approximation order of bivariate spline interpolation for arbitrary smoothness by using the algorithm of nurnberger and riessinger 1995 we construct hermite interpolation sets for spaces of bivariate splines sqr 1 of arbitrary smoothness defined on the uniform type triangulations it is shown that our hermite interpolation method yields optimal approximation order for q 3 5r 1 in order to prove this we use the concept of weak interpolation and arguments of birkhoff interpolation

tyz expansion for the kepler manifold the main goal of the paper is to address the issue of the existence of kempf s distortion function and the tian yau zelditch tyz asymptotic expansion for the kepler manifold an important example of non compact manifold motivated by the recent results for compact manifolds we construct kempf s distortion function and derive a precise tyz asymptotic expansion for the kepler manifold we get an exact formula finite asymptotic expansion of n 1 terms and exponentially small error terms uniformly with respect to the discrete quantization parameter m rightarrow infty hbar m 1 rightarrow 0 standing for planck s constant and x rightarrow infty x in mathbb c n moreover the coefficients are calculated explicitly and they turned out to be homogeneous functions with respect to the polar radius in the kepler manifold we show that our estimates are sharp by analyzing the nonharmonic behaviour of tm for m rightarrow infty the arguments of the proofs combine geometrical methods quantization tools and functional analytic techniques for investigating asymptotic expansions in the framework of analytic gevrey spaces

syzygies of abelian varieties let a be an ample line bundle on an abelian variety x over an algebraically closed field a theorem of koizumi ko s developing mumford s ideas and results m states that if m 3 the line bundle l ao embeds x in projective space as a projectively normal variety moreover a celebrated theorem of mumford m2 slightly refined by kempf k4 asserts that the homogeneous ideal of x is generated by quadrics as soon as m 4 such results turn out to be particular cases of a statement conjectured by rob lazarsfeld concerning the minimal resolution of the graded algebra rl 0 ho x l h over the polynomial ring sl do symhho x l the purpose of this paper is to prove lazarsfeld s conjecture to put such matters into perspective it is useful to review the case of projective curves a classical theorem of castelnuovo states that a curve x embedded in projective space by a complete linear system ili is projectively normal as soon as deg l 2g x 1 and a theorem of mattuck fujita and saint donat states that if deg l 2g x 2 then the homogeneous ideal of x is generated by quadrics green g1 unified re interpreted and generalized these results to a statement about syzygies specifically given a smooth projective variety x and a very ample line bundle l on x a minimal resolution of rl as a graded sl module notation as above looks like

involutions on moduli spaces and refinements of the verlinde formula the moduli space m of semi stable rank 2 bundles with trivial deter minant over a complex curvecarries involutions naturally associated to 2 torsion points on the jacobian of the curve for every lift of a 2 torsion point to a 4 torsion point we define a lift of the involution to the determinant line bundle l we ob tain an explicit presentation of the group generated by these lifts in terms of the order 4 weil pairing this is related to the triple intersections of the components of the fixed point sets in m which we also determine completely using the order 4 weil pairing the lifted involutions act on the spaces of holomorphic sections of powers of l whose dimensions are given by the verlinde formula we compute the characters of these vector spaces as representations of the group generated by our lifts and we obtain an explicit isomorphism as group representations with the combinatorial topological tqft vector spaces of bhmv as an application we describe a brick decomposition with explicit dimension formulas of the verlinde vector spaces we also obtain similar results in the twisted i e degree one case

projective normality and syzygies of algebraic surfaces in this work we develop new techniques to compute koszul cohomology groups for several classes of varieties as applications we prove results on projective normality and syzygies for algebraic surfaces from more general results we obtain in particular the following a mukai s conjecture and stronger variants of it regarding projective normality and normal presentation for surfaces with kodaira dimension 0 and uniform bounds for higher syzygies associated to adjoint linear series b effective bounds along the lines of mukai s conjecture regarding projective normality and normal presentation for surfaces of positive kodaira dimension and c results on projective normality for pluricanonical models of surfaces of general type recovering and strengthening results by ciliberto and generalizations of them to higher syzygies in addition we also extend the above results to singular surfaces

regularity on abelian varieties i we introduce the notion of mukai regularity m regularity for coherent sheaves on abelian varieties the definition is based on the fourier mukai transform and in a special case depending on the choice of a polarization it parallels and strengthens the usual castelnuovo mumford regularity mukai regularity has a large number of applications ranging from basic properties of linear series on abelian varieties and defining equations for their subvarieties to higher dimensional type statements and to a study of special classes of vector bundles some of these applications are explained here while others are the subject of upcoming sequels

simple waves in quasilinear hyperbolic systems i theory of simple waves and simple states examples of applications this paper presents a new method of construction of solutions to nonlinear nonelliptic systems of partial differential equations and especially nonhomogeneous ones these equations have been considered from the point of view of integral elements in particular the connections between the structure of the set of integral elements and the possibility of a construction of special classes of solutions have been studied these classes consist of what is called simple waves and k waves for homogeneous systems and simple states in the case of nonhomogeneous systems they provide us with a possibility for a selection of simple integral elements from the set of all integral elements analyses have been performed using differential forms and cartan theory of system in involution the problem has been reduced to examining pfaff forms the cauchy problem for pfaff systems has been formulated and solved using the riemann function some remarks concerning the notion of backlund transformations for the case of k waves have been formulated it is shown that in contrast to simple wave the simple state has no gradient catastrophy the technique presented of constructing the solutions in form of simple states has been illustrated by the examples of korteweg and de vries and four dimensional klein gordon sine gordon and liouville equations it has been shown that the known soliton equations are closely connected with the elliptical functions and especially with the p weierstrass functions

on the integrability of 2 1 dimensional quasilinear systems a 2 1 dimensional quasilinear system is said to be integrable if it can be decoupled in infinitely many ways into a pair of compatible n component one dimensional systems in riemann invariants exact solutions described by these reductions known as nonlinear interactions of planar simple waves can be viewed as natural dispersionless analogues of n gap solutions it is demonstrated that the requirement of the existence of sufficiently many n component reductions provides the effective classification criterion as an example of this approach we classify integrable 2 1 dimensional systems of conservation laws possessing a convex quadratic entropy

reductions and hodograph solutions of the dispersionless kp hierarchy a general scheme for analysing reductions of dispersionless integrable hierarchies is presented it is based on a method for determining the s function by means of a system of first order differential equations compatibility systems of nonlinear partial differential equations of bourlet type characterizing both reductions and hodograph solutions of the dkp hierarchy are obtained wide classes of illustrative explicit examples are exhibited

on the solutions of the second heavenly and pavlov equations we have recently solved the inverse scattering problem for one parameter families of vector fields and used this result to construct the formal solution of the cauchy problem for a class of integrable nonlinear partial differential equations connected with the commutation of multidimensional vector fields such as the heavenly equation of plebanski the dispersionless kadomtsev petviashvili dkp equation and the two dimensional dispersionless toda 2ddt equation as well as with the commutation of one dimensional vector fields such as the pavlov equation we also showed that the associated riemann hilbert inverse problems are powerful tools to establish if the solutions of the cauchy problem break at finite time to construct their long time behaviour and characterize classes of implicit solutions in this paper using the above theory we concentrate on the heavenly and pavlov equations i establishing that their localized solutions evolve without breaking unlike the cases of dkp and 2ddt ii constructing the long time behaviour of the solutions of their cauchy problems iii characterizing a distinguished class of implicit solutions of the heavenly equation

asymptotic syzygies of algebraic varieties we study the asymptotic behavior of the syzygies of a smooth projective variety as the positivity of the embedding line bundle grows the main result asserts that the syzygy modules are non zero in almost all degrees allowed by castelnuovo mumford regularity we also give an effective statement for veronese varieties that we conjecture to be optimal

on the canonical rings of covers of surfaces of minimal degree let s be a regular surface of general type with at worst canonical singularities and with basepoint free canonical system let x be its canonical image it is well known that x must be a canonical surface or a minimal degree surface the main result of the authors completely describes the number and degree of the generators of the canonical ring of s in the second case more concretely if r deg x and n is the degree of the canonical map then 1 if n 2 and r 1 the canonical ring is generated in degree 1 plus one generator in degree 4 2 in the other cases the canonical ring is generated in degree 1 plus r n 2 generators in degree 2 and r 1 generators in degree 3 this result together with previous results of ciliberto and green describes when the canonical ring of s is generated in degree less than or equal to 2 x is not a surface of minimal degree other than the plane and in this last case n 6 2 the authors also construct a series of non trivial examples of the theorem and prove that some expected ones do not exist finally the authors apply their results to calabi yau threefolds obtaining analogous results the key point here is that for a calabi yau threefold the general member of a big and base point free linear system is a surface of general type

dispersionless scalar integrable hierarchies whitham hierarchy and the quasiclassical dressing method the quasiclassical limit of the scalar nonlocal problem is derived and a quasiclassical version of the dressing method is presented dispersionless kadomtsev petviashvili kp modified kp and dispersionless two dimensional toda lattice 2dtl hierarchies are discussed as illustrative examples it is shown that the universal whitham hierarchy is nothing but the ring of symmetries for the quasiclassical problem the reduction problem is discussed and in particular the d2dtl equation of b type is derived

three dimensional reductions of four dimensional quasilinear systems in this paper we show that four dimensional quasilinear systems of first order integrable by the method of two dimensional hydrodynamic reductions possess infinitely many three dimensional hydrodynamic reductions which are also integrable systems these three dimensional multi component integrable systems are irreducible to two dimensional hydrodynamic reductions in a generic case

on integrability in grassmann geometries integrable systems associated with fourfolds in gr 3 5 let gr d n be the grassmannian of d dimensional linear subspaces of an n dimensional vector space v n a submanifold x gr d n gives rise to a differential system x that governs d dimensional submanifolds of v n whose gaussian image is contained in x systems of the form x appear in numerous applications in continuum mechanics theory of integrable systems general relativity and differential geometry they include such wellknown examples as the dispersionless kadomtsev petviashvili equation the boyer finley equation plebansky s heavenly equations and so on in this paper we concentrate on the particularly interesting case of this construction where x is a fourfold in gr 3 5 our main goal is to investigate differential geometric and integrability aspects of the corresponding systems x we demonstrate the equivalence of several approaches to dispersionless integrability such as the method of hydrodynamic reductions the method of dispersionless lax pairs integrability on solutions based on the requirement that the characteristic variety of system x defines an einstein weyl geometry on every solution integrability on equation meaning integrability in twistor theoretic sense of the canonical gl 2 r structure induced on a fourfold x gr 3 5 all these seemingly different approaches lead to one and the same class of integrable systems x we prove that the moduli space of such systems is 6 dimensional we give a complete description of linearisable systems the corresponding fourfold x is a linear section of gr 3 5 and linearly degenerate systems the corresponding fourfold x is the image of a quadratic map p4 99k gr 3 5 the fourfolds corresponding to generic integrable systems are not algebraic and can be parametrised by generalised hypergeometric functions

the approach to the dispersionless kp hierarchy the dispersionless limit of the scalar nonlocal problem is derived it is given by a special class of nonlinear first order equations a quasiclassical version of the dressing method is presented it is shown that the algebraic formulation of dispersionless hierarchies can be expressed in terms of properties of beltrami type equations the universal whitham hierarchy and in particular the dispersionless kp hierarchy turn out to be rings of symmetries for the quasiclassical problem pacs number 02 30 uk

finite tycz expansions and csck metrics abstract let m g be a kahler manifold whose associated kahler form is integral and let l h m be a quantization hermitian line bundle in this paper we study those kahler manifolds m g admitting a finite tycz expansion namely those for which the associated kempf distortion function t m g is of the form t m g p f s p m s f s 1 p m s 1 f r p m r f j c m s r z we show that if the tycz expansion is finite then t m g is indeed a polynomial in m of degree n n dim c m and the log term of the szego kernel of the disc bundle d l vanishes where l is the dual bundle of l moreover we provide a complete classification of the kahler manifolds admitting finite tycz expansion either when m is a complex curve or when m is a complex surface with a csck metric which admits a radial kahler potential

on irregular threefolds and fourfolds with numerically trivial canonical bundle we prove that for a smooth projective irregular threefold x with k x equiv 0 and a nef and big divisor l on x ml p gives a birational map for all m ge 3 and all p in mathrm pic 0 x we also use the same method to deal with fourfolds and prove that for a smooth projective irregular fourfold x with k x equiv 0 and an ample divisor l on x ml p gives a birational map for all m ge 5 and all p in mathrm pic 0 x these results are optimal

filling polygonal holes with minimal energy surfaces on powell sabin type triangulations in this paper we present two different methods for filling in a hole in an explicit 3d surface defined by a smooth function f in a part of a polygonal domain email protected r 2 we obtain the final reconstructed surface over the whole domain d we do the filling in two different ways discontinuous and continuous in the discontinuous case we fill the hole with a function in a powell sabin spline space that minimizes a linear combination of the usual seminorms in an adequate sobolev space and approximates in the least squares sense the values of f and those of its normal derivatives at an adequate set of points in the continuous case we will first replace f outside the hole by a smoothing bivariate spline s f and then we fill the hole also with a powell sabin spline minimizing a linear combination of given seminorms in both cases we obtain existence and uniqueness of solutions and we present some graphical examples and in the continuous case we also give a local convergence result

szego kernel regular quantizations and spherical cr structures we compute the szego kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact kahler manifold as a corollary we provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds disk bundles over homogeneous hodge manifolds for which the log terms in the fefferman expansion of the szego kernel vanish and which are not locally cr equivalent to the sphere we also give a proof of the fact that for homogeneous hodge manifolds the existence of a locally spherical cr structure on the unit circle bundle alone implies that the manifold is biholomorphic to a projective space our results generalize those obtained by englis math z 264 4 901 912 2010 for hermitian symmetric spaces of compact type

minimal energy cr surfaces on uniform powell sabin type meshes for noisy data in this paper we present a method to obtain for noisy data a c r surface for any r 1 on a polygonal domain which approximates a lagrangian data set and minimizes a quadratic functional that contains some terms associated with sobolev semi norms weighted by some smoothing parameters the minimization space is composed of bivariate spline functions constructed on a uniform powell sabin type triangulation of the domain we prove a result of almost sure convergence and we choose optimal values of the smoothing parameters by adapting the generalized cross validation method we finish with some numerical and graphical examples

on syzygies of calabi yau varieties and varieties of general type abstract we establish projective normality property n p and effective very ampleness for line bundles on a calabi yau variety as well as for pluricanonical bundles of a variety of general type we also prove a result on the degrees of generators of canonical rings under the condition that the image of the canonical map has minimal degree

the τ function of the universal whitham hierarchy matrix models and topological field theories the universal witham hierarchy is considered from the point of view of topological field theories the function for this hierarchy is defined it is proved that the algebraic orbits of whitham hierarchy can be identified with various topological matter models coupled with topological gravity

inverse free recursive multiresolution algorithms for a data approximation problem we present inverse free recursive multiresolution algorithms for data approximation problems based on energy functionals minimization during the multiresolution process a linear system needs to be solved at each different resolution level which can be solved with direct or iterative methods numerical results are reported using the sparse cholesky factorization for two applications one concerning the localization of regions in which the energy of a given surface is mostly concentrated and another one regarding noise reduction of a given dataset in addition for large scale data approximation problems that require a very fine resolution we discuss the use of the preconditioned conjugate gradient pcg iterative method coupled with a specialized monolithic preconditioner for which one preconditioner is built for the highest resolution level and then the corresponding blocks of that preconditioner are used as preconditioners for the forthcoming lower levels

the level rank duality for non abelian theta functions we prove that the strange duality conjecture of beauville donagi tu holds for all curves we establish first a more extended level rank duality interesting in its own right from which the standard level rank duality follows by restriction

integrable equations in 2 1 dimensions deformations of dispersionless limits we classify integrable third order equations in 2 1 dimensions which generalize the examples of kadomtsev petviashvili veselov novikov and harry dym equations our approach is based on the observation that dispersionless limits of integrable systems in 2 1 dimensions possess infinitely many multi phase solutions coming from the so called hydrodynamic reductions in this paper we adopt a novel perturbative approach to the classification problem based on the method of hydrodynamic reductions we first classify integrable quasilinear systems which may potentially occur as dispersionless limits of soliton equations in 2 1 dimensions to reconstruct dispersive deformations we require that all hydrodynamic reductions of the dispersionless limit be inherited by the corresponding dispersive counterpart this procedure leads to a complete list of integrable third order equations some of which are apparently new

dispersionless integrable systems in 3d and einstein weyl geometry for several classes of second order dispersionless pdes we show that the symbols of their formal linearizations define conformal structures which must be einstein weyl in 3d or self dual in 4d if and only if the pde is integrable by the method of hydrodynamic reductions this demonstrates that the integrability of these dispersionless pdes can be seen from the geometry of their formal linearizations

integrability via geometry dispersionless differential equations in three and four dimensions we prove that the existence of a dispersionless lax pair with spectral parameter for a nondegenerate hyperbolic second order partial differential equation pde is equivalent to the canonical conformal structure defined by the symbol being einstein weyl on any solution in 3d and self dual on any solution in 4d the first main ingredient in the proof is a characteristic property for dispersionless lax pairs the second is the projective behaviour of the lax pair with respect to the spectral parameter both are established for nondegenerate determined systems of pdes of any order thus our main result applies more generally to any such pde system whose characteristic variety is a quadric hypersurface

the strange duality conjecture for generic curves let sux r be the moduli space of semi stable vector bundles of rank r with trivial determinant over a connected smooth projective algebraic curve x of genus g 1 over c recall that a vector bundle e on x is called semi stable if for any subbundle v deg v rk v deg e rk e points of sux r correspond to isomorphism classes of semi stable rank r vector bundles with trivial determinant up to an equivalence relation for any line bundle l of degree g 1 onx define l e sux r h e l 1 this turns out be a non zero cartier divisor whose associated line bundle l o l does not depend upon l it is known that l generates the picard group of sux r for this and the precise definition of l in terms of determinant of cohomology see dn let u x k be the moduli space of semi stable rank k and degree k g 1 bundles on x recall that on u x k there is a canonical non zero theta cartier divisor k whose underlying set is f u x k h x f 0 put m o k consider the natural map k r sux r u x k u x kr given by tensor product from the theorem of the square it follows that k rm is isomorphic to l m the canonical element kr h0 u x kr m and the kunneth theorem give a map well defined up to scalars

bergman kernels tyz expansions and hankel operators on the kepler manifold for a class of o n 1 r invariant measures on the kepler manifold possessing finite moments of all orders we describe the reproducing kernels of the associated bergman spaces discuss the corresponding asymptotic expansions of tian yau zelditch and study the relevant hankel operators with conjugate holomorphic symbols related reproducing kernels on the minimal ball are also discussed finally we observe that the kepler manifold either does not admit balanced metrics or such metrics are not unique

integrable systems and modular forms of level 2 a set of nonlinear differential equations associated with the eisenstein series of the congruent subgroup 0 2 of the modular group is constructed these nonlinear equations are analogues of the well known ramanujan equations as well as the chazy and darboux halphen equations associated with the modular group the general solutions of these equations can be realized in terms of the schwarz triangle function s 0 0 1 2 z

an infinite number of hidden variables in hyper kahler metrics three types of hidden variables both independent and dependent are shown to underlie the hyper kahler geometry in a complexified setting the variables satisfy an infinite set of differential equations hierarchy just as in the case of most nonlinear integrable systems the notion of the plebanski key functions j math phys 16 2395 1975 is extended to this hierarchy to give an analog of the notion of the function two examples of special solutions which are reminiscent of several solution techniques in the theory of nonlinear integrable systems are presented for illustration

verlinde bundles and generalized theta linear series in this paper we approach the study of generalized theta linear series on moduli of vector bundles on curves via vector bundle techniques on abelian varieties we study a naturally defined class of vector bundles on a jacobian called verlinde bundles in order to obtain information about duality between theta functions and effective global and normal generation on these moduli spaces

killing vectors in self dual euclidean einstein spaces using the formalism of complex h spaces we show that all real euclidean self dual spaces that admit at least one killing vector may be gauged so that only two distinct types of killing vectors appear in kahler coordinates these are the generators of a translational or a rotational symmetry we give explicit forms both for the killing vectors and for the constraint on the kahler potential function which allows for such a killing vector in the translational case we show how all such spaces are determined by the general solution of the three dimensional flat laplace s equation and how these are related to the multi taub nut metrics of gibbons and hawking in the rotational case we simplify the equation determining but this is not sufficient to obtain the general solution

preconditioned conjugate gradient method for finding minimal energy surfaces on powell sabin triangulations abstract we present an iterative proposal based on the preconditioned conjugate gradient method to solve the linear system associated to the problem of approximating a data set by a minimal energy surface constructed through a powell sabin finite element over a 1 type triangulation defined on a polygonal domain these approximation problems give rise to symmetric banded and positive definite matrices with a very special block structure that depends on the basis functions of the associated vector space and also on the numeration of the nodes in the triangulation in practice the associated sparse matrices are large and ill conditioned the special structure of these matrices allows us to adapt and explore several known preconditioned strategies to improve the performance of the conjugate gradient method we adapt and explore five different preconditioning strategies including some well known direct and also some recent inverse strategies special attention is paid to the delicate and difficult task of choosing the related parameters in each case the quality of each preconditioner is evaluated by observing the clustering of the preconditioned matrix eigenvalues and the obtained reduction in number of iterations we report on seven different surfaces and our results indicate that the best preconditioning strategies for this application are the ones based on incomplete factorizations nevertheless from a computational cost point of view all the explored strategies are competitive

hydrodynamic reductions of multidimensional dispersionless pdes the test for integrability a d 1 dimensional dispersionless pde is said to be integrable if its n component hydrodynamic reductions are locally parametrized by d 1 n arbitrary functions of one variable the most important examples include the four dimensional heavenly equation descriptive of self dual ricci flat metrics and its six dimensional generalization arising in the context of sdiff 2 self dual yang mills equations given a multidimensional pde which does not pass the integrability test the method of hydrodynamic reductions allows one to effectively reconstruct additional differential constraints which when added to the equation make it an integrable system in fewer dimensions as an example of this phenomenon we discuss the second commuting flow of the dispersionless kp hierarchy considered separately this is a four dimensional pde which does not pass the integrability test however the method of hydrodynamic reductions generates additional differential constraints which reconstruct the full 2 1 dimensional dis

equations of kummer varieties let x 0 be a principally polarized abelian variety over a field k of characteristic 2 we may assume that 0 1 0 and that we have an involution i ox 0 ox 0 covering the 1 morphism of x thus we have iln yx n0 yx n0 for all n which has global effect i f x yx n0 r f x yx n0 let f x yx no and f x yx n0 be the 1 and 1 eigenspace of i clearly d3n 0 f x yx n0 is a ring and d3n 0 f x yx n0 is a module over this ring in this paper we will prove

the characterization of two component 2 1 dimensional integrable systems of hydrodynamic type we obtain the necessary and sufficient conditions for a two component 2 1 dimensional system of hydrodynamic type to possess infinitely many hydrodynamic reductions these conditions are in involution implying that the systems in question are locally parametrized by 15 arbitrary constants it is proved that all such systems possess three conservation laws of hydrodynamic type and therefore are symmetrizable in godunov s sense moreover all such systems are proved to possess a scalar pseudopotential which plays the role of the dispersionless lax pair we demonstrate that the class of two component systems possessing a scalar pseudopotential is in fact identical with the class of systems possessing infinitely many hydrodynamic reductions thus establishing the equivalence of the two possible definitions of integrability explicit linearly degenerate examples are constructed

on the projective normality and normal presentation on higher dimensional varieties with nef canonical bundle abstract in this article we prove new results on projective normality and normal presentation of adjunction bundle associated to an ample and globally generated line bundle on higher dimensional smooth projective varieties with nef canonical bundle as one of the consequences of the main theorem we give bounds on very ampleness and projective normality of pluricanonical linear systems on varieties of general type in dimensions three four and five these improve known such results

simple waves in quasilinear hyperbolic systems ii riemann invariants for the problem of simple wave interactions in this paper a generalization of the riemann invariant method to the case of a nonhomogeneous system of equations has been formulated we have discussed in detail the necessary and sufficient conditions for the existence of riemann invariants we perform the analysis using the apparatus of differential forms and cartan theory of systems in involution the problem has been reduced to examining pfaff forms we have considered the connections between the structure of the set of integral elements and the possibility of a construction of special classes of solutions depending on k arbitrary functions of one variable these solutions can be interpreted physically as the interactions between k simple waves on a simple state we have proven that in the case of interaction of many simple waves described by riemann invariants a conservation law for the type and quantity of waves holds it has been also shown that such a solution resulting from the interaction of many simple waves propagating on the simple state

very ampleness and higher syzygies for calabi yau threefolds the authors prove various results concerning multiples of ample base point free linear systems on calabi yau threefolds suppose that b is an ample divisor on a calabi yau threefold x and that b has no base points then the authors prove that 3b is very ample and embeds x as a projectively normal variety if and only if b does not map x 2 1 onto p3 similarly they prove that 2b enjoys the same properties if and only if b does not map x onto a variety of minimal degree other than p3 nor maps x 2 1 onto p3 further results are proved giving conditions for when the linear system nb satisfies the condition np

integrable structure of the dirichlet boundary problem in multiply connected domains we study the integrable structure of the dirichlet boundary problem in two dimensions and extend the approach to the case of planar multiply connected domains the solution to the dirichlet boundary problem in the multiply connected case is given through a quasiclassical tau function which generalizes the tau function of the dispersionless toda hierarchy it is shown to obey an infinite hierarchy of hirota like equations which directly follow from properties of the dirichlet green function and from the fay identities the relation to multi support solutions of matrix models is briefly discussed

multidimensional integrable systems and deformations of lie algebra homomorphisms we use deformations of lie algebra homomorphisms to construct deformations of dispersionless integrable systems arising as symmetry reductions of anti self dual yang mills equations with a gauge group diff s1

projective normality of the moduli space of rank 2 vector bundles on a generic curve we prove that the embedding to the projective space by the generalized theta divisors of the moduli space of rank 2 vector bundles on a generic curve is projective normal

radial balanced metrics on the unit ball of the kepler manifold abstract we show that there is no radial balanced metric on the unit ball of the kepler manifold with not too wild boundary behavior additionally we identify explicitly the weights corresponding to radial metrics with such boundary behavior which satisfy the balanced condition as far as germs at the boundary are concerned related results for poincare metrics are also established

integrable hydrodynamic chains a new approach for derivation of benney type moment chains and integrable hydrodynamic type systems is presented new integrable hydrodynamic chains are constructed all their hydrodynamical reductions are described and integrated new 2 1 integrable hydrodynamic type systems are found