embeddings of spherical homogeneous spaces in characteristic p let g be a reductive group over an algebraically closed field of characteristic p 0 we study properties of embeddings of spherical homogeneous g spaces we look at frobenius splittings canonical or by a p 1 th power compatible with certain subvarieties we show the existence of rational g equivariant resolutions by toroidal embeddings and give results about cohomology vanishing and surjectivity of restriction maps of global sections of line bundles we show that the class of homogeneous spaces for which our results hold contains the symmetric homogeneous spaces in characteristic ne 2 and is closed under parabolic induction

monodromy group for a strongly semistable principal bundle over a curve let x be a geometrically irreducible smooth projective curve defined over a field k assume that x has a k rational point fix a k rational point x e x from these data we construct an affine group scheme x defined over the field k as well as a principal x bundle s1755069607000151inline1 over the curve x the group scheme x is given by a xs0211a graded neutral tannakian category built out of all strongly semistable vector bundles over x the principal bundle s1755069607000151inline1 is tautological let g be a linear algebraic group defined over k that does not admit any nontrivial character which is trivial on the connected component containing the identity element of the reduced center of g let e g be a strongly semistable principal g bundle over x we associate to e g a group scheme m defined over k which we call the monodromy group scheme of e g and a principal m bundle e m over x which we call the monodromy bundle of e g the group scheme m is canonically a quotient of x and e m is the extension of structure group of s1755069607000151inline1 the group scheme m is also canonically embedded in the fiber ad e g x over x of the adjoint bundle

the melnikov theory for subharmonics and their bifurcations in forced oscillations the subharmonic melnikov theory for periodic perturbations of planar hamiltonian systems is improved an approximation to the associated poincare map in action angle coordinates is explicitly constructed and existence stability and bifurcation theorems for subharmonics are obtained in particular simple formulas for determining the stability of subharmonics and invariant circles bifurcating from them at hopf bifurcations are obtained and a degenerate resonance case which was not appropriately treated in previous references is discussed furthermore the weak nonlinearity case in which the unperturbed system is linear is studied the results are also useful to describe dynamics near the unperturbed centers in strongly nonlinear systems several examples are given to illustrate our theory

global f regularity of schubert varieties with applications to d modules a projective algebraic variety x over an algebraically closed field k of positive characteristic is called globally f regular if the section ring s c 0n o h x cn of an ample line bundle c on x is strongly f regular in the sense of h chster and huneke 9 cf definition 1 1 this important notion was introduced by karen smith in 18 in this paper we prove that schubert varieties are globally f regular an im mediate consequence is that local rings of schubert varieties are strongly f regular and thereby f rational another consequence is that local rings of varieties like de terminantal varieties that can be identified with open subsets of schubert varieties cf 13 are strongly f regular let x denote a flag variety and y c x a schubert variety over k then the local cohomology sheaves hjy ox are equivariant for the action of the borel subgroup and holonomic in the sense of 4 x x niodules as an application of ir rationality of schubert varieties we apply recent results of blickle cf 2 to prove that the simple objects in the category of equivariant and holonomic px modules are precisely the local cohomology sheaves hy ox where c is the codimension of y in x using a local grothendieck cousin complex from 11 we prove that the decomposition of the local cohomology modules with support in bruhat cells is multiplicity free see 4 2 in characteristic zero the local cohomology modules with support in bruhat cells correspond to dual verma modules in this setting the decomposition behavior and the simple t x modules arise from intersection cohomology complexes of schubert varieties by the riemann hilbert correspondence if we pick the singular codi mension one schubert variety y in the full flag variety z for sl4 in characteristic zero computations in kazhdan lusztig theory show that hy oz is not a simple z module see 4 1

frobenius amplitude and strong vanishing theorems for vector bundles the primary goal of this paper is to systematically exploit the method of deligne illusie to obtain kodaira type vanishing theorems for vector bundles and more generally coherent sheaves on algebraic varieties the key idea is to introduce a number which provides a cohomological measure of the positivity of a coherent sheaf called the frobenius or f amplitude the f amplitude enters into the statement of the basic vanishing theorem and this leads to the problem of calculating or at least estimating this number most of the work in this paper is devoted to doing this various situations

on the flatness of local models for the symplectic group abstract we investigate the bad reduction of certain shimura varieties associated to the symplectic group more precisely we look at a model of the shimura variety at a prime p with parahoric level structure at p we show that this model is flat as conjectured by rapoport and zink ann of math stud 141 1996 and that its special fibre is reduced a crucial ingredient is faltings theorem on the normality of schubert varieties in the affine flag variety

on the coherence conjecture of pappas and rapoport we prove the generalized coherence conjecture proposed by pappas and rapoport as a corollary one of their theorems which describes the geometry of the special bers of the local models for ramied unitary groups holds unconditionally our proof is based on the study of the geometry in particular certain line bundles and adic sheaves of the global schubert varieties which are the equal characteristic counterparts of the local models

f regularity of large schubert varieties let g denote a connected reductive algebraic group over an algebraically closed field k and let x denote a projective g g equivariant embedding of g the large schubert varieties in x are the closures of the double cosets bgb where b denotes a borel subgroup of g and g g we prove that these varieties are globally f regular in positive characteristic resp of globally f regular type in characteristic 0 as a consequence the large schubert varieties are normal and cohen macaulay 1 introduction the class of globally f regular varieties was introduced by smith in 21 these are projective algebraic varieties in positive character istics such that all the ideals in their homogeneous coordinate rings are tightly closed the globally f regular varieties and their analogues in characteristic 0 the varieties of globally f regular type have remarkable properties e g they are normal and cohen macaulay and the higher cohomology groups of all nef invertible sheaves are trivial examples of globally f regular varieties include the projective toric varieties prop 6 4 in 21 in this note we obtain the global f regularity of a wider class of varieties with algebraic group action the g g equivariant projective embeddings of any connected reductive group g and the closures in any such embedding of the double cosets bgb where b denotes a borel subgroup of g and g g is arbitrary by the bruhat decomposition these large schubert varieties are parametrized by the weyl group of g examples are the closures of parabolic subgroups we also show that the large schubert varieties are of globally f regular type in characteristic 0 for this we exploit the close relation between global f regularity and frobe nius splitting established in 21 and the frobenius splitting properties of the large schubert varieties proved in 5 and 19 another key ingredient is the global f regularity of the flag varieties and their schubert varieties 12 note that unlike for schubert varieties no desingularization of large schubert varieties is known in general this makes our arguments somewhat indirect as a consequence of our result the large schubert varieties in any equivariant embedding of g are normal and cohen macaulay this was first proved in the case of the canonical compactification of a semisimple adjoint group by frobenius

on the conjecture o of ggi for g p abstract in this paper we show that general homogeneous manifolds g p satisfy conjecture o of galkin golyshev and iritani which underlies gamma conjectures i and ii of them our main tools are the quantum chevalley formula for g p and a theory on nonnegative matrices including perron frobenius theorem

sufficient stability conditions for the weighted minimum uncertainty prediction controller sufficient bibo stability conditions are provided for the weighted minimum uncertainty prediction controller this controller minimizes a cost functional which penalizes the predicted output uncertainty the control effort and the tracking error the bibo stability is addressed by examining the worst case time variant interval matrix representation of the closed loop system

on the flatness of models of certain shimura varieties of pel type consider a pel shimura variety associated to a unitary group that splits over an unramified extension of q p rapoport and zink have defined a model of the shimura variety over the ring of integers of the completion of the reflex field at a place lying over p with parahoric level structures at p we show that this model is flat as conjectured by rapoport and zink and that its special fibre is reduced

melnikov s method and codimension two bifurcations in forced oscillations abstract using a melnikov type technique we study codimension two bifurcations called the bogdanov takens bifurcations for subharmonics in periodic perturbations of planar hamiltonian systems we give a criterion for the occurrence of the bogdanov takens bifurcations and present approximate expressions for saddle node hopf and homoclinic bifurcation sets near the bogdanov takens bifurcation points we illustrate the theoretical result with an example

frobenius splitting and cohomology vanishing for schubert varieties let k be an algebraically closed field of characteristic p 0 and x be a projective variety over k we then have the absolute frobenius f x x and an injection ox f 6x given by f fp f e o9x if this makes ox a direct summand in f ox as an x module we call x a frobenius split variety for such a variety the vanishing theorem for ample line bundles follows trivially from serre s vanishing theorem for tensoring ox f ox by an ample line bundle l and noting that l f x f f l f lp projection formula we get that the map in cohomology h x l h x f lp h x lp is an injection iterating this we see that h x l injects into h x lp for every p but for large v the latter is zero thus frobenius split varieties have quite pleasant properties it also turns out that using duality for the frobenius morphism or equivalently the cartier operator one can give a manageable criterion for frobenius splitting the point here is the local nature of duality the relevance of the compatibility of local and global duality was suggested to us by grothendieck s proof of h x y 0 for a noncomplete variety x of dimension n 4 theorem 6 9 and by kempf s paper 10 by this criterion and the bott samelson demazure desingularisation of schubert varieties it follows very easily that schubert varieties in characteristic p are frobenius split the following vanishing theorem is then an immediate consequence let g be a reductive group over the field k of arbitrary characteristic zero or positive let q be a parabolic subgroup and x c g q a schubert variety let l be an ample line bundle on g q then hi x l 0 for i 0 and the restriction map h0 g q l h0 x l is surjective if char k 0 this is a consequence of the compatible frobenius splitting of x in g q and the char k 0 case is handled by semicontinuity the above result for special schubert varieties has been proved by seshadrimusili lakshmibai 12 13 and by kempf 9 by using characteristic free methods for x g q andersen 1 and haboush 5 have given simple proofs using characteristic p methods

brief paper simultaneous schur stability of interval matrices interval matrix structures are ubiquitous in nature and engineering ordinarily in an uncertain system there is associated with a set of coupled interval matrices a basic issue of exploring its asymptotic stability here we introduce the notion of simultaneous schur stability by linking the concepts of the majorant and the joint spectral radius and prove the asymptotic stability of a set of interval matrices governed by simultaneous schur stability the present result may lead to the stability analysis of discrete dynamical interval systems

bifurcations in a discrete predator prey model with nonmonotonic functional response abstract the predator prey consumer resource interaction is the most fundamental and important process in population dynamics many species such as monocarpic plants and semelparous animals have discrete nonoverlapping generations and their births occur in regular breeding seasons their interactions are described by difference equations or formulated as discrete time mappings in this paper we study bifurcations in a discrete predator prey model with nonmonotone functional response described by a simplified holling iv function it is shown that the model exhibits various bifurcations of codimension 1 including fold bifurcation transcritical bifurcation flip bifurcations and neimark sacker bifurcation as the values of parameters vary moreover we establish the existence of bogdanov takens bifurcation of codimension 2 and calculate the approximate expressions of bifurcation curves numerical simulations are also presented to illustrate the theoretical analysis these results demonstrate that the bogdanov takens bifurcation of codimension 2 at the degenerate singularity persists in all three versions of the predator prey model with nonmonotone functional response continuous time time delayed and discrete time

lines in g p our goal is reached as follows fix a maximal torus t in g and a borel subgroup b t so that we can consider the corresponding root lattice q weight lattice set of simple roots 1 n and set of fundamental weights 1 n we then choose p b so that p corresponds to a subset s of or equivalently or also equivalently of the dynkin diagram of g recall that the picard group of g p can be identified with a sublattice of taken an ample l pic g p and setting

stability robustness bound estimates of discrete systems analysis and comparison robustness bound estimates based on the direct lyapunov method for discrete time nominally linear systems are analysed and compared although various robustness bound estimates were introduced recently little effort has been made to compare them we develop a scheme for obtaining the estimates which brings a new robustness bound estimate and provides a useful tool for the study and comparison of a variety of the estimates a novel technique is presented to compare two superior robustness estimates the results obtained are illustrated on several examples

frobenius splitting of cotangent bundles of flag varieties let g be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic p 0 let u be the unipotent radical of a borel subgroup b g and u the lie algebra of u springer 16 has shown for good primes that there is a b equivariant isomorphism u u where b acts through conjugation on u and through the adjoint action on u for g sln one has the well known equivariant isomorphism a 7 a i between unipotent and nilpotent upper triangular matrices let p be a good prime for g then there is an isomorphism of homogeneous bundles x g b u g b u where the latter can be identified with the cotangent bundle t g b of g b motivated in part by 12 we establish a link between the g invariant form on the steinberg module st h0 g b p 1 cf 1 8 and frobenius splittings 15 of the cotangent bundle of g b the representation h0 g b 2 p 1 is a quotient of the space of functions h x ox on x here h0 g b m denotes the g module induced from the b module m and half the sum of the roots r opposite to the roots

lengths of chains of minimal rational curves on fano manifolds abstract in this paper we consider a natural question how many minimal rational curves are needed to join two general points on a fano manifold x of picard number 1 in particular we study the minimal length of such chains in the cases where the dimension of x is at most 5 the coindex of x is at most 3 and x equips with a structure of a double cover as an application we give a better bound on the degree of fano 5 folds of picard number 1

discreteness and rationality of f jumping numbers on singular varieties we prove that the f jumping numbers of the test ideal tau x delta mathfrak a t are discrete and rational under the assumptions that x is a normal and f finite scheme over a field of positive characteristic p kx is mathbb q cartier of index not divisible p and either x is essentially of finite type over a field or the sheaf of ideals mathfrak a is locally principal this is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero

centers of f purity in this paper we study a positive characteristic analogue of the centers of log canonicity of a pair r we call these analogues centers of f purity we prove positive characteristic analogues of subadjunction like results prove new stronger subadjunction like results and in some cases lift these new results to characteristic zero using a generalization of centers of f purity which we call uniformly f compatible ideals we give a characterization of the test ideal which unifies several previous characterizations finally in the case that 0 we show that uniformly f compatible ideals coincide with the annihilators of the mathcal f e r k submodules of e r k as defined by lyubeznik and smith

borel weil theorem for configuration varieties and schur modules we present a generalization of the classical schur modules ofgl n exhibiting the same interplay among algebra geometry and combinatorics a generalized young diagramdis an arbitrary finite subset ofn n for eachd we define the schur modulesdofgl n we introduce a projective variety fdand a line bundle ld and describe the schur module in terms of sections of ld for diagrams with the northeast property i1 j1 i2 j2 d min i1 i2 max j1 j2 d which includes the skew diagrams we resolve the singularities of fdand show analogues of bott s and kempf s vanishing theorems finally we apply the atiyah bott fixed point theorem to establish a weyl type character formula of the form charsd x txwt t i j 1 xix 1j dij t wheretruns over certain standard tableaux ofd our results are valid over fields of arbitrary characteristic

frobenius splitting and geometry of g schubert varieties let x be an equivariant embedding of a connected reductive group g over an algebraically closed field k of positive characteristic let b denote a borel subgroup of g a g schubert variety in x is a subvariety of the form diag g v where v is a b b orbit closure in x in the case where x is the wonderful compactification of a group of adjoint type the g schubert varieties are the closures of lusztig s g stable pieces we prove that x admits a frobenius splitting which is compatible with all g schubert varieties moreover when x is smooth projective and toroidal then any g schubert variety in x admits a stable frobenius splitting along an ample divisors although this indicates that g schubert varieties have nice singularities we present an example of a nonnormal g schubert variety in the wonderful compactification of a group of type g2 finally we also extend the frobenius splitting results to the more general class of schubert varieties

globally f regular and log fano varieties abstract we prove that every globally f regular variety is log fano in other words if a prime characteristic variety x is globally f regular then it admits an effective q divisor such that k x is ample and x has controlled kawamata log terminal in fact globally f regular singularities a weak form of this result can be viewed as a prime characteristic analog of de fernex and hacon s new point of view on kawamata log terminal singularities in the non q gorenstein case we also prove a converse statement in characteristic zero every log fano variety has globally f regular type our techniques apply also to f split varieties which we show to satisfy a log calabi yau condition we also prove a kawamata viehweg vanishing theorem for globally f regular pairs

degenerate flag varieties of type a frobenius splitting and bw theorem let mathcal f a lambda be the pbw degeneration of the flag varieties of type a n 1 these varieties are singular and are acted upon with the degenerate lie group sl n a we prove that mathcal f a lambda have rational singularities are normal and locally complete intersections and construct a desingularization r lambda of mathcal f a lambda the varieties r lambda can be viewed as towers of successive mathbb p 1 fibrations thus providing an analogue of the classical bott samelson demazure hansen desingularization we prove that the varieties r lambda are frobenius split this gives us frobenius splitting for the degenerate flag varieties and allows to prove the borel weil type theorem for mathcal f a lambda using the atiyah bott lefschetz formula for r lambda we compute the q characters of the highest weight mathfrak sl n modules

on compactifications of the steinberg zero fiber abstract let g be a connected semisimple linear algebraic group over an algebraically closed field k of positive characteristic and let x denote an equivariant embedding of g we define a distinguished steinberg fiber n in g called the zero fiber and prove that the closure of n within x is normal and cohen macaulay furthermore when x is smooth we prove that the closure of n is a local complete intersection

quasi frobenius splitting and lifting of calabi yau varieties in characteristic p generalizing the notion of frobenius splitting we prove that every finite height calabi yau variety defined over an algebraically closed field of positive characteristic can be lifted to the ring of witt vectors of length two

affine demazure modules and t fixed point subschemes in the affine grassmannian let g be a simple algebraic group defined over c and t be a maximal torus of g for a dominant coweight of g the t fixed point subscheme formula of the schubert variety formula in the affine grassmannian grg is a finite scheme we prove that for all such if g is of type a or d and for many of them if g is of type e there is a natural isomorphism between the dual of the level one affine demazure module corresponding to and the ring of functions twisted by certain line bundle on grg of formula we use this fact to give a geometrical proof of the frenkel kac segal isomorphism between basic representations of affine algebras of a d e type and lattice vertex algebras

a remark on ulrich and acm bundles abstract i show that on any smooth projective ordinary curve of genus at least two and a projective embedding there is a natural example of a stable ulrich bundle for this embedding namely the sheaf b x 1 of locally exact differentials twisted by o x 1 given by this embedding and in particular there exist ordinary varieties of any dimension which carry ulrich bundles in higher dimensions assuming x is frobenius split variety i show that b x 1 is an acm bundle and if x is also a calabi yau variety and p 2 then b x 1 is not a direct sum of line bundles in particular i show that b x 1 is an acm bundle on any ordinary calabi yau variety i also prove a characterization of projective varieties with trivial canonical bundle such that b x 1 is acm for some projective embedding datum all such varieties are frobenius split with trivial canonical bundle

another strongly exceptional collection of coherent sheaves on a grassmannian abstract on the grassmannian of 2 dimensional subspaces in a finite dimensional linear space we construct a karoubian complete strongly exceptional po set of coherent sheaves parametrized by the cosets of the weyl group of the general linear group of the linear space modulo the weyl group of the parabolic subgroup stabilizing the subspace from subquotients of the frobenius direct image of the structure sheaf of the grassmannian defined over a field of large positive characteristic our collection diverges from the one discovered by kapranov we also show in the general setting over any field of positive characteristic that the sheaf corresponding to the longest element of the cosets is a direct summand of the frobenius direct image of the structure sheaf of the homogeneous space

bifurcations from an attracting heteroclinic cycle under periodic forcing abstract there are few examples of non autonomous vector fields exhibiting complex dynamics that may be proven analytically we analyse a family of periodic perturbations of a weakly attracting robust heteroclinic network defined on the two sphere we derive the first return map near the heteroclinic cycle for small amplitude of the perturbing term and we reduce the analysis of the non autonomous system to that of a two dimensional map on a cylinder interesting dynamical features arise from a discrete time bogdanov takens bifurcation when the perturbation strength is small the first return map has an attracting invariant closed curve that is not contractible on the cylinder near the centre of frequency locking there are parameter values with bistability the invariant curve coexists with an attracting fixed point increasing the perturbation strength there are periodic solutions that bifurcate into a closed contractible invariant curve and into a region where the dynamics is conjugate to a full shift on two symbols

moduli of vector bundles frobenius splitting and invariant theory let x be an irreducible projective curve of genus g over an algebraically closed field of positive characteristic 2 3 in part i we prove adapting the degeneration arguments of n tr that moduli spaces of rank two parabolic bundles on x are frobenius split m r for generic smooth x a similar result holds for x nodal a consequence is the verlinde formula in positive characteristic in part ii we give a direct proof of the fact that the local rings of the moduli spaces are f split and further that they are cohen macaulay this involves showing that the ring of invariants of g copies of 2 x 2 matrices under the adjoint action of sl 2 is f split and cohen macaulay

rees algebras of f regular type abstract we study the f regularity of rees algebras r i a it in terms of the global f regularity of the blowing up x proj r i of spec a as it reads global f regularity is a global analog of strong f regularity defined via splitting of frobenius maps in prime characteristic and these notions are extended to characteristic zero by reduction modulo p 0 we study in detail the case where a m is a two dimensional local ring and i is an m primary ideal in characteristic zero the condition for r i to have f regular type is described in terms of the dual graph of a resolution x on which i o x is invertible we also prove some miscellaneous results concerning singularities of rees algebras and extended rees algebras of higher dimension

polytopes associated to demazure modules of symmetrizable kac moody algebras of rank two let 1 2 be the two fundamental weights of a symmetrizable kac moody algebra g of rank two hence necessarily affine or finite and an element of the weyl group in this paper we construct polytopes p 1 p 2 rl and a linear map rl h such that for any dominant weight k1 1 k2 2 we have char e e e x where the sum is over all the integral points x of the polytope k1p 1 k2p 2 furthermore we show that there exists a flat deformation of the schubert variety s into the toric variety defined by p 1 p 2

hankel determinantal rings have rational singularities abstract hankel determinantal rings i e determinantal rings defined by minors of hankel matrices of indeterminates arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves they may also be viewed as linear specializations of generic determinantal rings we prove that over fields of characteristic zero hankel determinantal rings have rational singularities in the case of positive prime characteristic we prove that they are f pure independent of the characteristic we give a complete description of the divisor class groups of these rings and show that each divisor class group element is the class of a maximal cohen macaulay module

on the b canonical splittings of flag varieties abstract let g be a semisimple algebraic group over an algebraically closed field of positive characteristic in this note we show that an irreducible closed subvariety of the flag variety of g is compatibly split by the unique canonical frobenius splitting if and only if it is a richardson variety i e an intersection of a schubert and an opposite schubert variety

frobenius splitting of hilbert schemes of points on surfaces let x be a quasiprojective smooth surface defined over an algebraically closed field of positive characteristic in this note we show that if x is frobenius split then so is the hilbert scheme hilb n x of n points inx in particular we get the higher cohomology vanishing for ample line bundles on hilb n x when x is projective and frobenius split

the space of triangles vanishing theorems and combinatorics abstract we consider compactifications of pn 3 ij the space of triples of distinct points in projective space one such space is a singular variety of configurations of points and lines another is the smooth compactification of fulton and macpherson and a third is the triangle space of schubert and semple we compute the sections of line bundles on these spaces and show that they are equal as gl n representations to the generalized schur modules associated to bad generalized young diagrams with three rows borel weil theorem on the one hand this yields weyl type character and dimension formulas for the schur modules on the other a combinatorial picture of the space of sections cohomology vanishing theorems play a key role in our analysis

f rational rings have rational singularities it is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational a key point in the proof is a characterization of f rational local rings as those cohen macaulay local rings r m in which the local cohomology module h d m r where d is the dimension of r have no submodules stable under the natural action of the frobenius map an analog for finitely generated algebras over a field of characteristic zero is developed which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i without reference to a desingularization

algebraic frobenius splitting of cotangent bundles of flag varieties abstract following the program of algebraic frobenius splitting begun by kumar and littelmann we use representation theoretic techniques to construct a frobenius splitting of the cotangent bundle of the flag variety of a semisimple algebraic group over an algebraically closed field of positive characteristic we also show that this splitting is the same as one of the splittings constructed by kumar lauritzen and thomsen