on finite alphabets and infinite bases van glabbeek presented the linear time branching time spectrum of behavioral semantics he studied these semantics in the setting of the basic process algebra bccsp and gave finite sound and ground complete axiomatizations for most of these semantics groote proved for some of van glabbeek s axiomatizations that they are w complete meaning that an equation can be derived if and only if all of its closed instantiations can be derived in this paper we settle the remaining open questions for all the semantics in the linear time branching time spectrum either positively by giving a finite sound and ground complete axiomatization that is w complete or negatively by proving that such a finite basis for the equational theory does not exist we prove that in case of a finite alphabet with at least two actions failure semantics affords a finite basis while for ready simulation completed simulation simulation possible worlds ready trace failure trace and ready semantics such a finite basis does not exist completed simulation semantics also lacks a finite basis in case of an infinite alphabet of actions

a vanishing theorem for log canonical pairs using inversion of adjunction we deduce from nadel s theorem a vanishing property for ideals sheaves on projective varieties a special case of which recovers a result due to bertram ein lazarsfeld this enables us to generalize to a large class of projective schemes certain bounds on castelnuovo mumford regularity previously obtained by bertram ein lazarsfeld in the smooth case and by chardin ulrich for locally complete intersection varieties with rational singularities our results are tested on several examples

some group actions on homotopy spheres of dimension seven and fifteen 0 introduction this paper is based on the simple observation that every 4 plane bundle over s4 admits a natural action of 50 3 by bundle maps and that every 8 plane bundle over 58 admits a natural action of the compact lie group g2 by bundle maps these actions are easy to see once one remembers that so 3 is the group of automorphisms of the quaternions and that g2 is the group of automorphisms of the cayley numbers the study of these actions is closely connected to several well known phenomena in differential topology and compact transformation groups the most obvious connection is to milnor s original construction of exotic 7 spheres as 3 sphere bundles over 54 26 milnor proved that if the euler class of such a bundle is a generator of h4 s4 z then its total space is homeomorphic to 57 he also defined a numerical invariant of the diffeomorphism type and used it to detect an exotic differential structure on some of these sphere bundles subsequently eells and kuiper 12 introduced a refinement of this invariant called the it invariant using the u invariant they proved that the sphere bundle m7 milnor s notation is a generator for the group of homotopy 7 spheres these constructions also work for 7 sphere bundles over 58 32 and the manifold m15 is a generator for bp16 the group of homotopy 15 spheres which bound wn manifolds it is a routine matter to check that milnor s arguments and their subsequent refinements work g equivariantly where g 5so 3 or g2 and we shall do this in sections 2 and 3 in particular each sphere bundle with the correct euler class is g homeomorphic to an orthogonal action on s2 1 where 2n 1 7 or 15 moreover distinct sphere bundles have distinct oriented g diffeomorphism types the proof of the second fact uses an equivariant version of the it invariant as originally defined this invariant takes values in q z however it is well known that in the presence of a g action with s c g its value in q is well defined using

geometric proofs of theorems of ax kochen and ersov we give an algebraic geometric proof of the theorem of ax and kochen on p adic diophantine equations in many variables unlike ax kochen s proof ours does not use any notions from mathematical logic and is based on weak toroidalization of morphisms we also show how this geometric approach yields new proofs of the ax kochen er v s ov transfer principle for local fields and of quantifier elimination theorems of basarab and pas

arcs cords and felts six instances of the linearization principle it is shown how a selection of prominent results in singularity theory and differential geometry can be deduced from it one theorem the rank theorem for maps between spaces of power series

vanishing of principal value integrals on surfaces principal value integrals are associated to multi valued rational differential forms with normal crossings support on a non singular algebraic variety we prove their vanishing on rational surfaces in the context of a conjecture of denef jacobs as an appli cation we obtain a strong vanishing result for candidate poles of p adic and motivic igusa zeta functions

towards the singular locus of the space of arcs we give a theory on the space of arcs x infty of a singular variety x based on the finiteness property of the stable points of x infty we also show how computations can be made in this theory as a consequence we obtain a process of determining invariants of x from its space of arcs x infty and from its spaces of wedges arcs in the space of arcs more precisely we give two concepts of regularity for the stable points of x infty and we show that we have regularity in codimension 1 and for all stable points of the space of arcs of a curve as another application we show concrete examples obtained from the spaces of arcs of noetherian 1 dimensional local rings which are analytically ramified

the stringy e function of the moduli space of rank 2 bundles over a riemann surface of genus 3 we compute the stringy e function or the motivic integral of the moduli space of rank 2 bundles over a riemann surface of genus 3 in doing so we answer a question of batyrev about the stringy e functions of the git quotients of linear representations

motivic height zeta functions we consider a motivic analogue of the height zeta function for integral points of equivariant partial compactifications of affine spaces we establish its rationality and determine its largest pole a major problem of diophantine geometry is to understand the distribution of rational or integral points of algebraic varieties defined over number fields for example a well studied question put forward by manin in 1 is that of an asymptotic expansion for the number of rational integral points of bounded height a basic tool is the height zeta function which is a dirichlet series around 2000 e peyre suggested to consider the analogous problem over function fields which has then an even more geometric flavor since it translates as a problem of enumerative geometry namely counting algebraic curves of given degree and establishing properties of the corresponding generating series in view of the developments of motivic integration by kontsevich denef loeser 11 etc it is natural to look at a more general generating series which not only counts the number of such algebraic curves but takes into account the space they constitute in a suitable hilbert scheme a natural coefficient ring for the generating series is the grothendieck ring of varieties kvark if k is a field this ring is generated as a group by the isomorphism classes of k schemes of finite type the addition being subject to obvious cut and paste relations and the product is induced by the product of k varieties one interest of this generalization is that it also makes sense in a purely geometric context where no counting is available such a situation has been first studied in a paper by kapranov in 15 where the analogy with the mentioned diophantine problem is not pointed out later bourqui made some progress on the motivic analogue of manin s problem see 4 as well as his survey report 5 manuscript received april 14 2014 revised march 3 2015 research of the first author supported by the institut universitaire de france by the national science foundation under agreement no dms 0635607 as well as by the positive project of agence nationale de la recherche under agreement anr 2010 blan 0119 01 research of the second author supported in part by the european research council under the european community s seventh framework programme fp7 2007 2013 erc grant agreement nr 246903 nmnag also partially supported by valcomo anr 13 bs01 0006 american journal of mathematics 138 2016 1 59 c 2016 by johns hopkins university press

nested semantics over finite trees are equationally hard this paper studies nested simulation and nested trace semantics over the language bccsp a basic formalism to express finite process behaviour it is shown that none of these semantics affords finite in equational axiomatizations over bccsp in particular for each of the nested semantics studied in this paper the collection of sound closed in equations over a singleton action set is not finitely based

poincare series of a toric variety abstract for an affine toric variety x we compute the poincare series of the multi index filtration defined by a finite number of monomial divisorial valuations on the ring o x 0 we give an alternative description of the poincare series as an integral with respect to the euler characteristic over the projectivization of the space of germs o x 0 in particular we study divisorial valuations on the ring o c d 0 that arise by considering toric constellations we give an explicit formula for the poincare series and a nice geometric description this generalizes an expression of the poincare series for curves and rational surface singularities

log canonical thresholds in positive characteristic in this paper we study the singularities of pairs in arbitrary characteristic via jet schemes for a smooth variety x in characteristic 0 ein lazarsfeld and musta showed that there is a correspondence between irreducible closed cylinders and divisorial valuations on x via this correspondence one can relate the codimension of a cylinder to the log discrepancy of the corresponding divisorial valuation we now extend this result to positive characteristic in particular we prove musta s log canonical threshold formula avoiding the use of log resolutions making the formula available also in positive characteristic as a consequence we get a comparison theorem via reduction modulo p and a version of inversion of adjunction in positive characteristic

elliptic genera of singular varieties the notions of orbifold elliptic genus and elliptic genus of singular varieties are introduced and the relation between them is studied the elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of calabi yau hypersurfaces in fano gorenstein toric varieties introduced earlier the orbifold elliptic genus is given in terms of the fixed point sets of the action we show that the generating function for the orbifold elliptic genus ellorb x 6n p for symmetric groups 6n acting on n fold products coincides with the one proposed by r dijkgraaf g moore e verlinde and h verlinde the two notions of elliptic genera are conjectured to coincide

the arc space of the grassmannian abstract we study the arc space of the grassmannian from the point of view of the singularities of schubert varieties our main tool is a decomposition of the arc space of the grassmannian that resembles the schubert cell decomposition of the grassmannian itself just as the combinatorics of schubert cells is controlled by partitions the combinatorics in the arc space is controlled by plane partitions sometimes also called 3d partitions a combination of a geometric analysis of the pieces in the decomposition and a combinatorial analysis of plane partitions leads to invariants of the singularities as an application we reduce the computation of log canonical thresholds of pairs involving schubert varieties to an easy linear programming problem we also study the nash problem for schubert varieties showing that the nash map is always bijective in this case

ideal adic semi continuity problem for minimal log discrepancies we discuss the ideal adic semi continuity problem for minimal log discrepancies by musta we study the purely log terminal case and prove the semi continuity of minimal log discrepancies when a kawamata log terminal triple deforms in the ideal adic topology

blakers massey elements and exotic diffeomorphisms of and via geodesics we use the geometry of the geodesics of a certain left invariant metric on the lie group sp 2 to find explicit related formulas for two topological objects the blakers massey element a generator of 6 s 3 and an exotic i e not isotopic to the identity diffeomorphism of s 6 c e duran 2001 these formulas depend on two quaternions and their conjugates and we produce their extensions to the octonions through formulas for a generator of 14 s 7 and exotic diffeomorphisms of s 14 thus giving explicit gluing maps for half of the 15 dimensional exotic spheres expressed as the union of two 15 disks

cohomology theory in birational geometry this is a continuation of 9 where it was shown that k equivalent complex projective manifolds have the same betti numbers by using the theory of p adic integrals and deligne s solution to the weil conjecture the aim of this note is to show that with a little more book keeping work namely by applying faltings p adic hodge theory our p adic method also leads to the equivalence of hodge numbers a result which was previously known via motivic integration

axiomatizing weak simulation semantics over bccsp this paper is devoted to the study of the in equational theory of the largest pre congruences over the language bccsp induced by variations on the classic simulation preorder and equivalence that abstract from internal steps in process behaviours in particular the article focuses on the pre congruences associated with the weak simulation the weak complete simulation and the weak ready simulation preorders we present results on the non existence of finite ground complete in equational axiomatizations for each of these behavioural semantics the axiomatization of those semantics using conditional equations is also discussed in some detail

terminal valuations and the nash problem let x be an algebraic variety of characteristic zero terminal valuations are defined in the sense of the minimal model program as those valuations given by the exceptional divisors on a minimal model over x we prove that every terminal valuation over x is in the image of the nash map and thus it corresponds to a maximal family of arcs through the singular locus of x in dimension two this result gives a new proof of the theorem of fernandez de bobadilla and pe pereira stating that for surfaces the nash map is a bijection

differential arcs and regular types in differential fields we introduce differential arc spaces in analogy to the algebraic arc spaces and show that a differential variety is determined by its arcs at a point using differential arcs we show that if k 1 n is a differentially closed field of characteristic zero with n commuting derivations and p s k is a regular type over k then either p is locally modular or there is a definable subgroup g k of the additive group having a regular generic type that is nonorthogonal to p

representations of symplectic reflection algebras and resolutions of deformations of symplectic quotient singularities we give an equivalence of triangulated categories between the derived category of finitely generated representations of symplectic reflection algebras associated with wreath products with parameter t 0 and the derived category of coherent sheaves on a crepant resolution of the spectrum of the centre of these algebras

partial evaluation and ω completeness of algebraic specifications abstract suppose p x y is a program with two arguments whose first argument has a known value c but whose second argument is not yet known partial evaluation of p c y results or rather should result in a specialized residual program p c y in which as much as possible has been computed on the basis of c in the literature on partial evaluation this is often more or less loosely expressed by saying that partial evaluation amounts to making maximal use of incomplete information in this paper a precise meaning is given to this notion in the context of equational logic initial algebra specification and term rewriting systems if maximal propagation of incomplete information is to be achieved within this context as a first step it is necessary to add equations to the algebraic specification in question until it is complete if ever the basic properties of complete specifications are discussed and some examples of complete specifications as well as of specifications that do not have a finite complete enrichment are given

arc valuations on smooth varieties abstract let x be a nonsingular variety with dim x 2 over an algebraically closed field k of characteristic zero let spec k t x be an arc on x and let v ord be the valuation given by the order of vanishing along we describe the maximal irreducible subset c v of the arc space of x such that val c v v we describe c v both algebraically in terms of the sequence of valuation ideals of v and geometrically in terms of the sequence of infinitely near points associated to v as a corollary we get that v is determined by its sequence of centers also when x is a surface our construction also applies to any divisorial valuation v and in this case c v coincides with the one introduced in l ein r lazarsfeld m musta contact loci in arc spaces compos math 140 2004 1229 1244 example 2 5

the nash problem on arc families of singularities nash 21 proved that every irreducible component of the space of arcs through a singularity corresponds to an exceptional divisor that appears on every resolution he asked if the converse also holds does every such exceptional divisor correspond to an arc family we prove that the converse holds for toric singularities but fails in general

dimensions of jet schemes of log singularities we characterize kawamata log terminal singularities and log canonical singularities by dimensions of jet schemes our main result is theorem 2 4

jet closures and the local isomorphism problem abstract if a morphism of germs of schemes induces isomorphisms of all local jet schemes does it follow that the morphism is an isomorphism this problem is called the local isomorphism problem in this paper we use jet schemes to introduce various closure operations among ideals and relate them to the local isomorphism problem this approach leads to a partial solution of the local isomorphism problem which is shown to have a negative answer in general and a positive one in several situations of geometric interest

germs of arcs on singular algebraic varieties and motivic integration we study the scheme of formal arcs on a singular algebraic variety and its images under truncations we prove a rationality result for the poincare series of these images which is an analogue of the rationality of the poincare series associated to p adic points on a p adic variety the main tools which are used are semi algebraic geometry in spaces of power series and motivic integration a notion introduced by m kontsevich in particular we develop the theory of motivic integration for semi algebraic sets of formal arcs on singular algebraic varieties we prove a change of variable formula for birational morphisms and we prove a geometric analogue of a result of oesterle

the arc space of a toric variety abstract the nash problem on arc families is affirmatively answered for a toric variety by ishii and kollar s paper which also shows the negative answer for general case the nash problem is one of questions about the relation between arc families and valuations in this paper the relation is described clearly for a toric variety the arc space of a toric variety admits an action of the group scheme determined by the torus each orbit on the arc space corresponds to a lattice point in the cone and therefore corresponds to a toric valuation the dominant relation among the orbits is described in terms of the lattice points as a corollary we obtain the answer to the embedded version of the nash problem for an invariant ideal on a toric variety

adjoint ideals along closed subvarieties of higher codimension in this paper we introduce a notion of adjoint ideal sheaves along closed subvarieties of higher codimension and study its local properties using char acteristic p methods when x is a normal gorenstein closed subvariety of a smooth complex variety a we formulate a restriction property of the adjoint ideal sheaf adj x a of a along x involving the l c i ideal sheaf dx of x the proof relies on a modification of generalized test ideals of hara and yoshida 11

birationally rigid hypersurfaces we prove that for n 4 all smooth hypersurfaces of degree n in n are birationally superrigid first discovered in the case n 4 by iskovskikh and manin in a work that started this whole direction of research this property was later conjectured to hold in general by pukhlikov the proof relies on the method of maximal singularities in combination with a formula on restrictions of multiplier ideals

on the existence of a crepant resolution of some moduli spaces of sheaves on an abelian surface let j be an abelian surface with a generic ample line bundle open image in new window for n 1 the moduli space mj 2 0 2n of open image in new window 1 semistable sheaves f of rank 2 with chern classes c1 f 0 c2 f 2n is a singular projective variety endowed with a holomorphic symplectic structure on the smooth locus in this paper we show that there does not exist a crepant resolution of mj 2 0 2n for n 2 this certainly implies that there is no symplectic desingularization of mj 2 0 2n for n 2

f singularities of pairs and inversion of adjunction of arbitrary codimension we generalize the notions of f regular and f pure rings to pairs r mathfrak a t of rings r and ideals mathfrak a subset r with real exponent t 0 and investigate these properties these f singularities of pairs correspond to singularities of pairs of arbitrary codimension in birational geometry via this correspondence we prove a sort of inversion of adjunction of arbitrary codimension which states that for a pair x y of a smooth variety x and a closed subscheme y subsetneq x if the restriction z y z to a normal gorenstein closed subvariety z subsetneq x is klt resp lc then the pair x y z is plt resp lc near z

constructible motivic functions and motivic integration 1 1 in this paper intended to be the first in a series we lay new general foundations for motivic integration and give answers to some important issues in the subject since its creation by maxim kontsevich 23 motivic integration developed quickly and has spread out in many directions in a nutshell in motivic integration numbers are replaced by geometric objects like virtual varieties or motives but classicaly not only numbers are defined using integrals but also interesting classes of functions the previous constructions of motivic integration were all quite geometric and it was quite unclear how they could be generalized to handle integrals depending on parameters the new approach we present here based on cell decomposition allows us to develop a very general theory of motivic integration taking parameters into account more precisely we define a natural class of functions constructible motivic functions which is stable under integration the basic idea underlying our approach is to construct more generally push forward morphisms f which are functorial they satisfy f g f g so that performing motivic integration corresponds to taking the push forward to the point this strategy has many technical advantages in essence it allows to reduce the construction of f to the case of closed immersions and projections and in the latter case we can perform induction on the relative dimension the basic case being that of relative dimension 1

towards boundedness of minimal log discrepancies by the riemann roch theorem we introduce an approach via the riemann roch theorem to the boundedness problem of minimal log discrepancies in fixed dimension after reducing it to the case of a gorenstein terminal singularity firstly we prove that the minimal log discrepancy is bounded if either multiplicity or embedding dimension is bounded secondly we recover the characterization of a gorenstein terminal three fold singularity by reid and the sharp bound for its minimal log discrepancy by markushevich without explicit classification finally we provide the sharp bound for a special four fold singularity whose general hyperplane section has a terminal piece

definable sets motives and p adic integrals 0 1 let x be a scheme reduced and separated of finite type over z for p a prime number one may consider the set x zp of its zp rational points for every n in n there is a natural map n x zp x z p assigning to a zp rational point its class modulo p the image yn p of x zp by n is exactly the set of z p rational points which can be lifted to zp rational points denote by nn p the cardinality of the finite set yn p by a result of the first author 7 the poincare series

proofs of the integral identity conjecture over algebraically closed fields recently it has become well known that the conjectural integral identity is of crucial importance in the motivic donaldson thomas invariants theory for noncommutative calabi yau threefolds the purpose of this article is to consider different versions of the identity for regular functions and formal functions and to give them the positive answer for the algebraically closed ground fields technically the result on the motivic milnor fiber by hrushovski loeser using hrushovski kazhdan s motivic integration and nicaise s computations on motivic integrals on special formal schemes are main tools

inversion of adjunction for local complete intersection varieties we extend inversion of adjunction and semicontinity of minimal log discrepancies from the case of a smooth ambient variety to the case of local complete intersection varieties

on grothendieck rings and algebraically constructible functions we investigate grothendieck rings appearing in real geometry notably for arc symmetric sets and focus on the relative case in analogy with the properties of the ring of algebraically constructible functions defined by mccrory and parusinski we study in particular the duality and link operators including their behaviour with respect to motivic milnor fibres with signs

formal loops iii additive functions and the radon transform to any algebraic variety x and closed 2 form on x we associate the symplectic action functional t which is a function on the formal loop space lx introduced by the authors earlier the correspondence t can be seen as a version of the radon transform we give a characterization of the functions of the form t in terms of factorizability infinitesimal analog of additivity in holomorphic pairs of pants as well as in terms of vertex operator algebras these results will be used in the subsequent paper which will relate the gerbe of chiral differential operators on x whose lien is the sheaf of closed 2 forms and the determinantal gerbe of the tangent bundle of lx whose lien is the sheaf of invertible functions on lx on the level of liens this relation associates to a closed 2 form the invertible function expt

curvature and symmetry of milnor spheres since milnor s discovery of exotic spheres mi one of the most intriguing problems in riemannian geometry has been whether there are exotic spheres with positive curvature it is well known that there are exotic spheres that do not even admit metrics with positive scalar curvature hi on the other hand there are many examples of exotic spheres with positive ricci curvature cf ch1 he po and na and this work recently culminated in wr where it is shown that every exotic sphere that bounds a parallelizable manifold has a metric of positive ricci curvature this includes all exotic spheres in dimension 7 so far however no example of an exotic sphere with positive sectional curvature has been found in fact until now only one example of an exotic sphere with nonnegative sectional curvature was known the so called gromoll meyer sphere gm in dimension 7 as one of our main results we prove

on the generalized nash problem for smooth germs and adjacencies of curve singularities abstract in this paper we explore the generalized nash problem for arcs on a germ of smooth surface given two prime divisors above its special point to determine whether the arc space of one of them is included in the arc space of the other one we prove that this problem is combinatorial and we explore its relation with several notions of adjacency of plane curve singularities

the volume of an isolated singularity we introduce a notion of volume of a normal isolated singularity that gener alizes wahl s characteristic number of surface singularities to arbitrary dimensions we prove a basic monotonicity property of this volume under finite morphisms we draw sev eral consequences regarding the existence of non invertible finite endomorphisms fixing an isolated singularity using a cone construction we deduce that the anticanonical di visor of any smooth projective variety carrying a non invertible polarized endomorphism is pseudoeffective our techniques build on shokurov s b divisors we define the notion of nef weil b divisors and of nef envelopes of b divisors we relate the latter notion to the pull back of weil divisors introduced by de fernex and hacon using the subadditivity theorem for multiplier ideals with respect to pairs recently obtained by takagi we carry over to the isolated singularity case the intersection theory of nef weil b divisors formerly developed by boucksom favre and jonsson in the smooth case

zeta functions and oscillatory integrals for meromorphic functions abstract in the 70 s igusa developed a uniform theory for local zeta functions and oscillatory integrals attached to polynomials with coefficients in a local field of characteristic zero in the present article this theory is extended to the case of rational functions or more generally meromorphic functions f g with coefficients in a local field of characteristic zero this generalization is far from being straightforward due to the fact that several new geometric phenomena appear also the oscillatory integrals have two different asymptotic expansions the usual one when the norm of the parameter tends to infinity and another one when the norm of the parameter tends to zero the first asymptotic expansion is controlled by the poles with negative real parts of all the twisted local zeta functions associated to the meromorphic functions f g c for certain special values c the second expansion is controlled by the poles with positive real parts of all the twisted local zeta functions associated to f g

supersymmetry and the formal loop space abstract for any algebraic super manifold m we define the super ind scheme lm of formal loops and study the transgression map radon transform on differential forms in this context applying this to the super manifold m s x the spectrum of the de rham complex of a manifold x we obtain in particular that the transgression map for x is a quasi isomorphism between the 2 3 truncated de rham complex of x and the additive part of the 1 2 truncated de rham complex of l x the proof uses the super manifold ss x and the action of the lie super algebra sl 1 2 on this manifold this quasi isomorphism result provides a crucial step in the classification of sheaves of chiral differential operators in terms of geometry of the formal loop space

non commutative periods and mirror symmetry in higher dimensions we study an analog for higher dimensional calabi yau manifolds of the standard predictions of mirror symmetry we introduce periods associated with non commutative deformations of calabi yau manifolds these periods define a map on the moduli space of such deformations which is a local isomorphism using these non commutative periods we introduce invariants of variations of semi infinite generalized hodge structures living over the moduli space it is shown that the generating function of such invariants satisfies the system of wdvv equations exactly as in the case of gromov witten invariants we prove that the total collection of rational gromov witten invariants of complete intersection calabi yau manifold can be identified with the collection of invariants of variations of generalized semi infinite hodge structures attached to the mirror variety the basic technical tool utilized is the deformation theory

tools for proving inductive equalities relative completeness and ω completeness abstract inductive theorems are properties valid in the initial algebra a now popular tool for proving them in equational theories or abstract data types is based on proof by consistency this method uses a completion procedure and requires two essential properties of the specification namely relative completeness and completeness this paper investigates ways of proving them for the first one the complement algorithm is presented it is based on unification and computation of coverings and complements for the second one a technique based on discrimination of pairs of normal forms is explained and illustrated through examples