generalized kahler manifolds commuting complex structures and split tangent bundles we study generalized kahler manifolds for which the corresponding complex structures commute and classify completely the compact four dimensional ones

lie bialgebroids and poisson groupoids lie bialgebras arise as infinitesimal invariants of poisson lie groups a lie bialgebra is a lie algebra g with a lie algebra structure on the dual g which is compatible with the lie algebra g in a certain sense for a poisson group g the multiplicative poisson structure induces a lie algebra structure on the lie algebra dual g which makes g g into a lie bialgebra in fact there is a one one correspondence between poisson lie groups and lie bialgebras if the lie groups are assumed to be simply connected 7 16 19 the importance of poisson lie groups themselves arises in part from their role as classical limits of quantum groups 8 and in part because they provide a class of poisson structures for which the realization problem is tractable 15 poisson groupoids were introduced by weinstein 24 as a generalization of both poisson lie groups and the symplectic groupoids which arise in the integration of arbitrary poisson manifolds 4 11 he noted that the lie algebroid dual a g of a poisson groupoid g itself has a lie algebroid structure but did not develop the infinitesimal structure further in this paper we introduce and study a natural infinitesimal invariant for poisson groupoids the lie bialgebroids of the title our ultimate purpose is to develop a lie theory for poisson groupoids parallel to that for poisson groups in this paper we are primarily concerned with the first half of this process that is with the construction of the lie bialgebroid of a poisson groupoid after the preliminary 2 in which we describe the generalization to arbitrary lie algebroids of the exterior calculus and schouten calculus in 3 we define a lie bialgebroid to be a lie algebroid a whose dual a is also equipped with a lie algebroid structure such that the coboundary operator d a a associated to a satisfies a cocycle equation with respect to a the lie algebra of sections of a this is clearly a straightforward extension of the concept of a lie bialgebra 16 but cannot be formalized in lie algebroid cohomological terms since there is no satisfactory adjoint representation for a general lie algebroid most of 3 is devoted to proving that this definition is self dual if a a is a lie bialgebroid then a a is also in 4 we briefly consider the special case of lie bialgebroids satisfying a triangularity condition which include some important examples such as the usual triangular lie bialgebras and lie bialgebroids associated to poisson manifolds the techniques used in 2 4 are similar to those known for lie bialgebras it would be possible by suitably generalizing the proof for poisson groups to prove

symmetries in generalized kahler geometry we define the notion of a moment map and reduction in both generalized complex geometry and generalized kahler geometry as an application we give very simple explicit constructions of bi hermitian structures on mathbb c mathbb p n hirzebruch surfaces the blow up of mathbb c mathbb p n at arbitrarily many points and other toric varieties as well as complex grassmannians

a chain code for representing 3d curves abstract a chain code for representing three dimensional 3d curves is defined any 3d continuous curve can be digitalized and represented as a 3d discrete curve this 3d discrete curve is composed of constant straight line segments thus the chain elements represent the orthogonal direction changes of the constant straight line segments of the discrete curve the proposed chain code only considers relative direction changes which allows us to have a curve descriptor invariant under translation and rotation also this curve descriptor may be starting point normalized for open and closed curves and invariant under mirroring transformation in the content of this work the main characteristics of this chain code are presented this chain code is inspired by the work of guzman mcc technical report number aca 254 87 july 13 1987 for representing 3d stick bodies finally we present some results of this chain code to represent and process 3d discrete curves as linear features over the terrain by means of digital elevation model dem data also we use this chain code for representing solids composed of voxels thus each solid represents a dem which is described by only one chain

local symmetries of digital contours from their chain codes in this work symmetry is evaluated as a numeric feature for each point of a contour using only the positions of a local vicinity of points a measurement is defined named as local symmetric deficiency lsd so that the lower this quantity is the higher the symmetry will be in the local region considered this approach is very simple and it is based on a suitable manipulation of the chain code of the curve its computational cost is very low and it has the advantages of a parallel algorithm since values for lsd can be computed for each point independently

unobstructed k deformations of generalized complex structures and bi hermitian structures abstract we introduce k deformations of generalized complex structures on a compact kahler manifold m x j with an effective anti canonical divisor and show that obstructions to k deformations of generalized complex structures on m always vanish applying unobstructed k deformations and the stability theorem for generalized kahler structures we construct deformations of bi hermitian structures in the form j j t h t on a compact kahler surface with a non zero holomorphic poisson structure then we prove that a compact kahler surface s admits a non trivial bi hermitian structure with the torsion condition and the same orientation if and only if s has a non zero holomorphic poisson structure we also obtain bi hermitian structures j j h on del pezzo surfaces degenerate del pezzo surfaces and some ruled surfaces for which the complex structure j is not equivalent to j under diffeomorphisms

a formal language approach for a 3d curve representation abstract a formal language approach for representing three dimensional 3d curves is presented based on the chain code for representing 3d curves defined in 1 we propose an approach for mapping 3d curves into strings this mapping allows us to have a unique curve descriptor which is invariant under translation and rotation also it is possible to use inverse and mirroring operators and the use of formal language techniques for 3d curve generation and analysis finally we present a result of this approach to represent and to generate polygonal sequences convergent to cube filling hilbert curves

geometry of maurer cartan elements on complex manifolds the semi classical data attached to stacks of algebroids in the sense of kashiwara and kontsevich are maurer cartan elements on complex manifolds which we call extended poisson structures as they generalize holomorphic poisson structures a canonical lie algebroid is associated to each maurer cartan element we study the geometry underlying these maurer cartan elements in the light of lie algebroid theory in particular we extend lichnerowicz poisson cohomology and koszul brylinski homology to the realm of extended poisson manifolds we establish a sufficient criterion for these to be finite dimensional we describe how homology and cohomology are related through the evens lu weinstein duality module and we describe a duality on koszul brylinski homology which generalizes the serre duality of dolbeault cohomology

integration of holomorphic lie algebroids we prove that a holomorphic lie algebroid is integrable if and only if its underlying real lie algebroid is integrable thus the integrability criteria of crainic fernandes theorem 4 1 in crainic fernandes in ann math 2 157 2003 do also apply in the holomorphic context without any modification as a consequence we prove that a holomorphic poisson manifold is integrable if and only if its real part or imaginary part is integrable as a real poisson manifold

generalized kahler manifolds and off shell supersymmetry we solve the long standing problem of finding an off shell supersymmetric formulation for a general n 2 2 nonlinear two dimensional sigma model geometrically the problem is equivalent to proving the existence of special coordinates these correspond to particular superfields that allow for a superspace description we construct and explain the geometric significance of the generalized kahler potential for any generalized kahler manifold this potential is the superspace lagrangian

reduction of courant algebroids and generalized complex structures we present a theory of reduction for courant algebroids as well as dirac structures generalized complex and generalized kahler structures which interpolates between holomorphic reduction of complex manifolds and symplectic reduction the enhanced symmetry group of a courant algebroid leads us to define extended actions and a generalized notion of moment map key examples of generalized kahler reduced spaces include new explicit bi hermitian metrics on cp2

instantons poisson structures and generalized kahler geometry using the idea of a generalized kahler structure we construct bihermitian metrics on cp2 and cp1 cp1 and show that any such structure on a compact 4 manifold m defines one on the moduli space of anti self dual connections on a fixed principal bundle over m we highlight the role of holomorphic poisson structures in all these constructions

generalized complex geometry generalized complex geometry encompasses complex and symplectic ge ometry as its extremal special cases we explore the basic properties of this geometry including its enhanced symmetry group elliptic deforma tion theory relation to poisson geometry and local structure theory we also dene and study generalized complex branes which interpolate be tween at bundles on lagrangian submanifolds and holomorphic bundles on complex submanifolds

a new chain code abstract a new chain code for shapes composed of regular cells is defined this boundary chain code is based on the numbers of cell vertices which are in touch with the bounding contour of the shape this boundary chain code is termed vertex chain code vcc the vcc is invariant under translation and rotation also it may be starting point normalized and invariant under mirroring transformation using this concept of chain code it is possible to relate the chain length to the contact perimeter which corresponds to the sum of the boundaries of neighboring cells of the shape bribiesca e comp math appl 33 11 1997 1 9 also to relate the chain nodes to the contact vertices which correspond to the vertices of neighboring cells so in this way these relations among the chain and the characteristics of interior of the shape allow us to obtain interesting properties this work is motivated by the idea of obtaining various shape features computed directly from the vcc without going to cartesian coordinate representation finally in order to illustrate the capabilities of the vcc we present some results using real shapes

a contour tracing and coding algorithm for generating 2d contour codes from 3d classified objects in 3d image data sets generated by voxel based classification each voxel is marked with a specific class label voxels of the same class label can form 3d objects of extremely complex shape interactively drawn regions are usually represented by their 2d region borders in order to combine automatically classified with interactively drawn regions a contour tracing and coding algorithm for generating optimized 2d contours from 3d classified objects is presented a special conversion algorithm allows a chain or a crack code representation an application to medical images shows the method s necessity and usefulness in dealing with highly complex regions

representations up to homotopy of lie algebroids we introduce and study the notion of representation up to homotopy of a lie algebroid paying special attention to examples we use representations up to homotopy to define the adjoint representation of a lie algebroid and show that the resulting cohomology controls the deformations of the structure the weil algebra of a lie algebroid is defined and shown to coincide with kalkman s brst model for equivariant cohomology in the case of group actions the relation of this algebra with the integration of poisson and dirac structures is explained in arias abad crainic ann inst fourier

on global deformation quantization in the algebraic case abstract we give a proof of yekutieli s global algebraic deformation quantization result which does not rely on the choice of local sections of the bundle of affine coordinate systems instead we use an argument inspired by algebraic de rham cohomology

manin triples for lie bialgebroids in his study of dirac structures a notion which includes both poisson structures and closed 2 forms t courant introduced a bracket on the direct sum of vector fields and 1 forms this bracket does not satisfy the jacobi identity except on certain subspaces in this paper we systematize the properties of this bracket in the definition of a courant algebroid this structure on a vector bundle e rightarrow m consists of an antisymmetric bracket on the sections of e whose jacobi anomaly has an explicit expression in terms of a bundle map e rightarrow tm and a field of symmetric bilinear forms on e when m is a point the definition reduces to that of a lie algebra carrying an invariant nondegenerate symmetric bilinear form for any lie bialgebroid a a over m a notion defined by mackenzie and xu there is a natural courant algebroid structure on a oplus a which is the drinfel d double of a lie bialgebra when m is a point conversely if a and a are complementary isotropic subbundles of a courant algebroid e closed under the bracket such a bundle with dimension half that of e is called a dirac structure there is a natural lie bialgebroid structure on a a whose double is isomorphic to e the theory of manin triples is thereby extended from lie algebras to lie algebroids our work gives a new approach to bihamiltonian structures and a new way of combining two poisson structures to obtain a third one we also take some tentative steps toward generalizing drinfel d s theory of poisson homogeneous spaces from groups to groupoids

a kobayashi hitchin correspondence for i holomorphic bundles abstract in this paper we introduce the notions of hermitian einstein metric and stability for i holomorphic vector bundles on bi hermitian manifolds moreover we establish a kobayashi hitchin correspondence for i holomorphic vector bundles on bi hermitian manifolds examples of such vector bundles include generalized holomorphic bundles on generalized kahler manifolds we also show that stability of a vector bundle in this sense can depend on the parameter

a one pass algorithm for local symmetry of contours from chain codes this paper presents a one pass algorithm for computing the local symmetry of contours from their chain code representation this algorithm embeds the local symmetry deficiency lsd computation into our chain codes extraction process the existing algorithms need to extract chain codes first and use them as input to compute lsd the new algorithm uses a binary image as input and can complete the code extraction and the lsd computation in one pass it can also extract multiple contours in an image in one pass the algorithm is suitable for parallel implementation

contour sequence moments for the classification of closed planar shapes abstract a technique for classifying closed planar shapes is described in which a shape is characterized by a one dimensional sequence that represents the euclidean distance between the centroid and all contour pixels of the shape contour sequences obtained from shapes belonging to the same class are similar and are considered to be different realizations of a stochastic process statistical moment functions derived from contour sequences are used to characterize the process and hence the shapes it is shown that these moment functions are invariant to shape translation rotation and scale transformations examples are presented to illustrate the performance of these moment functions in the classification of noisy shapes

on the thickness of graphs of given degree abstract the results presented here refer to the determination of the thickness of a graph that is the minimum number of planar subgraphs into which the graph can be decomposed a useful general preliminary result obtained here is theorem 8 a planar graph has a planar representation for any arbitrary placement of its nodes in the plane it is then proved that if we have positive integers d and t such that any graph of degree at most d has thickness at most t the following hold theorem 9 any graph of degree d has thickness at most t roof d 1 d theorem 10 any graph of degree d can always be embedded in a regular graph g of any degree d corollary 5 any graph of degree d has thickness at most roof d 2 theorem 12 with d and t defined as above we have d 4t 2 corollary 6 if t 2 then d 6 we further conjecture that indeed the thickness of any graph of degree not exceeding 6 is never more than 2 and that more generally t roof d 2 4 since the design and fabrication of vlsi computer chips is essentially a concrete representation of the planar decomposition of a graph all these results are of direct practical interest

generalized kahler geometry generalized kahler geometry is the natural analogue of kahler geometry in the context of generalized complex geometry just as we may require a complex structure to be compatible with a riemannian metric in a way which gives rise to a symplectic form we may require a generalized complex structure to be compatible with a metric so that it defines a second generalized complex structure we prove that generalized kahler geometry is equivalent to the bi hermitian geometry on the target of a 2 dimensional sigma model with 2 2 supersymmetry we also prove the existence of natural holomorphic courant algebroids for each of the underlying complex structures and that these split into a sum of transverse holomorphic dirac structures finally we explore the analogy between pre quantum line bundles and gerbes in the context of generalized kahler geometry

morita equivalence of poisson manifolds poisson manifolds are the classical analogue of associative algebras for poisson manifolds symplectic realizations play a similar role as representations do for associative algebras in this paper the notion of morita equivalence of poisson manifolds a classical analogue of morita equivalence of algebras is introduced and studied it is proved that morita equivalent poisson manifolds have equivalent categories of complete symplectic realizations for certain types of poisson manifolds the geometric invariants of morita equivalence are also investigated