error analysis and comparative study of numerical methods for the parabolic equation applied to tunnel propagation modeling parabolic equation pe methods have been widely applied to the modeling of wireless propagation in tunnel environments however the relevant literature does not include concrete guidelines for the choice of the parameters of these methods and the tradeoffs involved this paper provides a comprehensive analysis of the two sources of error that arise when pe methods are employed for the modeling of radio wave propagation scenarios the well known numerical dispersion error stemming from the finite difference solvers for pe and the approximation error stemming from the use of pe for the solution of wave propagation problems that are subject to maxwell s equations the analysis is performed for four methods three of which have been already used in pe based propagation studies namely the crank nicolson cn scheme the alternative direction implicit adi method and its locally one dimensional lod adi version the fourth method is the mitchell fairweather mf adi scheme that has been recently shown to be a promising alternative technique for tunnel propagation modeling the proposed method leads to robust criteria for the choice of spatial discretization in realistic propagation scenarios as shown via numerical examples

absolutely trianalytic tori in the generalized kummer variety we prove that a generic complex deformation of a generalized kummer variety contains no complex analytic tori

moduli spaces of polarized symplectic o grady varieties and borcherds products we study moduli spaces of o grady s ten dimensional irreducible symplectic manifolds these moduli spaces are covers of modular varieties of dimension 21 namely quotients of hermitian symmetric domains by a suitable arithmetic group the interesting and new aspect of this case is that the group in question is strictly bigger than the stable orthogonal group this makes it different from both the k3 and the k3 n case which are of dimension 19 and 20 respectively

topological invariants of o grady s six dimensional irreducible symplectic variety we study o grady examples of irreducible symplectic varieties we establish that both of them can be deformed into lagrangian fibrations we analyze in detail the topology of the six dimensional example in particular we compute its euler characteristic and determine its beauville form

on monodromies of a degeneration of irreducible symplectic kahler manifolds we study the monodromy operators on the betti cohomologies associated to a good degeneration of irreducible symplectic manifold and we show that the unipotency of the monodromy operator on the middle cohomology is at least the half of the dimension this implies that the mildest singular fiber of a good degeneration with non trivial monodromy of irreducible symplectic manifolds is quite different from the generic degeneration of abelian varieties or calabi yau manifolds

the 2 factoriality of the o grady moduli spaces the aim of this work is to show that the moduli space m10 introduced by o grady is a 2 factorial variety namely m10 is the moduli space of semistable sheaves with mukai vector v 2 0 2 in h ev x mathbb z on a projective k3 surface x as a corollary to our construction we show that the donaldson morphism gives a hodge isometry between v perp sublattice of the mukai lattice of x and its image in h 2 widetilde m 10 mathbb z lattice with respect to the beauville form of the 10 dimensional irreducible symplectic manifold widetilde m 10 obtained as symplectic resolution of m10 similar results are shown for the moduli space m6 introduced by o grady to produce its 6 dimensional example of irreducible symplectic variety

a hyper kahler compactification of the intermediate jacobian fibration associated with a cubic 4 fold let x be a general cubic fourfold it was observed by donagi and markman that the relative intermediate jacobian fibration j u with u p5 x associated to the smooth hyperplane sections of x carries a natural holomorphic symplectic form making the fibration lagrangian in this paper we obtain a smooth projective compactification j of j with the property that the holomorphic symplectic form on j extends to a holomorphic symplectic form on j in particular j is a 10 dimensional compact hyper kahler manifold which we show to be deformation equivalent to the exceptional example of o grady this proves a conjecture by markushevich

hyper kahler fourfolds and grassmann geometry we construct a new 20 dimensional family of projective hyper kahler fourfolds and prove that they are deformationequivalent to the second punctual hilbert scheme of a k3 surface of genus 12

higher direct images of dualizing sheaves of lagrangian fibrations let f x s be a lagrangian fibration between projective varieties we prove that inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i i s if s is smooth suppose that x is an irreducible symplectic manifold or a certain moduli space of semistable torsion free sheaves on a k 3 surface the hodge numbers satisfy h p q s h p q inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i where n dim s if s inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i and x is an irreducible symplectic manifold there exists a hypersurface m f of the kuranishi space of x such that every member of the kuranishi family over m f admits a lagrangian fibration over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i

on the beauville form of the known irreducible symplectic varieties we study the global geometry of o grady s ten dimensional irreducible symplectic variety we determine its second betti number its beauville form and its fujiki constant

deformation theory of singular symplectic n folds by a symplectic manifold or a symplectic n fold we mean a compact kaehler manifold of even dimension n with a non degenerate holomorphic 2form i e is a nowhere vanishing n form this notion is generalized to a variety with singularities we call x a projective symplectic variety if x is a normal projective variety with rational gorenstein singularities and if the regular locus u of x admits a non degenerate holomorphic 2 form a symplectic variety will play an important role together with a singular calabi yau variety in the generalized bogomolov decomposition conjecture now that essentially a few examples of symplectic manifolds are discovered it seems an important task to seek new symplectic manifolds by deforming symplectic varieties in this paper we shall study a projective symplectic variety from a view point of deformation theory if x has a resolution x x such that x is a symplectic manifold we say that x has a symplectic resolution our first results are concerned with a birational contraction map of a symplectic manifold

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

birational geometry of o grady s six dimensional example over the donaldson uhlenbeck compactification we determine the birational geometry of o grady s six dimensional example over the donaldson uhlenbeck compactification by looking at the locus of non locally free sheaves on the relevant moduli space

moduli of sheaves fourier mukai transform and partial desingularization we study birational maps among 1 the moduli space of semistable sheaves of hilbert polynomial 4m 2 on a smooth quadric surface 2 the moduli space of semistable sheaves of hilbert polynomial m 2 3m 2 on mathbb p 3 3 kontsevich s moduli space of genus zero stable maps of degree 2 to the grassmannian gr 2 4 a regular birational morphism from 1 to 2 is described in terms of fourier mukai transforms the map from 3 to 2 is kirwan s partial desingularization we also investigate several geometric properties of 1 by using the variation of moduli spaces of stable pairs

desingularizations of the moduli space of rank 2 bundles over a curve let x be a smooth projective curve of genus g 3 and m0 be the moduli space of rank 2 semistable bundles over x with trivial determinant there are three desingularizations of this singular moduli space constructed by narasimhan ramanan nr78 seshadri ses77 and kirwan kir86b respectively the relationship between them has not been understood so far the purpose of this paper is to show that there is a morphism from kirwan s desingularization to seshadri s which turns out to be the composition of two blow downs in doing so we will show that the singularities of m0 are terminal and the plurigenera are all trivial as an application we compute the betti numbers of the cohomology of seshadri s desingularization in all degrees this generalizes the result of bs90 which computes the betti numbers in low degrees another application is the computation of the stringy e function see bat98 for definition of m0 for any genus g 3 which generalizes the result of kie03

projectivity and birational geometry of bridgeland moduli spaces we construct a family of nef divisor classes on every moduli space of stable complexes in the sense of bridgeland this divisor class varies naturally with the bridgeland stability condition for a generic stability condition on a k3 surface we prove that this class is ample thereby generalizing a result of minamide yanagida and yoshioka our result also gives a systematic explanation of the relation between wall crossing for bridgeland stability and the minimal model program for the moduli space we give three applications of our method for classical moduli spaces of sheaves on a k3 surface 1 we obtain a region in the ample cone in the moduli space of gieseker stable sheaves only depending on the lattice of the k3 2 we determine the nef cone of the hilbert scheme of n points on a k3 surface of picard rank one when n is large compared to the genus 3 we verify the hassett tschinkel huybrechts sawon conjecture on the existence of a birational lagrangian fibration for the hilbert scheme in a new family of cases

singularities of moduli spaces of sheaves on k3 surfaces and nakajima quiver varieties abstract the aim of this paper is to study the singularities of certain moduli spaces of sheaves on k3 surfaces by means of nakajima quiver varieties the singularities in question arise from the choice of a non generic polarization with respect to which we consider stability and admit natural symplectic resolutions corresponding to choices of general polarizations for sheaves that are pure of dimension one we show that these moduli spaces are locally around a singular point isomorphic to a quiver variety and that via this isomorphism the natural symplectic resolutions correspond to variations of git quotients of the quiver variety

picard number of the generic fiber of an abelian fibered hyperkahler manifold we shall show that the picard number of the generic fiber of an abelian fibered hyperkahler manifold over the projective space is always one we then give a few applications for the mordell weil group in particular by deforming o grady s 10 dimensional manifold we construct an abelian fibered hyperkahler manifold of mordell weil rank 20 which is the maximum possible among all known ones

tautological classes on moduli spaces of hyper kahler manifolds we study algebraic cycles on moduli spaces f h of h polarized hyperkahler manifolds following previous work of marian oprea and pandharipande on the tautological conjecture on moduli spaces of k3 surfaces we first define the tautological ring on f h we then study the images of these tautological classes in the cohomology groups of f h and prove that most of them are linear combinations of noether lefschetz cycle classes in particular we prove the cohomological version of the tautological conjecture on moduli space of k3 n type hyperkahler man ifolds with n 2 secondly we prove the cohomological generalized franchetta conjecture on universal family of these hyperkahler mani folds

mmp for moduli of sheaves on k3s via wall crossing nef and movable cones lagrangian fibrations we use wall crossing with respect to bridgeland stability conditions to systematically study the birational geometry of a moduli space m of stable sheaves on a k3 surface x a we describe the nef cone the movable cone and the effective cone of m in terms of the mukai lattice of x b we establish a long standing conjecture that predicts the existence of a birational lagrangian fibration on m whenever m admits an integral divisor class d of square zero with respect to the beauville bogomolov form these results are proved using a natural map from the space of bridgeland stability conditions mathop mathrm stab nolimits x to the cone mathop mathrm mov nolimits x of movable divisors on m this map relates wall crossing in mathop mathrm stab nolimits x to birational transformations of m in particular every minimal model of m appears as a moduli space of bridgeland stable objects on x

singular symplectic moduli spaces moduli spaces of semistable sheaves on a k3 or abelian surface with respect to a general ample divisor are shown to be locally factorial with the exception of symmetric products of a k3 or abelian surface and the class of moduli spaces found by o grady consequently since singular moduli space that do not belong to these exceptional cases have singularities in codimension 4 they do no admit projective symplectic resolutions

a necessary and sufficient condition for the existence of positive solutions to a kind of singular three point boundary value problem abstract in this paper we are concerned with the following three point boundary value problem u t f t u t 0 0 t 1 u 0 u 0 u 1 0 where 0 1 and the nonlinear term f may be singular at t 1 or and t 0 a necessary and sufficient condition for the existence of c 0 1 as well as c 1 0 1 positive solution is given by constructing lower and upper solutions and with the maximal theorem also the uniqueness iterative methods convergence rate and computational methods of the positive solutions are considered