twistor spaces of hypercomplex manifolds are balanced abstract a hypercomplex structure on a differentiable manifold consists of three integrable almost complex structures that satisfy quaternionic relations if in addition there exists a metric on the manifold which is hermitian with respect to the three structures and such that the corresponding hermitian forms are closed the manifold is said to be hyperkahler in the paper non hermitian yang mills connections 13 kaledin and verbitsky proved that the twistor space of a hyperkahler manifold admits a balanced metric these were first studied in the article on the existence of special metrics in complex geometry 17 by michelsohn in the present article we review the proof of this result and then generalize it and show that twistor spaces of general compact hypercomplex manifolds are balanced

flat g bundles with canonical metrics on considere la relation entre la theorie invariante des actions de groupes algebriques semi simples sur des varietes algebriques complexes et le comportement de l application de moment et des donnees quantiques fournies par une structure symplectique associee on etudie un exemple de dimension infinie comprenant des fibres plats sur des varietes de riemann compactes

integrable systems and reductions of the self dual yang mills equations many integrable equations are known to be reductions of the self dual yang mills equations this article discusses some of the well known reductions including the standard soliton equations the classical painleve equations and integrable generalizations of the darboux halphen system and chazy equations the chazy equation first derived in 1909 is shown to correspond to the equations studied independently by ramanujan in 1916

vortices in holomorphic line bundles over closed kahler manifolds we apply a modified yang mills higgs functional to unitary bundles over closed kahler manifolds and study the equations which govern the global minima the solutions represent vortices in holomorphic bundles and are direct analogs of the vortices overr2 we obtain a complete description of the moduli space of these new vortices where the bundle is of rank one the description is in terms of a class of divisors in the base manifold there is also a dependence on a real valued parameter which can be attributed to the compactness of the base manifold

remarks on the geodesic einstein metrics of a relative ample line bundle in this paper we introduce the associated geodesic einstein flow for a relative ample line bundle l over the total space mathcal x of a holomorphic fibration and obtain a few properties of that flow in particular we prove that the pair mathcal x l is nonlinear semistable if the associated donaldson type functional is bounded from below and the geodesic einstein flow has long time existence property we also define the associated s classes and c classes for mathcal x l and obtain two inequalities between them when l admits a geodesic einstein metric finally in the appendix of this paper we prove that a relative ample line bundle is geodesic einstein if and only if an associated infinite rank bundle is hermitian einstein

the yang mills flow near the boundary of teichmuller space we study the behavior of the yang mills flow for unitary connections on compact and non compact oriented surfaces with varying metrics the flow can be used to define a one dimensional foliation on the space of su 2 representations of a once punctured surface this foliation universalizes over teichmuller space and is equivariant with respect to the action of the mapping class group it is shown how to extend the foliation as a singular foliation over the augmented boundary of teichmuller space obtained by adding nodal riemann surfaces continuity of this extension is the main result of the paper

the 2 1 dimensional nonisospectral relativistic toda hierarchy related to the generalized discrete painleve hierarchy in this paper we will concentrate on the topic of integrable discrete hierarchies in 2 1 dimensions and their connection with discrete painleve hierarchies by considering a 2 1 dimensional nonisospectral discrete linear problem two new 2 1 dimensional nonisospectral integrable lattice hierarchies the 2 1 nonisospectral relativistic toda lattice hierarchy and the 2 1 nonisospectral negative relativistic toda lattice hierarchy are constructed it is shown that the reductions of the two new 2 1 nonisospectral lattice hierarchies lead to the 2 1 dimensional nonisospectral volterra lattice hierarchy and the 2 1 dimensional nonisospectral negative volterra lattice hierarchy we also obtain two new 1 1 dimensional nonisospectral integrable lattice hierarchies and two new ordinary difference hierarchies which are direct reductions of the two 2 1 nonisospectral integrable lattice hierarchies one of the two difference hierarchies yields our previously obtained generalized discrete first painleve dpi hierarchy and another one yields a generalized alternative discrete second painleve alt dpii hierarchy

completely convex formulation of the chan vese image segmentation model the active contours without edges model of chan and vese ieee transactions on image processing 10 2 266 277 2001 is a popular method for computing the segmentation of an image into two phases based on the piecewise constant mumford shah model the minimization problem is non convex even when the optimal region constants are known a priori in siam journal of applied mathematics 66 5 1632 1648 2006 chan esedo lu and nikolova provided a method to compute global minimizers by showing that solutions could be obtained from a convex relaxation in this paper we propose a convex relaxation approach to solve the case in which both the segmentation and the optimal constants are unknown for two phases and multiple phases in other words we propose a convex relaxation of the popular k means algorithm our approach is based on the vector valued relaxation technique developed by goldstein et al ucla cam report 09 77 2009 and brown et al ucla cam report 10 43 2010 the idea is to consider the optimal constants as functions subject to a constraint on their gradient although the proposed relaxation technique is not guaranteed to find exact global minimizers of the original problem our experiments show that our method computes tight approximations of the optimal solutions particularly we provide numerical examples in which our method finds better solutions than the method proposed by chan et al siam journal of applied mathematics 66 5 1632 1648 2006 whose quality of solutions depends on the choice of the initial condition

parabolic higgs bundles and representations of the fundamental group of a punctured surface into a real group abstract we study parabolic g higgs bundles over a compact riemann surface with fixed punctures when g is a real reductive lie group and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in g with arbitrary holonomy around the punctures this generalizes simpson s results for gl n c to arbitrary complex and real reductive lie groups three interesting features are the relation between the parabolic degree and the tits geometry of the boundary at infinity of the symmetric space the treatment of the case when the logarithm of the monodromy is on the boundary of a weyl alcove and the correspondence of the orbits encoding the singularity via the kostant sekiguchi correspondence we also describe some special features of the moduli spaces when g is a split real form or a group of hermitian type

central extensions of current groups in two dimensions the paper is devoted to generalization of the theory of loop groups to the two dimensional case to every complex riemann surface we assign a central extension of the group of smooth maps from this surface to a simple complex lie group g by the jacobian of this surface this extension is topologically nontrivial as in the loop group case orbits of coadjoint representation of this extension correspond to equivalence classes of holomorphic principalg bundles over the surface when the surface is the torus elliptic curve classification of coadjoint orbits is related to linear difference equations in one variable and to classification of conjugacy classes in the loop group we study integral orbits for which the kirillov kostant form is a curvature form for some principal torus bundle the number of such orbits for a given level is finite as in the loop group case conjecturedly they correspond to analogues of integrable modules occurring in conformal field theory

chern simons states at genus one we present a rigorous analysis of the schrodinger picture quantization for thesu 2 chern simons theory on 3 manifold torusxline with insertions of wilson lines the quantum states defined as gauge covariant holomorphic functionals ofsu 2 connections on the torus are expressed by degree 2k theta functions satisfying additional conditions the conditions are obtained by splitting the space of semistablesu 2 connections into nine submanifolds and by analyzing the behavior of states at four codimension 1 strata we construct the knizhnik zamolodchikov bernard connection allowing to compare the states for different complex structures of the torus and different positions of the wilson lines by letting two wilson lines come together we prove a recursion relation for the dimensions of the spaces of states which together with the unproven absence of states for spins 1 2 level implies the verlinde dimension formula

the yang mills heat flow on the moduli space of framed bundles on a surface we study the analog of the yang mills heat flow on the moduli space of framed bundles on a cut surface existence and convergence of the heat flow give a stratification of morse type invariant under the action of the loop group we use the stratification to prove versions of kahler quantization commutes with reduction and kirwan surjectivity

curvature of the l2 metric on the direct image of a family of hermitian einstein vector bundles for a holomorphic family of simple hermitian einstein holomorphic vector bundles over a compact kahler manifold the locally free part of the associated direct image sheaf over the parameter space forms a holomorphic vector bundle and it is endowed with a hermitian metric given by the l 2 pairing using the hermitian einstein metrics our main result in this paper is to compute the curvature of the l 2 metric in the case of a family of hermitian holomorphic line bundles with fixed positive first chern form and under certain curvature conditions we show that the l 2 metric is conformally equivalent to a hermitian einstein metric as applications this proves the semi stability of certain picard bundles and it leads to an alternative proof of a theorem of kempf 0 introduction given a holomorphic family of simple hermitian einstein holomorphic vector bundles over a fixed compact kahler manifold and paramet rized by a complex manifold one obtains a coherent analytic sheaf over the parameter space by taking direct image the locally free part of the direct image sheaf forms a holomorphic vector bundle over a zariski open subset of the pa rameter space and is endowed with a hermitian metric given by the l 2 pairings on global holomorphic sections of the hermitian einstein bundles with respect to the hermitian einstein metrics and the kahler metric on the kahler manifold our objective in this paper is to compute the curvature of this l 2 metric this work is partly motivated by earlier works of various authors on the analogous problem of computing the curvature of the weil petersson metric on the moduli space of hermitian einstein vector bundles over a compact kahler manifold c st zt where the weil petersson metric is defined similarly by the l 2 pairing on the harmonic representatives of the kodaira spencer classes associated to tangent vectors of the moduli space in particular the approach of schumacher toma st is especially inspiring to us and their observation that the curvature form of the total space contains the harmonic representatives of the kodaira spencer classes also plays an important role in our computations in the case of a family of ample line bundles over a compact kahler manifold of semi positive ricci cuvature the curvature of the l 2 metric takes a simple form see theorem 2 when the parameter space is a compact kahler manifold the l 2 metric is conformally equivalent to a hermitian einstein metric which

l 2 metrics projective flatness and families of polarized abelian varieties we compute the curvature of the l 2 metric on the direct image of a family of hermitian holomorphic vector bundles over a family of compact kahler manifolds as an application we show that the l 2 metric on the direct image of a family of ample line bundles over a family of abelian varieties and equipped with a family of canonical hermitian metrics is always projectively flat when the parameter space is a compact kahler manifold this leads to the poly stability of the direct image with respect to any kahler form on the parameter space

the integrability criterion in su 2 chern simons gauge theory we prove that the multiplicity spaces appearing in chern simons theory as defioned by segal vanish unless they are associated to integrable representations this and other links with conformal field theory are examined

the limit of the hermitian yang mills flow on reflexive sheaves abstract in this paper we study the asymptotic behavior of the hermitian yang mills flow on a reflexive sheaf we prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the harder narasimhan seshadri filtration this answers a question by bando and siu

extremal kahler metrics on projective bundles over a curve let m p e be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle e over a compact complex curve of genus 2 building on ideas of fujiki 1992 27 we prove that m admits a kahler metric of constant scalar curvature if and only if e is polystable we also address the more general existence problem of extremal kahler metrics on such bundles and prove that the splitting of e as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal kahler metrics in kahler classes sufficiently far from the boundary of the kahler cone the methods used to prove the above results apply to a wider class of manifolds called rigid toric bundles over a semisimple base which are fibrations associated to a principal torus bundle over a product of constant scalar curvature kahler manifolds with fibres isomorphic to a given toric kahler variety we discuss various ramifications of our approach to this class of manifolds

2 1 dimensional integrable lattice hierarchies related to discrete fourth order nonisospectral problems in this paper we consider the 2 1 dimensional discrete fourth order nonisospectral problem by using the lax technique three new 2 1 dimensional nonisospectral four field integrable lattice hierarchies are constructed their reductions yield three 1 1 dimensional isospectral four field integrable lattice hierarchies due to mlaszak marciniak we make a comparison between the 1 1 dimensional discrete fourth order nonisospectral problem and the third order nonisospectral problem we found that the integrable lattice hierarchies related to the discrete fourth order nonisospectral problem have new characteristics

a kahler structure on the moduli space of isometric maps of a circle into euclidean space we study the spaces and and lip of smooth resp non degenerate lipschitz isometric maps of a circle into euclidean space modulo orientation preserving euclidean motions we prove that and lip are infinite dimensional kahler manifolds in particular they are complex frechet resp banach manifolds this is proved by an infinite dimensional version of the kirwan kempf ness theorem kir84 kn78 nes84 relating symplectic quotients to holomorphic quotients applied to the action ofpsl2 on the free loop space ofs2

instanton moduli as a novel map from tori to k3 surfaces a map is constructed from the moduli of hyper kahler tori to hyper kahler k3 surfaces which does not coincide with the kummer map the map takes a torus to the moduli space of so 3 connections on a bundle with nontrivial first stiefel whitney class and first pontrjagin class equal to 4 this map is shown to intersect the kummer moduli and also certain subvarieties of singular k3 surfaces our map is shown to satisfy the local torelli theorem and the k3 surfaces in its image are shown to carry a natural metric which is calabi yau

an instanton invariant for 3 manifolds to an oriented closed 3 dimensional manifoldm withh1 m 0 we assign a 8 graded homology groupi m whose euler characteristic is twice casson s invariant the definition uses a construction on the space of instantons onm

the yang mills flow and the atiyah bott formula on compact kahler manifolds we study the yang mills flow on a holomorphic vector bundle e over a compact k ahler manifold x along a solution of the flow we show that the curvature endomorphism i lambda f a t approaches in l 2 an endomorphism with constant eigenvalues given by the slopes of the quotients from the harder narasimhan filtration of e this proves a sharp lower bound for the hermitian yang mills functional and thus the yang mills functional generalizing to arbitrary dimension a formula of atiyah and bott first proven on riemann surfaces furthermore we show any reflexive extension to all of x of the limiting bundle e infty is isomorphic to rm gr rm hns e verifying a conjecture of bando and siu

euler number of instanton moduli space and seiberg witten invariants we show that a partition function of topological twisted n 4 yang mills theory is given by seiberg witten invariants on a riemannian four manifolds under the condition that the sum of the euler number and the signature of the four manifolds vanishes the partition function is the sum of the euler number of instanton moduli space when it is possible to apply the vanishing theorem also we obtain a relation of the euler number labeled by the instanton number k with seiberg witten invariants all calculations in this article are done without assuming duality

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

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

around polygons in ℝ3 and s3 we survey certain moduli spaces in low dimensions and some of the geometric structures that they carry and then construct identifications among all of these spaces in particular we identify the moduli spaces of polygons in 3 and s 3 the moduli space of restricted representations of the fundamental group of a punctured 2 sphere the moduli space of flat connections on a punctured sphere the moduli space of parabolic bundles on a sphere the moduli space of weighted points on 1 and the symplectic quotient of so 3 acting diagonally on s 2 n all of these spaces depend on parameters and some the above identifications require the parameters to be small one consequence of this work is that these spaces are all biholomorphic with respect to the most natural complex structures they can each be given

geodesic einstein metrics and nonlinear stabilities in this paper we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic einstein metric on this line bundle which generalize the classical stabilities and hermitian einstein metrics of holomorphic vector bundles we introduce a donaldson type functional and show that this functional attains its absolute minimum at geodesic einstein metrics and we also discuss the relations between the existence of geodesic einstein metrics and the nonlinear stabilities of the line bundle as an application we will prove that a holomorphic vector bundle admits a finsler einstein metric if and only if it admits a hermitian einstein metric which answers a problem posed by s kobayashi

generalised monopole equations on kahler surfaces in this article we establish a hitchin kobayashi type correspondence for generalised seiberg witten monopole equations on kahler surfaces we show that the stability criterion we obtain for the existence of solutions coincides with that of the usual seiberg witten monopole equations this enables us to construct a map from the moduli space of solutions to the generalised equations to effective divisors in this article we establish a hitchin kobayashi type correspondence for generalised seiberg witten monopole equations on kahler surfaces we show that the stability criterion we obtain for the existence of solutions coincides with that of the usual seiberg witten monopole equations this enables us to construct a map from the moduli space of solutions to the generalised equations to effective divisors

morse theory interpretation of topological quantum field theories topological quantum field theories are interpreted as a generalized form of morse theory this interpretation is applied to formulate the simplest topological quantum field theory topological quantum mechanics the only non trivial topological invariant corresponding to this theory is computed and identified with the euler characteristic using field theoretical methods this topological invariant is calculated in different ways and in the process a proof of the gauss bonnet chern avez formula as well as some results of degenerate morse theory are obtained

seiberg witten monopole equations on noncommutative r4 it is well known that due to vanishing theorems there are no nontrivial finite action solutions to the abelian seiberg witten sw monopole equations on euclidean four dimensional space r4 we show that this is no longer true for the noncommutative version of these equations i e on a noncommutative deformation r 4 of r4 there exist smooth solutions to the sw equations having nonzero topological charge we introduce action functionals for the noncommutative sw equations and construct explicit regular solutions all our solutions have finite energy we also suggest a possible interpretation of the obtained solutions as codimension four vortex like solitons representing d p 4 and d p 4 branes in a dp dp brane system in type ii superstring theory

the first chern form on moduli of parabolic bundles for moduli space of stable parabolic bundles on a compact riemann surface we derive an explicit formula for the curvature of its canonical line bundle with respect to quillen s metric and interprete it as a local index theorem for the family of overline partial operators in the associated parabolic endomorphism bundles the formula consists of two terms one standard proportional to the canonical kahler form on the moduli space and one nonstandard called a cuspidal defect that is defined by means of special values of the eisenstein maass series the cuspidal defect is explicitly expressed through the curvature forms of certain natural line bundles on the moduli space related to the parabolic structure we also compare our result with witten s volume computation

extremal metrics on ruled manifolds abstract in this paper we consider a compact kahler manifold with extremal kahler metric and a mumford stable holomorphic bundle over it we proved that if the holomorphic vector field defining the extremal kahler metric is liftable to the bundle and if the bundle is relatively stable with respect to the action of automorphisms of the manifold then there exist extremal kahler metrics on the projectivization of the dual vector bundle

negatively curved metrics on kodaira surfaces it has been known that kodaira surfaces of general type have negative holomorphic tangent bundles cf 1 this definition of negativity is algebraic geometric in nature i e negative in the sense of grauert 5 griffiths 7 discussed the relations between various notions of negativity in algebraic geometry and those in differential geometry and raised the question of whether a grauert negative holomorphic vector bundle also admits a hermitian metric of negative holomorphic bisectional curvature if it does we would say that this metric is negatively curved and the vector bundle is negative in griffiths sense the aim of this work is to show that the tangent bundles of certain type of surfaces considered in 11 by schneider including kodaira surfaces of general type are negative in griffiths sense we also remark here that cheung 2 constructed on kodaira surfaces of general type a k ihler metric of negative holomorphic sectional curvature

a complex frobenius theorem multiplier ideal sheaves and hermitian einstein metrics on stable bundles a complex frobenius theorem is proved for subsheaves of a holomorphic vector bundle satisfying a finite generation condition and a differential inclusion relation a notion of multiplier ideal sheaf for a sequence of hermitian metrics is defined the complex frobenius theorem is applied to the multiplier ideal sheaf of a sequence of metrics along donaldson s heat flow to give a construction of the destabilizing subsheaf appearing in the donaldson uhlenbeck yau theorem in the case of algebraic surfaces

the witten reshetikhin turaev invariant for links in finite order mapping tori i abstract we state asymptotic expansion and growth rate conjectures for the witten reshetikhin turaev invariants of arbitrary framed links in 3 manifolds and we prove these conjectures for the natural links in mapping tori of finite order automorphisms of marked surfaces our approach is based upon geometric quantisation of the moduli space of parabolic bundles on the surface which we show coincides with the construction of the witten reshetikhin turaev invariants using conformal field theory as was recently completed by andersen and ueno

exotic smoothness and physics the essential role played by differentiable structures in physics is reviewed in light of recent mathematical discoveries that topologically trivial space time models especially the simplest one r4 possess a rich multiplicity of such structures no two of which are diffeomorphic to each other and thus to the standard one this means that physics has available to it a new panoply of structures for space time models these can be thought of as a source of new global but not properly topological features this article reviews some background differential topology together with a discussion of the role which a differentiable structure necessarily plays in the statement of any physical theory recalling that diffeomorphisms are at the heart of the principle of general relativity some of the history of the discovery of exotic i e nonstandard differentiable structures is reviewed some new results suggesting the spatial localization of such exotic structures are described and speculations are made on the possible opportunities that such structures present for the further development of physical theories

scaling behavior of two dimensional domain growth computer simulation of vertex models computer simulations are performed for vertex models which are coarse grained models for dynamical cellular patterns in two dimensions by simulating large systems we obtain conclusive evidence of scaling behavior that is a power law for the growth of the average cell size and the scaling properties for the distribution functions of edge number and size of cells several versions of the vertex models are obtained by making some approximations for the equation of motion of a vertex and we compare the statistical properties of the patterns in the scaling regime

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

the reverse yang mills higgs flow in a neighbourhood of a critical point the main result of this paper is a construction of solutions to the reverse yang mills higgs flow converging in the c topology to a critical point the construction uses only the complex gauge group action which leads to an algebraic classification of the isomorphis m classes of points in the unstable set of a critical point in terms of a filtration of the underlying higgs bundle analysing the compatibility of this filtration with the hard er narasimhan seshadri double filtration gives an algebraic criterion for two critical points to be connect ed by a flow line as an application we can use this to construct hecke modifications of higgs bundles via the yan g mills higgs flow when the higgs field is zero corresponding to the yang mills flow this criterion has a geometric interpretation in terms of secant varieties of the projectivisation of the underlying bundle inside the unstable manifold of a critical point which gives a precise description of broken and unbroken flow lines connec ting two critical points for non zero higgs field at generic critical points the analogous interpretation in volves the secant varieties of the spectral curve of the higgs bundle

calabi yau manifolds as complete intersections in products of complex projective spaces we consider constructions of manifolds withsu 3 holonomy as embedded in products of complex projective spaces by imposing certain homogeneous holomorphic constraints we prove that every such construction leads to one deformation class of manifolds withsu 3 holonomy for a subset of these manifolds we prove simple connectedness address the problem of calculating the second betti number and explicitly calculate it for a class of constructions this establishes a very wide class of manifolds withsu 3 holonomy that can give rise to yet many more constructions via dividing out the action of suitably chosen discrete groups some of the examples studied may yield phenomenologically acceptable models

surgery in cusp neighborhoods and the geography of irreducible 4 manifolds since its inception the theory of smooth 4 manifolds has relied upon complex surface theory to provide its basic examples this is especially true for simply connected 4 manifolds a longstanding conjecture held that every smooth simply connected closed 4 manifold is the connected sum of manifolds each admitting a complex structure with one of its orientations for this conjecture the 4 sphere was considered as the empty connected sum quite recently this was disproved by gompf and mrowka gm1 who produced infinite families of examples of 4 manifolds homeomorphic to the k3 surface and which admit no complex structure with either orientation the discovery of these dramatic counterexamples to the h cobordism theorem in dimension 4 was the most recent success in a long line of applications of donaldson invariants to the study of smooth structures on 4 manifolds beginning with donaldson s d4 and followed by fm1 ko ov1 ov2 which depended on donaldson s f invariant or a suitable generalization and fmm dk e2 eo1 mn2 sa using donaldson s polynomial invariant a common thread running through many of these counterexamples to the h cobordism conjecture is the process of logarithmic transformation which introduces multiple fibres in an elliptic surface in controlled situations this process will change the diffeomorphism type of a given elliptic surface while keeping its homotopy type fixed taking advantage of the complex structure on an elliptic surface this change can often be detected by using the relationship between anti self dual connections over a complex surface and stable holomorphic bundles d2 an alternative and more differential geometric

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

topological field theory and rational curves we analyze the quantum field theory corresponding to a string propagating on a calabi yau threefold this theory naturally leads to the consideration of witten s topological non linear model and the structure of rational curves on the calabi yau manifold we study in detail the case of the world sheet of the string being mapped to a multiple cover of an isolated rational curve and we show that a natural compactification of the moduli space of such a multiple cover leads to a formula in agreement with a conjecture by candelas de la ossa green and parkes

the moduli space of stable vector bundles over a real algebraic curve we study the spaces of stable real and quaternionic vector bundles on a real algebraic curve the basic relationship is established with unitary representations of an extension of mathbb z 2 by the fundamental group by comparison with the space of real or quaternionic connections some of the basic topological invariants of these spaces are calculated