleff herrera current acting on more generally let f 1 and f 2 be sections of some vector bundles and assume that f 1 f 2 defines a complete intersection there are associated principal value currents u f and u g and residue currents r f and r g the residue currents equal the coleff herrera residue currents locally one can give meaning to formal expressions such as e g u f r g in such a way that formal leibnitz rules hold our results generalize to products of these currents as well 2006 elsevier inc all rights reserved

explicit serre duality on complex spaces abstract in this paper we use recently developed calculus of residue currents together with integral formulas to give a new explicit analytic realization as well as a new analytic proof of serre duality on any reduced pure n dimensional paracompact complex space x at the core of the paper is the introduction of certain fine sheaves b x n q of currents on x of bidegree n q such that the dolbeault complex b x n becomes in a certain sense a dualizing complex in particular if x is cohen macaulay then b x n is an explicit fine resolution of the grothendieck dualizing sheaf

infinitesimal deformations of restricted simple lie algebras i abstract we compute the infinitesimal deformations of two families of restricted simple modular lie algebras of cartan type the witt jacobson and the special lie algebras

pure hodge structure on the l_2 cohomology of varieties with isolated singularities cheeger goresky and macpherson conjectured in cgm an l2 de rham theorem that the intersection cohomology of a projective variety v is naturally isomorphic to the l2 cohomology of the incomplete manifold v singv with metric induced by a projective embedding the early interest in this conjecture was motivated in large part by the hope that one could then put a pure hodge structure on the intersection cohomology of v and even extend the rest of the kahler package cgm to this context saito s1 s2 eventually established the kahler package for intersection cohomology without recourse to l2 cohomology techniques however interest in l2 cohomology did not disappear with this result since among other things l2 cohomology provides intrinsic geometric invariants of an arbitrary complex projective variety which are not apparent from the point of view of d modules for instance l2 coholomology groups depend on boundary conditions ps which as we show here must be treated carefully in order to give the correct hodge components for the l2 cohomology of a singular variety a related fact is that for incomplete manifolds the pure hodge structure and lefschetz decompositions are not direct consequences of the kahler condition as they are in the compact case indeed the primary obstruction to obtaining a hodge structure on the l2 cohomology is the following apparent technicality on an incomplete kahler manifold there are several potentially distinct definitions of a square integrable harmonic form for example a form h might be considered harmonic if dh 0 h or if h 0 h or simply if h 0 moreover there are further domain considerations one imposes boundary conditions which turn out to have no effect on cohomology in the case of d but are crucial for cohomology on a compact or even complete manifold all these definitions of harmonics coincide and one obtains the pure hodge structure by decomposing harmonic forms into their p q components the p q components are harmonic in the weakest sense they are in the kernel of the equality of the different notions of harmonic then allows one to realize these p q components as spaces of both and d cohomology classes the equivalence of the different definitions of harmonic is also required in order to obtain the lefschetz decomposition a local computation shows that interior product with the kahler form preserves the kernel of but one requires the equivalence to see that this also induces an endomorphism on the l2 cohomology

simple lie algebras of small characteristic vi completion of the classification let l be a finite dimensional simple lie algebra over an algebraically closed field of characteristic p 3 it is proved in this paper that if the p envelope of adl in derl contains a torus of maximal dimension whose centralizer in adl acts nontriangulably on l then p 5 and l is isomorphic to one of the melikian algebras m m n in conjunction with a premet h strade simple lie algebras of small characteristic v the non melikian case j algebra 314 2007 664 692 theorem 1 2 this implies that up to isomorphism any finite dimensional simple lie algebra over an algebraically closed field of characteristic p 3 is either classical or a filtered lie algebra of cartan type or a melikian algebra of characteristic 5 this result finally settles the classification problem for finite dimensional simple lie algebras over algebraically closed fields of characteristic 2 3

on special zeros of p adic l functions of hilbert modular forms let e be a modular elliptic curve over a totally real number field f we prove the weak exceptional zero conjecture which links a higher derivative of the p adic l function attached to e to certain p adic periods attached to the corresponding hilbert modular form at the places above p where e has split multiplicative reduction under some mild restrictions on p and the conductor of e we deduce the exceptional zero conjecture in the strong form i e where the automorphic p adic periods are replaced by the mathcal l invariants of e defined in terms of tate periods from a special case proved earlier by mok crucial for our method is a new construction of the p adic l function of e in terms of local data

abelian differentials on singular varieties and variations on a theorem of lie griffiths we compare various notions of meromorphic and holomorphic differential forms on singular analytic varieties in particular we find that every meromorphic form gives rise to a canonical principal value current with support in the variety demanding that this current be closed we obtain a useful analytic description of the dualizing sheaf we then go on to generalize the lie griffiths theorem on one hand it is shown that a rational trace is enough to ensure algebraicity of the data on the other hand we prove that zero trace implies that the form is holomorphic in the sense of currents in fact we prove a more general local version of the theorem

on the effective nullstellensatz let mathbb k be an algebraically closed field and let x subset mathbb k m be an n dimensional affine variety assume that f1 fk are polynomials which have no common zeros on x we estimate the degrees of polynomials a i in mathbb k x such that 1 ki 1aifi on x our estimate is sharp for k n and nearly sharp for k n now assume that f1 fk are polynomials on x let i f 1 dots f k subset mathbb k x be the ideal generated by fi it is well known that there is a number e i the noether exponent such that ie i i we give a sharp estimate of e i in terms of n deg x and deg fi we also give similar estimates in the projective case finally we obtain a result from the elimination theory if f 1 dots f n in mathbb k x 1 dots x n is a system of polynomials with a finite number of common zeros then we have the following optimal elimination phi i x i sum n j 1 f jg ij quad i 1 dots n where deg f jg ij le prod n i 1 deg f i

simple lie algebras of small characteristic iv solvable and classical roots abstract let l be a finite dimensional simple lie algebra over an algebraically closed field f of characteristic p 3 and t a torus of maximal dimension in the p envelope of l in der l in this paper we describe the t semisimple quotients of the 2 sections of l relative to t and prove that if all 1 sections of l relative to t are compositionally classical or solvable then l is either classical or a block algebra or a filtered lie algebra of type s

l 2 properties of the overline partial and the overline partial neumann operator on spaces with isolated singularities let x be a hermitian complex space of pure dimension with only isolated singularities and pi m rightarrow x a resolution of singularities let omega subset subset x be a domain with no singularities in the boundary omega omega setminus mathrm sing x and omega pi 1 omega we relate l 2 properties of the overline partial and the overline partial neumann operator on omega to properties of the corresponding operators on omega where the situation is classically well understood outside some middle degrees there are compact solution operators for the overline partial equation on omega exactly if there are such operators on the resolution omega and the overline partial neumann operator is compact on omega exactly if it is compact on omega

the bar partial equation on a non reduced analytic space let x be a possibly non reduced analytic space of pure dimension we introduce a notion of overline partial equation on x and prove a dolbeault grothendieck lemma we obtain fine sheaves mathcal a x q of 0 q currents so that the associated dolbeault complex yields a resolution of the structure sheaf mathscr o x our construction is based on intrinsic semi global koppelman formulas on x

various approaches to products of residue currents abstract we describe various approaches to coleff herrera products of residue currents r j of cauchy fantappie leray type associated to holomorphic mappings f j more precisely we study to which extent exterior products of natural regularizations of the individual currents r j yield regularizations of the corresponding coleff herrera products our results hold globally on an arbitrary pure dimensional complex space

residue currents with prescribed annihilator ideals on singular varieties given an ideal mathcal j on a complex manifold andersson and wulcan constructed explicitly a current r mathcal j such that the annihilator of r mathcal j is mathcal j generalizing the duality theorem for coleff herrera products we describe a way to generalize this construction to ideals on singular varieties

integral formulas and an explicit version of the fundamental principle soit p c n c m une application polynomiale et on considere le systeme d equations differentielles p j d f t 0 j 1 m ou d i t et f est une fonction lisse dans un domaine convexe borne r n on presente une version explicite qui emploie des formules integrales et des courants residus du principe fondamental de ehrenpreis et palamodov

decomposition of residue currents given a submodule j o r 0 and a free resolution of j one can de ne a certain vector valued residue current whose annihilator is j we make a decomposition of the current with respect to assj that corresponds to a primary decomposition of j as a tool we introduce a class of currents that includes usual residue and principal value currents in particular these currents admit a certain type of restriction to analytic varieties and more generally to constructible sets

rank one lie algebras let l denote a finite dimensional lie algebra over an algebraically closed field such an algebra is called a rank one lie algebra provided l has a one dimensional cartan subalgebra when d is of characteristic zero a fundamental result in the theory of simple lie algebras over d is that the only rank one algebra is the 3 dimensional lie algebra b t 2 of traceless 2 x 2 matrices the representation theory of t t 2 plays a critical role in the structure and classification of simple algebras over as well as in the study of their representations when the field d is of prime characteristic p 3 in addition to b 2 the albert zassenhaus lie algebras 1 see also 3 are known to be rank one algebras by an albert zassenhaus lie algebra is meant an algebra over d with basis gala e g where g is a finite additive subgroup of d and with multiplication defined by

a geometric effective nullstellensatz the purpose of this paper is to present a geometric theorem which clarifies and extends in several directions work of brownawell kollar and others on the effective nullstellensatz specifically we work on an arbitrary smooth complex projective variety x with the previous classical results corresponding to the case when x is projective space in this setting we prove a local effective nullstellensatz for ideal sheaves and a corresponding global division theorem for adjoint type bundles we also make explicit the connection with the intersection theory of fulton and macpherson finally constructions involving products of prime ideals that appear in earlier work are replaced by geometrically more natural conditions involving orders of vanishing along subvarieties much of the previous activity in this area has been algebraic in nature and seems perhaps not well known in detail among geometers therefore we have felt it worthwhile to include here a rather extended introduction we start with an overview of the questions and earlier work on them then we present the set up and statement of our main theorem we conclude with a series of examples which can be read before the general result of what it yields in special cases

shintani cocycles and the order of vanishing of p adic hecke l series at s 0 let chi be a hecke character of finite order of a totally real number field f by using hill s shintani cocycle we provide a cohomological construction of the p adic l series l p chi s associated to chi this is used to show that l p chi s has a trivial zero at s 0 of order at least equal to the number of places of f above p where the local component of chi is trivial

simple lie algebras of small characteristic v the non melikian case abstract let l be a finite dimensional simple lie algebra over an algebraically closed field f of characteristic p 3 we prove in this paper that if for every torus t of maximal dimension in the p envelope of adl in derl the centralizer of t in adl acts triangulably on l then l is either classical or of cartan type as a consequence we obtain that any finite dimensional simple lie algebra over an algebraically closed field of characteristic p 5 is either classical or of cartan type this settles the last remaining case of the generalized kostrikin shafarevich conjecture the case where p 7

simple lie algebras of small characteristic iii the toral rank 2 case abstract it is proved that every finite dimensional simple lie algebra of absolute toral rank 2 over an algebraically closed field of characteristic p 3 is of classical or cartan type or a melikian algebra

weighted koppelman formulas and the equation on an analytic space let x be an analytic space of pure dimension we introduce a formalism to generate intrinsic weighted koppelman formulas on x that provide solutions to the partial derivative over bar equation we obtain new existence results for the partial derivative over bar equation as well as new proofs of various known results c 2011 elsevier inc all rights reserved

sharp effective nullstellensatz the usual proofs of this result however give no information about the g1 s for instance they give no bound on their degrees this question was first considered by g hermann h she used elimination theory to get a bound on the degree of the gi s which was doubly exponential in the number of variables her results were later improved in mw and in th all these produce bounds that are doubly exponential in the number of variables a major breakthrough was achieved by brownawell b1 who proved the following result

hilbert modular forms and the gross stark conjecture let f be a totally real eld and an abelian totally odd character of f in 1988 gross stated a p adic analogue of stark s conjecture that relates the value of the derivative of the p adic l function associated to and the p adic logarithm of a p unit in the extension of f cut out by in this paper we prove gross s conjecture when f is a real quadratic eld and is a narrow ring class character the main result also applies to general totally real elds for which leopoldt s conjecture holds assuming that either there are at least two primes above p in f or that a certain condition relating the l invariants of and 1 holds this condition on l invariants is always satised when is quadratic

sparse effective membership problems via residue currents we use residue currents on toric varieties to obtain bounds on the degrees of solutions to polynomial ideal membership problems our bounds depend on the volume of the newton polytope of the polynomial system and are therefore well adjusted to sparse polynomial systems we present sparse versions of max nother s af bg theorem macaulay s theorem and kollar s effective nullstellensatz as well as recent results by hickel and andersson gotmark

some l2 results for overline partial on projective varieties with general singularities let x be an irreducible n dimensional projective variety in bbb c p n with arbitrary singular locus we prove that the l 2 overline partial p 1 cohomology groups with respect to the fubini study metric of the regular part of x are finite dimensional

residue currents constructed from resolutions of monomial ideals given a free resolution of an ideal j of holomorphic functions one can construct a vector valued residue current r whose annihilator is precisely j in this paper we compute r in case j is a monomial ideal and the resolution is a cellular resolution in the sense of bayer and sturmfels a description of r is given in terms of the underlying polyhedral cell complex and it is related to irreducible decompositions of j

on the duality theorem on an analytic variety the duality theorem for coleff herrera products on a complex manifold says that if f f 1 f p defines a complete intersection then the annihilator of the coleff herrera product f equals locally the ideal generated by f this does not hold unrestrictedly on an analytic variety z we give necessary and in many cases sufficient conditions for when the duality theorem holds these conditions are related to how the zero set of f intersects certain singularity subvarieties of the sheaf mathcal o z

koppelman formulas on grassmannians we construct koppelman formulas on grassmannians for forms with values in any holomorphic line bundle as well as in the tautological vector bundle and its dual as an application we obtain new explicit proofs of some vanishing theorems of the bott borel weil type by solving the corresponding equation we also relate the projection part of our formulas to the bergman kernels associated to the line bundles

division formulas on projective varieties we introduce a division formula on a possibly singular projective subvariety x of complex projective space mathbb p n which e g provides explicit representations of solutions to various polynomial division problems on the affine part of x especially we consider a global effective version of the briancon skoda huneke theorem

global effective versions of the briancon skoda huneke theorem we prove global effective versions of the briancon skoda huneke theorem our results extend to singular varieties a result of hickel on the membership problem in polynomial ideals in cn and a related theorem of ein and lazarsfeld for smooth projective varieties the proofs rely on known geometric estimates and new results on multivariable residue calculus

koppelman formulas on the a1 singularity in the present paper we study the regularity of the andersson samuelsson koppelman integral operator on the a 1 singularity particularly we prove l p and c 0 estimates as applications we obtain l p homotopy formulas for the dbar equation on the a 1 singularity and we prove that the a forms introduced by andersson samuelsson are continuous on the a 1 singularity

contributions to the classification of simple modular lie algebras we develop results directed towards the problem of classifying the finite dimensional simple lie algebras over an algebraically closed field of characteristic p 7 a 1 section of such a lie algebra relative to a torus t of maximal absolute toral rank possesses a unique subalgebra maximal with respect to having a composition series with factors which are abelian or classical simple in this paper we show that the sum q of those compositionally classical subalgebras is a subalgebra this extends to the general case a crucial step in the classification by block and wilson of the restricted simple lie algebras we derive properties of the filtration which can be constructed using q and obtain structural information about the 1 sections and 2 sections of q relative to t we further classify all those algebras in which q is solvable

integral representation with weights ii division and interpolation let f be a r m matrix of holomorphic functions that is generically surjective we provide explicit integral representation of holomorphic such that f provided that is holomorphic and annihilates a certain residue current with support on the set where f is not surjective we also consider formulas for interpolation as applications we obtain generalizations of various results previously known for the case r 1

the classification of the simple modular lie algebras ii the toral structure abstract let l be a simple lie algebra over an algebraically closed field of characteristic p 7 and t an optimal torus in some p envelope l p we determine the action of t on the two sections of l which have been described in st4 we also give some new and noncomputational proofs to determine the conjugacy classes of the tori in w n 1 and of the cartan subalgebras of w 1 n

integral representation with weights i we describe a new approach to representation formulas for holomorphic functions including the cauchy fantappie leray formula and provide a general method to generate weighted formulas

explicit representation of membership in polynomial ideals we introduce a new division formula on projective space which provides explicit solutions to various polynomial division problems with sharp degree estimates we consider simple examples as the classical macaulay theorem as well as a quite recent result by hickel related to the effective nullstellensatz we also obtain a related result that generalizes max noether s classical af bg theorem

shintani zeta functions and gross stark units for totally real fields let f be a totally real number field and let p be a finite prime of f such that p splits completely in the finite abelian extension h of f tate has proposed a conjecture 22 conjecture 5 4 stating the existence of a p unit u in h with absolute values at the places above p specified in terms of the values at zero of the partial zeta functions associated to h f this conjecture is an analogue of stark s conjecture which tate called the brumer stark conjecture gross 12 conjecture 7 6 proposed a refinement of the brumer stark conjecture that gives a conjectural formula for the image of u in fp e where fp denotes the completion of f at p and e denotes the topological closure of the group of totally positive units e of f we present a further refinement of gross s conjecture by proposing a conjectural formula for the exact value of u in fp

segre numbers a generalized king formula and local intersections let mathcal j be an ideal sheaf on a reduced analytic space x with zero set z we show that the lelong numbers of the restrictions to z of certain generalized monge ampere products dd c log f 2 k where f is a tuple of generators of mathcal j coincide with the so called segre numbers of mathcal j introduced independently by tworzewski achilles manaresi and gaffney gassler more generally we show that these currents satisfy a generalization of the classical king formula that takes into account fixed and moving components of vogel cycles associated with mathcal j a basic tool is a new calculus for products of positive currents of bochner martinelli type we also discuss connections to intersection theory

the classification of the simple modular lie algebras iii solution of the classical case we investigate the structure of simple modular lie algebras over an algebraically closed field of characteristic p 7 if the distinguished subalgebra q l t with respect to an optimal torus t in some p envelope lp of l equals l then either l has a two section of type h 2 1 z or l is of classical type

a dolbeault grothendieck lemma on complex spaces via koppelman formulas let x be a complex space of pure dimension we introduce fine sheaves a x q of 0 q currents which coincides with the sheaves of smooth forms on the regular part of x so that the associated dolbeault complex yields a resolution of the structure sheaf hol x our construction is based on intrinsic and quite explicit semi global koppelman formulas

the complexity of the word problems for commutative semigroups and polynomial ideals abstract any decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied this bound is achieved by a simple procedure for the semigroup problem