quivers of sections on toric orbifolds abstract starting from a collection of line bundles on a projective toric orbifold x we introduce a stacky analogue of the classical linear series our first main result extends work of king by building moduli stacks of refined representations of labelled quivers we associate one such stack to any collection of line bundles on x to obtain our notion of a stacky linear series as in the classical case x maps to the ambient stack by evaluating sections of line bundles in the collection as a further application we describe a finite sequence of git wall crossings between a n g and g hilb a n for a finite abelian subgroup g sl n k where n 3

orbifold quasimap theory we extend to orbifolds the quasimap theory of ciocan fontanine and kim adv math 225 6 3022 3051 2010 j geom phys 75 17 47 2014 as well as the genus zero wall crossing results from algebr geom 1 4 400 448 2014 proceedings of the conference on the occasion of mukai s 60th birthday 2015 as a consequence we obtain generalizations of orbifold mirror theorems in particular of the mirror theorem for toric orbifolds recently proved independently by coates et al a mirror theorem for toric stacks

non archimedean geometry of artin fans abstract the purpose of this article is to study the role of artin fans in tropical and non archimedean geometry artin fans are logarithmic algebraic stacks that can be described completely in terms of combinatorial objects so called kato stacks a stack theoretic generalization of k kato s notion of a fan every logarithmic algebraic stack admits a tautological strict morphism x x a x to an associated artin fan the main result of this article is that on the level of underlying topological spaces the natural functorial tropicalization map of x is nothing but the non archimedean analytic map associated to x by applying thuillier s generic fiber functor using this framework we give a reinterpretation of the main result of abramovich caporaso payne identifying the moduli space of tropical curves with the non archimedean skeleton of the corresponding algebraic moduli space

ehrhart polynomials and stringy betti numbers we study the connection between stringy betti numbers of gorenstein toric varieties and the generating functions of the ehrhart polynomials of certain polyhedral regions we use this point of view to give counterexamples to hibi s conjecture on the unimodality of vectors of reflexive polytopes

the orbifold chow ring of toric deligne mumford stacks the orbifold chow ring of a deligne mumford stack defined by abramovich graber and vistoli 2 is the algebraic version of the orbifold cohomology ring in troduced by w chen and ruan 7 8 by design this ring incorporates numerical invariants such as the orbifold euler characteristic and the orbifold hodge num bers of the underlying variety the product structure is induced by the degree zero part of the quantum product in particular it involves gromov witten invariants inspired by string theory and results in batyrev 3 and yasuda 28 one expects that in nice situations the orbifold chow ring coincides with the chow ring of a resolution of singularities fantechi and g ttsche 14 and uribe 25 verify this conjecture when the orbifold is symn 5 where 5 is a smooth projective surface with ks 0 and the resolution is hilbn the initial motivation for this project was to compare the orbifold chow ring of a simplicial toric variety with the chow ring of a cr pant resolution to achieve this goal we first develop the theory of toric deligne mumford stacks modeled on simplicial toric varieties a toric deligne mumford stack corresponds to a combinatorial object called a stacky fan as a first approximation this object is a simplicial fan with a distinguished lattice point on each ray in the fan more precisely a stacky fan s is a triple consisting of a finitely generated abelian group n a simplicial fan e in q z n with n rays and a map zn n where the image of the standard basis in zn generates the rays in e a rational simplicial fan e produces a canonical stacky fan s n e where n is the distinguished lattice and is the map defined by the minimal lattice points on the rays hence there is a natural toric deligne mumford stack associated to every simplicial toric variety a stacky fan encodes a group action on a quasi affine variety and the toric deligne mumford stack is the quotient if e corresponds to a smooth toric variety x e and s is the canonical stacky fan associated to e then we simply have x z we show that many of the basic concepts such as open and closed toric substacks line bundles and maps between toric deligne mumford stacks correspond to combinatorial notions we expect that many more results about toric varieties lift to the realm of stacks and hope that toric deligne mumford stacks will serve as a useful testing ground for general theories

applications of homological mirror symmetry to hypergeometric systems duality conjectures abstract homological mirror symmetry for crepant resolutions of gorenstein toric singularities leads to a pair of conjectures on certain hypergeometric systems of pdes we explain these conjectures and verify them in some cases

gross fibrations syz mirror symmetry and open gromov witten invariants for toric calabi yau orbifolds given a toric calabi yau orbifold x whose underlying toric variety is semi projective we construct and study a non toric lagrangian torus bration on x which we call the gross bration we apply the strominger yau zaslow recipe to the gross bration of a toric modication of x to construct its instanton corrected mirror where the instanton corrections come from genus 0 open orbifold gromov witten invariants which are virtual counts of holomorphic orbi disks inx bounded by bers of the gross bration we explicitly evaluate all these invariants by rst proving an open closed equality and then employing the toric mirror theorem for suitable toric parital compactications of x our calculations are then applied to 1 prove a conjecture of gross siebert on a relation between genus 0 open orbifold gromov witten invariants and mirror maps ofx this is called the open mirror theorem which leads to an enumerative meaning of mirror maps and 2 demonstrate how open orbifold gromov witten invariants for toric calabi yau orb ifolds change under toric crepant resolutions this is an open analogue of ruan s crepant resolution conjecture

mellin barnes integrals as fourier mukai transforms we study the generalized hypergeometric system introduced by gelfand kapranov and zelevinsky and its relationship with the toric deligne mumford dm stacks recently studied by borisov chen and smith we construct series solutions with values in a combinatorial version of the chen ruan orbifold cohomology and in the k theory of the associated dm stacks in the spirit of the homological mirror symmetry conjecture of kontsevich we show that the k theory action of the fourier mukai functors associated to basic toric birational maps of dm stacks are mirrored by analytic continuation transformations of mellin barnes type

the mori program and non fano toric homological mirror symmetry in the case of toric varieties we continue the pursuit of kontsevich s fundamental insight homological mirror symmetry by unifying it with the mori program we give a refined conjectural version of homological mirror symmetry relating semi orthogonal decompositions of the b model on toric varieties to semi orthogonal decompositions on the a model on the mirror landau ginzburg models as evidence we prove a new case of homological mirror symmetry for a toric surface whose anticanonical bundle is not nef namely a certain blow up of p at three infinitesimally near points

toric stacks ii intrinsic characterization of toric stacks the purpose of this paper and its prequel is to introduce and develop a theory of toric stacks which encompasses and extends several notions of toric stacks defined in the literature as well as classical toric varieties while the focus of the prequel is on how to work with toric stacks the focus of this paper is how to show a stack is toric for toric varieties a classical result says that a finite type scheme with an action of a dense open torus arises from a fan if and only if it is normal and separated in the same spirit the main result of this paper is that any artin stack with an action of a dense open torus arises from a stacky fan under reasonable hypotheses

lagrangian floer superpotentials and crepant resolutions for toric orbifolds we investigate the relationship between the lagrangian floer superpotentials for a toric orbifold and its toric crepant resolutions more specifically we study an open string version of the crepant resolution conjecture crc which states that the lagrangian floer superpotential of a gorenstein toric orbifold mathcal x and that of its toric crepant resolution y coincide after analytic continuation of quantum parameters and a change of variables relating this conjecture with the closed crc we find that the change of variable formula which appears in closed crc can be explained by relations between open orbifold gromov witten invariants we also discover a geometric explanation in terms of virtual counting of stable orbi discs for the specialization of quantum parameters to roots of unity which appears in ruan s original crc gromov witten theory of spin curves and orbifolds contemp math amer math soc providence ri pp 117 126 2006 we prove the open crc for the weighted projective spaces mathcal x mathbb p 1 ldots 1 n using an equality between open and closed orbifold gromov witten invariants along the way we also prove an open mirror theorem for these toric orbifolds

blow up of h1 solution for the nonlinear schrodinger equation we consider the blow up of the solution in h1 for the following nonlinear schrodinger equation i tu u u p 1u x rn t 0 u 0 x u0 x x rn t 0 where n 2 and 1 4 n p min n 2 n 2 5 we prove that if the initial data u0 in h1 are radially symmetric and have negative energy then the solution of in h1 blows up in finite time we do not assume that xu0 l2 and therefore our result is the generalization of the results of glassey 4 and m tsutsumi 18 for the radially symmetric case

on the defect of compactness for the strichartz estimates of the schrodinger equations abstract in this paper we prove that every sequence of solutions to the linear schrodinger equation with bounded data in h1 r d d 3 can be written up to a subsequence as an almost orthogonal sum of sequences of the type h d 2 2nv t tn h2n x xn hn where v is a solution of the linear schrodinger equation with a small remainder term in strichartz norms using this decomposition we prove a similar one for the defocusing h1 critical nonlinear schrodinger equation assuming that the initial data belong to a ball in the energy space where the equation is solvable this implies in particular the existence of an a priori estimate of the strichartz norms in terms of the energy

on the blow up phenomenon of the critical nonlinear schrodinger equation abstract in this paper we consider the blow up phenomenon of critical nonlinear schrodinger equations in dimension 1d and 2d we define the minimal mass as the l 2 norm necessary to ignite a wave collapse and we stress its role in the blow up mechanism asymptotic compactness properties and l 2 concentration are proved the proof relies on linear and nonlinear profile decompositions

the focusing energy critical nonlinear schrodinger equation in dimensions five and higher we consider the focusing energy critical nonlinear schr odinger equation iu t delta u u 4 over d 2 u in dimensions d geq 5 we prove that if a maximal lifespan solution u colon i times bbb r d to bbb c obeys sup t in i nabla u t 2 nabla w 2 then it is global and scatters both forward and backward in time here w denotes the ground state which is a stationary solution of the equation in particular if a solution has both energy and kinetic energy less than those of the ground state w at some point in time then the solution is global and scatters we also show that any solution that blows up with bounded kinetic energy must concentrate at least the kinetic energy of the ground state similar results were obtained by kenig and merle for spherically symmetric initial data and dimensions d 3 4 5

global wellposedness of defocusing critical nonlinear schrodinger equation in the radial case is globally wellposed in time more precisely we obtain a unique solution u u ch1 0 such that for all time u t depends continuously on the data in fact the dependence is even real analytic here moreover there is scattering for t the same statement holds for radial data h s 1 and proves in particular global existence of classical solutions in the radially symmetric case also this issue was open thus this is the analogue for nls of the result for the wave equation with quintic nonlinearity obtained by m struwe str in the radial case and by m grillakis gr s s in general in the case of the wave equation the proof is based on the following two different facts

global spectra polytopes and stacky invariants given a convex polytope we define its geometric spectrum a stacky version of batyrev s stringy e functions and we prove a stacky version of a formula of libgober and wood about the e polynomial of a smooth projective variety as an application we get a closed formula for the variance of the geometric spectrum and a noether s formula for two dimensional fano polytopes polytopes whose vertices are primitive lattice points a fano polytope is not necessarily smooth we also show that this geometric spectrum is equal to the algebraic spectrum the spectrum at infinity of a tame laurent polynomial whose newton polytope is the polytope alluded to this gives an explanation and some positive answers to hertling s conjecture about the variance of the spectrum of tame regular functions

the quantum orbifold cohomology of weighted projective spaces we calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces we generalize givental s heuristic argument which relates small quantum cohomology to s1 equivariant floer cohomology of loop space to weighted projective spaces and use this to conjecture an explicit formula for the small j function a generating function for certain genus zero gromov witten invariants we prove this conjecture using a method due to bertram this provides the first non trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known in addition we obtain formulas for the small j functions of weighted projective complete intersections satisfying a combinatorial condition this condition naturally singles out the class of orbifolds with terminal singularities

global well posedness scattering and blow up for the energy critical focusing non linear wave equation we study the energy critical focusing non linear wave equation with data in the energy space in dimensions 3 4 and 5 we prove that for cauchy data of energy smaller than the one of the static solution w which gives the best constant in the sobolev embedding the following alternative holds if the initial data has smaller norm in the homogeneous sobolev space h1 than the one of w then we have global well posedness and scattering if the norm is larger than the one of w then we have break down in finite time

canonical artin stacks over log smooth schemes we develop a theory of toric artin stacks extending the theories of toric deligne mumford stacks developed by borisov chen smith fantechi mann nironi and iwanari we also generalize the chevalley shephard todd theorem to the case of diagonalizable group schemes these are both applications of our main theorem which shows that a toroidal embedding x is canonically the good moduli space in the sense of alper of a smooth log smooth artin stack whose stacky structure is supported on the singular locus of x

holomorphic orbi discs and lagrangian floer cohomology of symplectic toric orbifolds we develop floer theory of lagrangian torus fibers in compact symplectic toric orbifolds we first classify holomorphic orbi discs with boundary on lagrangian torus fibers we show that there exists a class of basic discs such that we have one to one correspondences between a smooth basic discs and facets of the moment polytope and b between basic orbi discs and twisted sectors of the toric orbifold we show that there is a smooth lagrangian floer theory of these torus fibers which has a bulk deformation by fundamental classes of twisted sectors of the toric orbifold we show by several examples that such bulk deformation can be used to illustrate the very rigid hamiltonian geometry of orbifolds we define its potential and bulk deformed potential and develop the notion of leading order potential we study leading term equations analogous to the case of toric manifolds by fukaya oh ohta and ono

on the better behaved version of the gkz hypergeometric system we consider a version of the generalized hypergeometric system introduced by gelfand kapranov and zelevinsky gkz suited for the case when the underlying lattice is replaced by a finitely generated abelian group in contrast to the usual gkz hypergeometric system the rank of the better behaved gkz hypergeometric system is always the expected one we give largely self contained proofs of many properties of this system the discussion is intimately related to the study of the variations of hodge structures of hypersurfaces in algebraic tori

frobenius morphisms and derived categories on two dimensional toric deligne mumford stacks abstract for a toric deligne mumford dm stack x we can consider a certain generalization of the frobenius endomorphism for such an endomorphism f x x on a 2 dimensional toric dm stack x we show that the push forward f o x of the structure sheaf generates the bounded derived category of coherent sheaves on x we also choose a full strong exceptional collection from the set of direct summands of f o x in several examples of two dimensional toric dm orbifolds x

z linear gale duality and poly weighted spaces pws abstract the present paper is devoted to discussing gale duality from the z linear algebraic point of view this allows us to isolate the class of q factorial complete toric varieties whose class group is torsion free here called poly weighted spaces pws it provides an interesting generalization of weighted projective spaces wps

pavages des simplexes schemas de graphes recolles et compactification des pglrn 1 pglr some equivariant compactifications of the quotients pglrn 1 pglr are constructed each one is decomposed into locally closed strata which are smooth are indexed by the entire convex pavings of the simplex of dimension n and admit a modular interpretation deduced from that of the grassmann varieties together they form a simplicial scheme which compactifies the classifying simplicial scheme of pglr consisting of all the quotients pglrn 1 pglr n 0

hamiltonian group actions on symplectic deligne mumford stacks and toric orbifolds abstract we develop differential and symplectic geometry of differentiable deligne mumford stacks orbifolds including hamiltonian group actions and symplectic reduction as an application we construct new examples of symplectic toric dm stacks

an integral structure in quantum cohomology and mirror symmetry for toric orbifolds abstract we introduce an integral structure in orbifold quantum cohomology associated to the k group and the class in the case of compact toric orbifolds we show that this integral structure matches with the natural integral structure for the landau ginzburg model under mirror symmetry by assuming the existence of an integral structure we give a natural explanation for the specialization to a root of unity in y ruan s crepant resolution conjecture yongbin ruan the cohomology ring of crepant resolutions of orbifolds in contemp math vol 403 amer math soc providence ri 2006 pp 117 126

quantum cohomology and toric minimal model programs abstract we give a quantum version of the danilov jurkiewicz presentation of the cohomology of a compact toric orbifold with projective coarse moduli space more precisely we construct a canonical isomorphism from a formal version of the batyrev ring from 4 to the quantum orbifold cohomology at a canonical bulk deformation this isomorphism generalizes results of givental 23 iritani 34 and fukaya oh ohta ono 21 for toric manifolds and coates lee corti tseng 11 for weighted projective spaces the proof uses a quantum version of kirwan surjectivity theorem 2 6 below and an equality of dimensions theorem 4 19 below deduced using a toric minimal model program tmmp we show that there is a natural decomposition of the quantum cohomology where summands correspond to singularities in the tmmp each of which gives rise to a collection of hamiltonian non displaceable lagrangian tori

the integral orbifold chow ring of toric deligne mumford stacks in this paper we study the integral chow ring of toric deligne mumford stacks we prove that the integral chow ring of a semi projective toric deligne mumford stack is isomorphic to the stanley reisner ring of the associated stacky fan the integral orbifold chow ring is also computed our results are illustrated with several examples

orbifold cohomology for global quotients let x be an orbifold that is a global quotient of a manifold y by a finite group g we construct a noncommutative ring h sp ast y g with a g action such that h sp y g sp g is the orbifold cohomology ring of x defined by w chen and y ruan cr when y s sp n with s a surface with trivial canonical class and g mathfrak s sb n we prove that a small modification of the orbifold cohomology of x is naturally isomorphic to the cohomology ring of the hilbert scheme s sp n computed by m lehn and c sorger ls2

a new cohomology theory of orbifold based on the orbifold string theory model in physics we construct a new cohomology ring for any almost complex orbifold the key theorem is the associativity of this new ring some examples are computed

global existence and scattering for rough solutions of a nonlinear schr odinger equation on r 3 we prove global existence and scattering for the defocusing cubic nonlinear schrodinger equation in h s r 3 for s 4 the main new estimate in the argument is a morawetz type inequality for the solution this estimate bounds k x t k l 4 r3 r whereas the well known morawetz type estimate of lin strauss controls r 1 0 r r3 x t 4 x dxdt

mori dream stacks we propose a generalisation of mori dream spaces to stacks we show that this notion is preserved under root constructions and taking abelian gerbes unlike the case of mori dream spaces such a stack is not always given as a quotient of the spectrum of its cox ring by the picard group we give a criterion when this is true in terms of mori dream spaces and root constructions finally we compare this notion with that of smooth toric deligne mumford stacks

toric stacks i the theory of stacky fans the purpose of this paper and its sequel is to introduce and develop a theory of toric stacks which encompasses and extends several notions of toric stacks defined in the literature as well as classical toric varieties in this paper we define a toric stack as the stack quotient of a toric variety by a subgroup of its torus we also define a generically stacky version any toric stack arises from a combinatorial gadget called a stacky fan we develop a dictionary between the combinatorics of stacky fans and the geometry of toric stacks stressing stacky phenomena such as canonical stacks and good moduli space morphisms we also show that smooth toric stacks carry a moduli interpretation extending the usual moduli interpretations of p n and a 1 g m indeed smooth toric stacks precisely solve moduli problems specified by generalized effective cartier divisors with given linear relations and given intersection relations smooth toric stacks therefore form a natural closure to the class of moduli problems introduced for smooth toric varieties and smooth toric dm stacks in papers by cox and perroni respectively we include a plethora of examples to illustrate the general theory we hope that this theory of toric stacks can serve as a companion to an introduction to stacks in much the same way that toric varieties can serve as a companion to an introduction to schemes

weighted ehrhart theory and orbifold cohomology we introduce the notion of a weighted d vector of a lattice polytope although the definition is motivated by motivic integration we study weighted d vectors from a combinatorial perspective we present a version of ehrhart reciprocity and prove a change of variables formula we deduce a new geometric interpretation of the coefficients of the ehrhart d vector more specifically they are sums of dimensions of orbifold cohomology groups of a toric stack

monodromy of monomially admissible fukaya seidel categories mirror to toric varieties abstract mirror symmetry for a toric variety involves laurent polynomials whose symplectic topology is related to the algebraic geometry of the toric variety we show that there is a monodromy action on the monomially admissible fukaya seidel categories of these laurent polynomials as the arguments of their coefficients vary that corresponds under homological mirror symmetry to tensoring by a line bundle naturally associated to the monomials whose coefficients are rotated in the process we introduce the monomially admissible fukaya seidel category as a new interpretation of the fukaya seidel category of a laurent polynomial on c n which has other potential applications and give evidence of homological mirror symmetry for non compact toric varieties

framed sheaves on root stacks and supersymmetric gauge theories on ale spaces we develop a new approach to the study of supersymmetric gauge theories on ale spaces using the theory of framed sheaves on root toric stacks which illuminates relations with gauge theories on r 4 and with two dimensional conformal field theory we construct a stacky compactification of the minimal resolution xk of the ak 1 toric singularity c 2 zk which is a projective toric orbifold xk such that xknxk is a zk gerbe we construct moduli spaces of tor sion free sheaves on xk which are framed along the compactification gerbe we prove that this moduli space is a smooth quasi projective variety compute its dimension and classify its fixed points under the natural induced toric action we use this construction to compute the partition functions and correlators of chiral bps operators forn 2 quiver gauge theories on xk with nontrivial holonomies at infinity the partition functions are computed with and without cou plings to bifundamental matter hypermultiplets and expressed in terms of toric blowup formulas which relate them to the corresponding nekrasov partition functions on the affine toric subsets of xk we compare our new partition functions with previous computations explore their con nections to the representation theory of affine lie algebras and find new constraints on fractional instanton charges in the coupling to fundamental matter we show that the partition functions in the low energy limit are characterised by the seiberg witten curves and in some cases also by suitable blowup equations involving riemann theta functions on the seiberg witten curve with characteristics related to the nontrivial holonomies

the orbifold topological vertex abstract we develop a topological vertex formalism for computing the donaldson thomas invariants of calabi yau orbifolds the basic combinatorial object is the orbifold vertex v g a generating function for the number of 3d partitions asymptotic to 2d partitions and colored by representations of a finite abelian group g acting on c 3 in the case where g z n acting on c 3 with transverse a n 1 quotient singularities we give an explicit formula for v g in terms of schur functions we discuss applications of our formalism to the donaldson thomas crepant resolution conjecture and to the orbifold donaldson thomas gromov witten correspondence we also explicitly compute the donaldson thomas partition function for some simple orbifold geometries the local football p a b 1 and the local b z 2 gerbe

tropical geometry and correspondence theorems via toric stacks in this paper we generalize correspondence theorems of mikhalkin and nishinou siebert providing a correspondence between algebraic and parameterized tropical curves we also give a description of a canonical tropicalization procedure for algebraic curves motivated by berkovich s construction of skeletons of analytic curves under certain assumptions we construct a one to one correspondence between algebraic curves satisfying toric constraints and certain combinatorially defined objects called stacky tropical reductions that can be enumerated in terms of tropical curves satisfying linear constraints similarly we construct a one to one correspondence between elliptic curves with fixed j invariant satisfying toric constraints and stacky tropical reductions that can be enumerated in terms of tropical elliptic curves with fixed tropical j invariant satisfying linear constraints our theorems generalize previously published correspondence theorems in tropical geometry and our proofs are algebra geometric in particular the theorems hold in large positive characteristic

smooth toric deligne mumford stacks we give a geometric definition of smooth toric deligne mumford stacks using the action of a torus we show that our definition is equivalent to the one of borisov chen and smith in terms of stacky fans in particular we give a geometric interpretation of the combinatorial data contained in a stacky fan we also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks

characteristic classes of singular toric varieties in this paper we compute the motivic chern classes and homology hirzebruch characteristic classes of possibly singular toric varieties which in the context of complete toric varieties fit nicely with a generalized hirzebruch riemann roch theorem as important special cases we obtain new or recover well known formulae for the baum fulton macpherson todd or macpherson s chern classes of toric varieties as well as for the thom milnor l classes of simplicial projective toric varieties we present two different perspectives for the computation of these characteristic classes of toric varieties first we take advantage of the torus orbit decomposition and the motivic properties of the motivic chern and respectively homology hirzebruch classes to express the latter in terms of dualizing sheaves and respectively the dual todd classes of closures of orbits this method even applies to torus invariant subspaces of a given toric variety the obtained formula is then applied to weighted lattice point counting in lattice polytopes and their subcomplexes yielding generalized pick type formulae second in the case of simplicial toric varieties we compute our characteristic classes by using the lefschetz riemann roch theorem of edidin graham in the context of the geometric quotient description of such varieties in this setting we define mock hirzebruch classes of simplicial toric varieties which specialize to the mock chern mock todd and mock l classes of such varieties and investigate the difference between the actual homology hirzebruch class and the mock hirzebruch class we show that this difference is localized on the singular locus and we obtain a formula for it in which the contribution of each singular cone is identified explicitly finally the two methods of computing characteristic classes are combined for proving several characteristic class formulae originally obtained by cappell and shaneson in the early 1990s 2015 wiley periodicals inc

mathbb c 2 mathbb z _ n fractional branes and monodromy we construct geometric representatives for the mathbb c 2 mathbb z n fractional branes in terms of branes wrapping certain exceptional cycles of the resolution in the process we use large radius and conifold type monodromies and also check some of the orbifold quantum symmetries we find the explicit seiberg duality which connects our fractional branes to the ones given by the mckay correspondence we also comment on the harvey moore bps algebras

on clifford double mirrors of toric complete intersections abstract we present a construction of noncommutative double mirrors to complete intersections in toric varieties this construction unifies existing sporadic examples and explains the underlying combinatorial and physical reasons for their existence

the crepant transformation conjecture for toric complete intersections abstract let x and y be k equivalent toric deligne mumford stacks related by a single toric wall crossing we prove the crepant transformation conjecture in this case fully equivariantly and in genus zero that is we show that the equivariant quantum connections for x and y become gauge equivalent after analytic continuation in quantum parameters furthermore we identify the gauge transformation involved which can be thought of as a linear symplectomorphism between the givental spaces for x and y with a fourier mukai transformation between the k groups of x and y via an equivariant version of the gamma integral structure on quantum cohomology we prove similar results for toric complete intersections we impose only very weak geometric hypotheses on x and y they can be non compact for example and need not be weak fano or have gorenstein coarse moduli space our main tools are the mirror theorems for toric deligne mumford stacks and toric complete intersections and the mellin barnes method for analytic continuation of hypergeometric functions

global well posedness and scattering for the defocusing energy critical nonlinear schrodinger equation in r1 4 we obtain global well posedness scattering uniform regularity and global l6t x spacetime bounds for energy space solutions to the defocusing energy critical nonlinear schrodinger equation in a 4 our arguments closely follow those of colliander keel et al though our derivation of the frequency localized interaction morawetz estimate is somewhat simpler as a consequence our method yields a better bound on the l6t x norm

on the conjecture of king for smooth toric deligne mumford stacks abstract we construct full strong exceptional collections of line bundles on smooth toric fano deligne mumford stacks of picard number at most two and of any picard number in dimension two it is hoped that the approach of this paper will eventually lead to the proof of the existence of such collections on all smooth toric nef fano deligne mumford stacks

a derham model for chen ruan cohomology ring of abelian orbifolds we present a derham model for chen ruan cohomology ring of abelian orbifolds we introduce the notion of twist factors so that formally the stringy cohomology ring can be defined without going through pseudo holomorphic orbifold curves thus our model can be viewed as the classical description of chen ruan cohomology for abelian orbifolds the model simplifies computation of chen ruan cohomology ring using our model we give a version of wall crossing formula

computing genus zero twisted gromov witten invariants twisted gromov witten invariants are intersection numbers in moduli spaces of stable maps to a manifold or orbifold x which depend in addition on a vector bundle over x and an invertible multiplicative characteristic class special cases are closely related to local gromov witten invariants of the bundle and to genus zero one point invariants of complete intersections in x we develop tools for computing genus zero twisted gromov witten invariants of orbifolds and apply them to several examples we prove a quantum lefschetz theorem that expresses genus zero one point gromov witten invariants of a complete intersection in terms of those of the ambient orbifold x we determine the genus zero gromov witten potential of the type a surface singularity c2 zn we also compute some genus zero invariants of c3 z3 verifying predictions of aganagic bouchard and klemm 3 in a self contained appendix we determine the relationship between the quantum cohomology of the an surface singularity and that of its crepant resolution thereby proving the crepant resolution conjectures of ruan and of bryan and graber 12 in this case

weighted blowups and mirror symmetry for toric surfaces abstract this paper explores homological mirror symmetry for weighted blowups of toric varietes it will be shown that both the a model and b model categories have natural semi orthogonal decompositions an explicit equivalence of the right orthogonal categories will be shown for the case of toric surfaces

framed sheaves on projective stacks abstract given a normal projective irreducible stack x over an algebraically closed field of characteristic zero we consider framed sheaves on x i e pairs e e where e is a coherent sheaf on x and e is a morphism from e to a fixed coherent sheaf f after introducing a suitable notion of semi stability we construct a projective scheme which is a moduli space for semistable framed sheaves with fixed hilbert polynomial and an open subset of it which is a fine moduli space for stable framed sheaves if x is a projective irreducible orbifold of dimension two and f a locally free sheaf on a smooth divisor d x satisfying certain conditions we consider d f framed sheaves i e framed sheaves e e with e a torsion free sheaf which is locally free in a neighbourhood of d and e d an isomorphism these pairs are stable for a suitable choice of a parameter entering the semi stability condition and of the polarization of x this implies the existence of a fine moduli space parameterizing isomorphism classes of d f framed sheaves on x with fixed hilbert polynomial which is a quasi projective scheme in an appendix we develop the example of stacky hirzebruch surfaces this is the first paper of a project aimed to provide an algebro geometric approach to the study of gauge theories on a wide class of 4 dimensional riemannian manifolds by means of framed sheaves on stacky compactifications of them in particular in a subsequent paper 20 these results are used to study gauge theories on ale spaces of type a k

multiopinion coevolving voter model with infinitely many phase transitions we consider an idealized model in which individuals changing opinions and their social network coevolve with disagreements between neighbors in the network resolved either through one imitating the opinion of the other or by reassignment of the discordant edge specifically an interaction between x and one of its neighbors y leads to x imitating y with probability 1 and otherwise i e with probability x cutting its tie to y in order to instead connect to a randomly chosen individual building on previous work about the two opinion case we study the multiple opinion situation finding that the model has infinitely many phase transitions in the large graph limit with infinitely many initial opinions moreover the formulas describing the end states of these processes are remarkably simple when expressed as a function of 1