topology of symplectomorphism groups of rational ruled surfaces let m be either s 2 s 2 or the one point blow up cp 2 cp 2 of cp 2 in both cases m carries a family of symplectic forms where 1 determines the cohomology class this paper calculates the rational co homology of the group g of symplectomorphisms of m as well as the rational homotopy type of its classifying space bg it turns out that each group g contains a nite collection kk k 0 of nite dimensional lie subgroups that generate its homotopy we show that these subgroups asymptotically commute i e all the higher whitehead products that they generate vanish as 1 however for each xed there is essentially one nonvanishing product that gives rise to a jumping generator w in h g and to a single relation in the rational cohomology ring h bg an analog of this generator w was also seen by kronheimer in his study of families of symplectic forms on 4 manifolds using seiberg witten theory our methods involve a close study of the space of compatible almost complex structures on m

multilinear calderon zygmund theory abstract a systematic treatment of multilinear calderon zygmund operators is presented the theory developed includes strong type and endpoint weak type estimates interpolation a multilinear t 1 theorem and a variety of results regarding multilinear multiplier operators

new maximal functions and multiple weights for the multilinear calderon zygmund theory a multi sub linear maximal operator that acts on the product of m lebesgue spaces and is smaller than the m fold product of the hardy littlewood maximal function is studied the operator is used to obtain a precise control on multilinear singular integral operators of calderon zygmund type and to build a theory of weights adapted to the multilinear setting a natural variant of the operator which is useful to control certain commutators of multilinear calderon zygmund operators with bmo functions is then considered the optimal range of strong type estimates a sharp end point estimate and weighted norm inequalities involving both the classical muckenhoupt weights and the new multilinear ones are also obtained for the commutators

a bilinear estimate and its application to a quadratic nonlinear klein gordon equation in two space dimensions abstract we study the existence of the wave operators for the nonlinear klein gordon equation with quadratic nonlinearity in two space dimensions t 2 1 u u 2 t x r r 2 we prove existence of wave operators in lower order sobolev spaces by using estimates of bilinear operators associated with klein gordon equation

compatible complex structures on symplectic rational ruled surfaces in this article we study the topology of the spacei of complex structures compatible with a fixed symplectic form using the framework of donaldson by comparing our analysis of the spacei with results of mcduff on the spacej of compatible almost complex structures on rational ruled surfaces we find that i is contractible in this case we then apply this result to study the topology of the symplectomorphism group of a rational ruled surface extending results of abreu and mcduff

a hormander type multiplier theorem for multilinear operators abstract in this paper we prove a hormander type multiplier theorem for multilinear operators as a corollary we can weaken the regularity assumption for multilinear fourier multipliers to assure the boundedness

conics in the grothendieck ring abstract the subring of the grothendieck ring of k varieties generated by smooth conics is described giving many zero divisors the proof uses only elementary projective geometry

poles of maximal order of motivic zeta functions we prove a 1999 conjecture of veys which says that the opposite of the log canonical threshold is the only possible pole of maximal order of denef and loeser s motivic zeta function associated with a germ of a regular function on a smooth variety over a field of characteristic zero we apply similar methods to study the weight function on the berkovich skeleton associated with a degeneration of calabi yau varieties our results suggest that the weight function induces a flow on the non archimedean analytification of the degeneration towards the kontsevich soibelman skeleton

hormander type theorems for multi linear and multi parameter fourier multiplier operators with limited smoothness abstract the main purpose of this paper is three fold first of all we are concerned with the limited smoothness conditions in the spirit of hormander on the multi linear and multi parameter coifman meyer type fourier multipliers studied by c muscalu j pipher t tao c thiele 2004 2006 where they established the l r estimates for the multiplier operators under the assumption that the multiplier has smoothness of sufficiently large order under our limited smoothness assumption we will prove the l p 1 l p n l r boundedness with 1 p 1 1 p n 1 r for 1 p 1 p n and 0 r second our proof of l r estimates also offers a different and more direct approach than the one given in muscalu et al 2004 2006 where they use the deep analysis of multi linear and multi parameter paraproducts third we also prove a hormander type multiplier theorem in the weighted lebesgue spaces for such operators when the fourier multiplier is only assumed with limited smoothness

neron models lie algebras and reduction of curves of genus one let k be a discrete valuation field let ok denote the ring of integers of k and let k be the residue field of ok of characteristic p 0 let s specok let x k be a smooth geometrically connected projective curve of genus 1 over k denote by ek the jacobian of x k let x s and e s be the minimal regular models of x k and ek respectively in this article we investigate the possible relationships between the special fibers xk and ek in doing so we are led to study the geometry of the picard functor pic x s when x s is not necessarily cohomologically flat as an application of this study we are able to prove in full generality a theorem of gordon on the equivalence between the artin tate and birch swinnertondyer conjectures recall that when k is algebraically closed the special fibers of elliptic curves are classified according to their kodaira type which is denoted by a symbol t in i n n z 0 ii ii iii iii iv iv given a type t and a positive integer m we denote by mt the new type obtained from t by multiplying all the multiplicities of t by m when k is algebraically closed the relationships between the type of a curve of genus 1 and the type of its jacobian can be summarized as follows

the grothendieck ring of varieties and piecewise isomorphisms let k0 var k be the grothendieck ring of algebraic varieties over a field k l et x y be two algebraic varieties over k which are piecewise isomorphic i e x and y admit finite partitions x1 xn y1 yn into locally closed subvarieties such that xi is iso morphic to yi for all i n then x y in k0 var k larsen and lunts ask whether the converse is true for characteristic zero and algebraically closedfield k we answer positively this question when dim x 1o rx is a smooth connected projective surface or if x contains only finitely many rational curves

on calderon s conjecture with constants cfi p1 p2 depending only on fi p1 p2 and p p1p2 p1 p2 hold the flrst result of this type is proved in 4 and the purpose of the current paper is to extend the range of exponents p1 and p2 for which 2 is known in particular the case p1 2 p2 1 is solved to the a rmative this was

vanishing cycles for formal schemes let k be a non archimedean field and let x be a formal scheme locally finitely presented over the ring of integers k see 1 in this work we construct and study the vanishing cycles functor from the category of etale sheaves on the generic fibre x of x which is a k analytic space to the category of etale sheaves on the closed fibre xs of x which is a scheme over the residue field of k we prove that if x is the formal completion x of a scheme x finitely presented over k along the closed fibre then the vanishing cycles sheaves of x are canonically isomorphic to those of x as defined in sga7 exp xiii in particular the vanishing cycles sheaves of x depend only on x and any morphism x induces a homomorphism from the pullback of the vanishing cycles sheaves of x under s ys xs to those of y furthermore we prove that for each x one can find a nontrivial ideal of k such that if two morphisms x coincide modulo this ideal then the homomorphisms between the vanishing cycles sheaves induced by and coincide these facts were conjectured by p deligne in 1 we associate with a formal scheme x locally finitely presented over k a k analytic space x in the sense of ber1 and ber2 in 2 we find that the morphism y x which is induced by an etale morphism of formal schemes y x possesses a certain property morphisms of k analytic spaces with this property are called quasi etale and they give rise to a quasi etale site xqet of a k analytic space x there is a canonical morphism of sites xqet xet where xet is the etale site introduced in ber2 we show that the inverse image functor identifies the category of etale sheaves x et with a full subcategory of

vanishing cycles and geometry of curves over a discrete valuation ring 0 introduction geometry of curves over a discrete valuation ring is reflected in the action of the monodromy on the f adic cohomology groups the stable reduction theorem d m gives an equivalent geometrical condition for the action of the inertia group to be unipotent in the global situation where global means that the curve in consideration is proper in this paper we will generalize it in two directions one is to the local situation with the isolated singularity the other is to consider the geometrical condition for the action of the wild ramification group to be trivial we will also give a purely cohomological proof without using picard schemes of the stable reduction theorem and a new proof of the formula of deligne la theorem 5 1 1 based only on the stable reduction theorem here we only treat a strictly local discrete valuation ring with algebraically closed residue field however the results apply to any discrete valuation ring with perfect residue field since they are etale local we introduce some notations and terminologies

a trace formula for varieties over a discretely valued field we study the motivic serre invariant of a smoothly bounded alge braic or rigid variety x over a complete discretely valued field k with perfect residue field k if k has characteristic zero we extend the definition to arbi trary k varieties using bittner s presentation of the grothendieck ring and a process of neron smoothening of pairs of varieties the motivic serre invariant can be considered as a measure for the set of unramified points on x under certain tameness conditions it admits a cohomological interpretation by means of a trace formula in the curve case we use t saito s geometric criterion for cohomological tameness to obtain more detailed results we discuss some applications to weil chatelet groups chow motives and the structure of the grothendieck ring of varieties msc 2000 11s15 14g05 14g22

motivic serre invariants and weil restriction we study the interactions between weil restriction for formal schemes and rigid varieties greenberg schemes and motivic serre invariants and their behavior with respect to finite extensions of the base ring r which is a complete discrete valuation ring with perfect residue field

multilinear littlewood paley estimates with applications to partial differential equations abstract we obtain a collection of multilinear littlewood paley estimates which we then apply to two problems in partial differential equations the first problem is the estimation of the square root of an elliptic operator in divergence form and the second is the estimation of solutions to the cauchy problem for nondivergence form parabolic equations

uber die lokale zetafunktion von shimuravarietaten monodromiefiltration und verschwindende zyklen in ungleicher charakteristik

motivic zeta functions of degenerating calabi yau varieties we study motivic zeta functions of degenerating families of calabi yau varieties our main result says that they satisfy an analog of igusa s monodromy conjecture if the family has a so called galois equivariant kulikov model we provide several classes of examples where this condition is verified we also establish a close relation between the zeta function and the skeleton that appeared in kontsevich and soibelman s non archimedean interpretation of the syz conjecture in mirror symmetry

well posedness for a quadratic derivative nonlinear schrodinger system at the critical regularity abstract we consider the cauchy problem for a quadratic derivative nonlinear schrodinger system in sobolev space h s in this paper under the null condition and the mass resonance relation we prove large data local well posedness if the space dimension d 3 and s s c d 2 and s s c or d 1 and s 0 where s c is the scaling critical regularity moreover under the same assumptions we also prove small data global well posedness and scattering in h s if d 3 and s s c our proof is based on a contraction argument using the u p and v p type function spaces

bilinear estimates in bmo and the navier stokes equations we prove that the bmo norm of the velocity and the vorticity controls the blow up phenomena of smooth solutions to the navier stokes equations our result is applied to the criterion on uniqueness and regularity of weak solutions in the marginal class

the hormander multiplier theorem ii the bilinear local l 2 case we use wavelets of tensor product type to obtain the boundedness of bilinear multiplier operators on mathbb r n times mathbb r n associated with hormander multipliers on mathbb r 2n with minimal smoothness we focus on the local l 2 case and we obtain boundedness under the minimal smoothness assumption of n 2 derivatives we also provide counterexamples to obtain necessary conditions for all sets of indices

discrete decompositions for bilinear operators and almost diagonal conditions this is the published version also available here http dx doi org 10 1090 s0002 9947 01 02912 9 first published in transaction of american mathematical society in 2002 published by the american mathematical society

motivic integration on smooth rigid varieties and invariants of degenerations we develop a theory of motivic integration for smooth rigid varieties as an application we obtain a motivic analogue for rigid varieties of serre s invariant for p adic varieties our construction provides new geometric birational invariants of degenerations of algebraic varieties for degenerations of calabi yau varieties our results take a stronger form

on multilinear littlewood paley operators abstract in this paper we established the weighted estimates for the multilinear littlewood paley operators including the multilinear marcinkiewicz integral and the multilinear area integral of lusin these operators are generalizations of the linear operators for higher dimension which were first defined and studied by e m stein in 1958 strong l p 1 1 l p m m l p estimates when each p i 1 and weak type l p 1 1 l p m m l p estimates if there is a p i 1 are obtained where i 1 m i p p i and each w i i 1 m is a nonnegative function on r n

improved bound for the bilinear bochner riesz operator we study l p times l q rightarrow l r bounds for the bilinear bochner riesz operator mathcal b alpha alpha 0 in mathbb r d d ge 2 which is defined by open image in new window we make use of a decomposition which relates the estimates for mathcal b alpha to the square function estimates for the classical bochner riesz operators in consequence we significantly improve the previously known bounds

algebraic independence in the grothendieck ring of varieties we give sufficient cohomological criteria for the classes of given varieties over a field k to be algebraically independent in the grothendieck ring of varieties over k and construct some examples

a trace formula for rigid varieties and motivic weil generating series for formal schemes we establish a trace formula for rigid varieties x over a complete discretely valued field which relates the set of unramified points on x to the galois action on its etale cohomology next we show that the analytic milnor fiber of a morphism f at a point x completely determines the formal germ of f at x we develop a theory of motivic integration for formal schemes of pseudo finite type over a complete discrete valuation ring r and we introduce the weil generating series of a regular formal r scheme mathfrak x of pseudo finite type via the construction of a gelfand leray form on its generic fiber when mathfrak x is the formal completion of a morphism f from a smooth irreducible variety to the affine line then its weil generating series coincides modulo normalization with the motivic zeta function of f when mathfrak x is the formal completion of f at a closed point x of the special fiber f 1 0 we obtain the local motivic zeta function of f at x

motivic zeta functions of abelian varieties and the monodromy conjecture we prove for abelian varieties a global form of denef and loeser s motivic monodromy conjecture in arbitrary characteristic more precisely we prove that for every tamely ramified abelian variety a over a complete discretely valued field with algebraically closed residue field its motivic zeta function has a unique pole at chai s base change conductor c a of a and that the order of this pole equals one plus the potential toric rank of a moreover we show that for every embedding of ql in c the value exp 2 ic a is an l adic tame monodromy eigenvalue of a the main tool in the paper is edixhoven s filtration on the special fiber of the neron model of a which measures the behavior of the neron model under tame base change

necessary and sufficient conditions for absolute continuity of elliptic harmonic measure our purpose in this paper is to give a necessary and sufficient condition on the modulus of continuity of the coefficients of an elliptic operator in divergence form in order that the corresponding poisson kernel for the dirichlet problem exist this condition on the coefficients is global continuity together with the additional property that the modulus of continuity along some nontangential direction at each boundary point be bounded uniformly in these points and directions by a function 7 t satisfying the dini type condition

the homotopy lie algebra of symplectomorphism groups of 3 fold blow ups of the projective plane by a result of kedra and pinsonnault we know that the topology of groups of symplectomorphisms of symplectic 4 manifolds is complicated in general however in all known very specific examples the rational cohomology rings of symplectomorphism groups are finitely generated in this paper we compute the rational homotopy lie algebra of symplectomorphism groups of the 3 point blow up of the projective plane with an arbitrary symplectic form and show that in some cases depending on the sizes of the blow ups it is infinite dimensional moreover we explain how the topology is generated by the toric structures one can put on the manifold our method involve the study of the space of almost complex structures compatible with the symplectic structure and it depends on the inflation technique of lalonde mcduff

a counterexample to weighted estimates for multilinear fourier multipliers with sobolev regularity abstract the problem whether the weighted estimates for multilinear fourier multipliers with sobolev regularity hold under weak condition on weights is considered

topology of symplectomorphism groups of s 2 s 2 in dimension 4 due to non existence of adequate tools very little is known about the topology of groups of diffeomorphisms for example it is unknown if the group of compactly supported diffeomorphisms of r is connected the situation is much better if one wants to study groups of symplectomorphisms this is due to the existence of powerful tools going by the name of pseudo holomorphic curve techniques and introduced in symplectic geometry by m gromov in his seminal paper of 1985 5 gromov proved in that paper among several other remarkable results the contractibility of the group of compactly supported symplectomorphisms of r with its standard symplectic form dx dy dx dy gromov also studied the following example letm be the symplectic manifold s s 1 0 0 where 0 r and 0 is a standard area form on s with total area equal to 1 denote by g the group of symplectomorphisms of m that act as the identity on h2 s s z gromov proved in 5 that g0 is homotopy equivalent to its subgroup of standard isometries so 3 so 3 he also showed why that would not be true for g with 0 and in 1987 d mcduff 10 constructed explicitly an element of infinite order in h1 g 0 in this paper we will give a more detailed description of g for 0 1 in particular we will prove the following on leave from instituto superior tecnico lisbon portugal partially supported by jnict programas ciencia and praxis fundacao luso americana para o desenvolvimento and a sloan doctoral dissertation fellowship current address school of mathematics institute for advanced study princeton nj 08540 usa e mail abreu math ias edu

geometric criteria for tame ramification we prove an a campo type formula for the tame monodromy zeta function of a smooth and proper variety over a discretely valued field k as a first application we relate the orders of the tame monodromy eigenvalues on the l adic cohomology of a k curve to the geometry of a relatively minimal sncd model and we show that the semi stable reduction theorem and saito s criterion for cohomological tameness are immediate consequences of this result as a second application we compute the error term in the trace formula for smooth and proper k varieties we see that the validity of the trace formula would imply a partial generalization of saito s criterion to arbitrary dimension

on the boundedness of bilinear operators on products of besov and lebesgue spaces abstract we prove mapping properties of the form t b p 1 1 q 1 l p 2 b p 3 2 q 2 and t b p 1 1 q 1 b p 2 2 q 2 l p 3 for certain related indices p 1 p 2 p 3 q 1 q 2 1 2 r where t is a bilinear hormander mihlin multiplier or a molecular paraproduct applications to bilinear littlewood paley theory are discussed

the topology of the space of symplectic balls in rational 4 manifolds we study in this paper the rational homotopy type of the space of symplectic embeddings of the standard ball b4 c r4 into 4 dimensional rational symplectic manifolds where c r2 is the capacity of the standard ball of radius r we compute the rational homotopy groups of that space when the 4 manifold has the form m s2 s2 0 0 where 0 is the area form on the sphere with total area 1 and belongs to the interval 1 2 we show that when is 1 this space retracts to the space of symplectic frames for any value of c however for any given 1 2 the rational homotopy type of that space changes as c crosses the critical parameter 1 which is the difference of areas between the two s2 factors we prove moreover that the full homotopy type of that space changes only at that value that is that the restriction map between these spaces is a homotopy equivalence as long as these values of c remain either below or above that critical value the same methods apply to all other values of and other rational 4 manifolds as well the methods rely on two different tools the study of the action of symplectic groups on the stratified space of almost complex structures developed by gromov abreu and mcduff and the analysis of the relations between the group corresponding to a manifold m the group corresponding to its blow up m and the space of symplectic embedded balls in m

the hormander multiplier theorem for multilinear operators in this paper we provide a version of the mikhlin hormander multiplier theorem for multilinear operators in the case where the target space is l p for pe 1 this extends a recent result of tomita 15 who proved an analogous result for p 1

examples of symplectic structures in this paper we construct symplectic forms tilde omega k k geqq 0 on a compact manifold tilde y which have the same homotopy theoretic invariants but which are not diffeomorphic

existence and boundedness of multilinear littlewood paley operators on campanato spaces abstract in this paper we established the boundedness of multilinear littlewood paley operators on campanato spaces including the multilinear g function multilinear lusin s area integral and multilinear littlewood paley g function we proved that if the above operators are finite for one point then they are finite almost everywhere moreover norm inequalities do hold for the above operators on the corresponding campanato spaces

a relative seidel morphism and the albers map in this note we introduce a relative or lagrangian version of the seidel homomorphism that assigns to each homotopy class of paths in ham m starting at the identity and ending on the subgroup that preserves a given lagrangian submanifold l an element in the floer homology of l we show that these elements are related to the absolute seidel elements by the albers map we also study for later use the effect of reversing the signs of the symplectic structure as well as the orientations of the generators and of the operations on the floer homologies

maximal estimates for the bilinear spherical averages and the bilinear bochner riesz operators abstract we study the maximal estimates for the bilinear spherical average and the bilinear bochner riesz operator first we obtain l p l q l r estimates for the bilinear spherical maximal function on the optimal range thus we settle the problem which was previously considered by geba greenleaf iosevich palsson and sawyer later barrionevo grafakos d he honzik and oliveira and recently heo hong and yang secondly we consider l p l q l r estimates for the maximal bilinear bochner riesz operators and improve the previous known results for the purpose we draw a connection between the maximal estimates and the square function estimates for the classical bochner riesz operators

l 2 times l 2 rightarrow l 1 l2 l2 l1 boundedness criteria we obtain a sharp l 2 times l 2 rightarrow l 1 boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function in terms of the l q integrability of this function precisely we show that boundedness holds if and only if q 4 we discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions our second result is an optimal l 2 times l 2 rightarrow l 1 boundedness criterion for bilinear operators associated with multipliers with l infty derivatives this result provides the main tool in the proof of the first theorem and is also manifested in terms of the l q integrability of the multiplier the optimal range is q 4 which in the absence of plancherel s identity on l 1 should be compared to q infty in the classical l 2 rightarrow l 2 boundedness for linear multiplier operators