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

betti numbers of holomorphic symplectic quotients via arithmetic fourier transform a fourier transform technique is introduced for counting the number of solutions of holomorphic moment map equations over a finite field this technique in turn gives information on betti numbers of holomorphic symplectic quotients as a consequence simple unified proofs are obtained for formulas of poincare polynomials of toric hyperkahler varieties recovering results of bielawski dancer and hausel sturmfels poincare polynomials of hilbert schemes of points and twisted atiyah drinfeld hitchin manin adhm spaces of instantons on 2 recovering results of nakajima yoshioka and poincare polynomials of all nakajima quiver varieties as an application a proof of a conjecture of kac on the number of absolutely indecomposable representations of a quiver is announced

moment maps scalar curvature and quantization of kahler manifolds building on donaldson s work on constant scalar curvature metrics we study the space of regular kahler metrics e i e those for which deformation quantization has been defined by cahen gutt and rawnsley after giving in sects 2 and 3 a review of donaldson s moment map approach we study the essential uniqueness of balanced basis i e of coherent states in a more general setting theorem 2 5 we then study the space e in sect 4 and we show in sect 5 how all the tools needed can be defined also in the case of non compact manifolds

modularity quaternion kahler spaces and mirror symmetry we provide an explicit twistorial construction of quaternion kahler manifolds ob tained by deformation of c map spaces and carrying an isometric action of the modular group sl 2 z the deformation is not assumed to preserve any continuous isometry and therefore this construction presents a general framework for describing ns5 brane instanton effects in string compactifications with n 2 supersymmetry in this context the modular invariant parametrization of twistor lines found in this work yields the complete non perturbative mirror map between type iia and type iib physical fields

kahlerian twistor spaces the work of r penrose has shown the deep relationship between the conformal structure of minkowski space and the complex geometry of lines in projective three space this transformation applies to a more general class of conformal structures and in particular there is a riemannian version described in 5 which is itself of interest to mathematical physicists basically to each oriented riemannian 4manifold x one associates canonically a 6 manifold z the twistor space of x together with an almost complex structure if this structure is integrable then z becomes a complex 3 manifold with a distinguished family of projective lines on it whose geometry describes the conformal geometry of the manifold x it is natural to ask what sort of complex manifold one obtains by this process in this paper we ask the more specific question which compact 4c manifolds have ka hlerian twistor spaces 1 the answer rather surprisingly is that there are just two examples the 4 sphere s and the complex projective plane p2 c each with their standard homogeneous conformal structures the twistor space of s 4 is the 3 dimensional projective space p3 c and that of p 2 q is the manifold f2 c of flags in c 3 the essential point of the proof of this result is the remark that if z is a twistor space with a kahler metric then the canonical bundle k is negative using vanishing theorems and the riemann roch theorem we then narrow down the possibilities for z to four cases projective space the flag manifold the intersection of two quadrics in p5 and the double covering of p3 branched over a non singular quartic surface the last two candidates cannot be realized however because the value of their euler characteristic is incompatible with their being twistor spaces throughout the proof we make essential use of a real structure on z which reduces considerably the occurrence of singularities at various stages aside from this our methods are the standard ones of algebraic geometry the four examples to which the proof leads us are all fano threefolds if we drop the requirement of compactness on x then it is quite possible for an open set of a fano threefold to be a twistor space in particular we show that the family of conies lying on the intersection of two

on the equality constraints tolerance of constrained optimization problems the tolerance value plays an important role when converting equality constraints into inequality constraints in solving constrained optimization problems many researchers use a fixed or dynamic setting directly based on trial or experiments without systematic study as a well known constraint handling technique deb s feasibility based rule is widely adopted but it has one drawback as the ranking is not consistent with the actual ranking after introducing the tolerance value after carefully analyzing how the tolerance value influences the ranking difference a novel strategy named ranking adjustment strategy ras is proposed which can be considered as a complement of deb s feasibility based rule the experiment has verified the effectiveness of the proposed strategy this is the first time to analyze the inner mechanism of the tolerance value for equality constraints systematically which can give some guide for future research

noncommutative instantons and twistor transform recently n nekrasov and a schwarz proposed a modification of the adhm construction of instantons which produces instantons on a noncommutative deformation of 4 in this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces we exhibit one to one correspondences between three classes of objects framed bundles on a noncommutative 2 certain complexes of sheaves on a noncommutative 3 and the modified adhm data the modified adhm construction itself is interpreted in terms of a noncommutative version of the twistor transform we also prove that the moduli space of framed bundles on the noncommutative 2 has a natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the moduli space of framed torsion free sheaves on the commutative 2 the natural complex structures on the two moduli spaces do not coincide but are related by an so 3 rotation finally we propose a construction of instantons on a more general noncommutative 4 than the one considered by nekrasov and schwarz a q deformed 4

ergodic complex structures on hyperkahler manifolds let m be a compact complex manifold the corresponding teichmuller space teich is the space of all complex structures on m up to the action of the group rm diff 0 m of isotopies the mapping class group gamma rm diff m rm diff 0 m acts on teich in a natural way an ergodic complex structure is a complex structure with a gamma orbit dense in teich let m be a complex torus of complex dimension ge 2 or a hyperkahler manifold with b 2 3 we prove that m is ergodic unless m has maximal picard rank there are countably many such m this is used to show that all hyperkahler manifolds are kobayashi non hyperbolic

hodge cohomology of gravitational instantons we study the space of l2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type these metrics generalize the geometric structures at infinity of several different well known classes of metrics including asymptotically locally euclidean manifolds the known types of gravitational instantons and also poincare metrics on q rank 1 ends of locally symmetric spaces and on the complements of smooth divisors in kahler manifolds the answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold the l2 signature formula implied by our result is closely related to the one proved by dai 25 and more generally by vaillant 67 and identifies dai s invariant directly in terms of intersection cohomology of differing perversities this work is also closely related to a recent paper of carron 12 and the forthcoming paper of cheeger and dai 17 we apply our results to a number of examples gravitational instantons among them arising in predictions about l2 harmonic forms in duality theories in string theory

on quantitative versions of theorems due to f e browder and r wittmann abstract this paper is another case study in the program of logically analyzing proofs to extract new typically effective information proof mining we extract explicit uniform rates of metastability in the sense of t tao from two ineffective proofs of a classical theorem of f e browder on the convergence of approximants to fixed points of nonexpansive mappings as well as from a proof of a theorem of r wittmann which can be viewed as a nonlinear extension of the mean ergodic theorem the first rate is extracted from browder s original proof that is based on an application of weak sequential compactness in addition to a projection argument wittmann s proof follows a similar line of reasoning and we adapt our analysis of browder s proof to get a quantitative version of wittmann s theorem as well in both cases one also obtains totally elementary proofs even for the strengthened quantitative forms of these theorems that neither use weak compactness nor the existence of projections anymore in this way the present article also discusses general features of extracting effective information from proofs based on weak compactness we then extract another rate of metastability of similar nature from an alternative proof of browder s theorem essentially due to halpern that already avoids any use of weak compactness the paper is concluded by general remarks concerning the logical analysis of proofs based on weak compactness as well as a quantitative form of the so called demiclosedness principle in a subsequent paper these results will be utilized in a quantitative analysis of baillon s nonlinear ergodic theorem

on the analysis of the 1 1 evolutionary algorithm many experimental results are reported on all types of evolutionary algorithms but only few results have been proved a step towards a theory on evolutionary algorithms in particular the so called 1 1 evolutionary algorithm is performed linear functions are proved to be optimized in expected time o nlnn but only mutation rates of size 1 n can ensure this behavior for some polynomial of degree 2 the optimization needs exponential time the same is proved for a unimodal function both results were not expected by several other authors finally a hierarchy result is proved moreover methods are presented to analyze the behavior of the 1 1 evolutionary algorithm

hyper kahler metrics and harmonic superspace the most general unconstrained superfield action for self interactingn 2 matter hypermultiplets in analyticn 2 superspace is argued to produce a most generaln 2 hyper kahler model after eliminating an infinite set of auxiliary fields this suggests a new possibility of classifying hyper kahler metrics according to then 2 analytic superfield self interactions and provides an effective tool to compute these metrics explicitly as the simplest example the u 2 invariant quartic self coupling of a singleq hypermultiplet is analyzed and is shown to yield the familiar taub nut metric to see the geometric pattern directly in terms ofn 2 superfields we introduce a new on shell representation ofq hypermultiplets inn 2 harmonic superspace similar to the description ofn 2 gauge theories for the u 2 example this formulation is checked to coincide with that by sierra and townsend

abcd of instantons we solve open image in new window 2 supersymmetric yang mills theories for an arbitrary classical gauge group i e su n so n sp n in particular we derive the prepotential of the low energy effective theory and the corresponding seiberg witten curves we manage to do this without resolving singularities of the compactified instanton moduli spaces

manifolds of g2 holonomy from the n 4 sigma model using the two dimensional 2d n 4 sigma model with u 1 r gauge symmetry and introducing the ade cartan matrices as gauge matrix charges we build toric hyper kahler eight real dimensional manifolds x8 dividing by one toric geometry circle action of x8 manifolds we present examples describing quotients x7 x8 u 1 of g2 holonomy in particular for the ar cartan matrix the quotient space is a cone on an s2 bundle over r intersecting wcp2 1 2 1 weighted projective spaces according to the ar dynkin diagram

hyperkahler manifolds and nonlinear supermultiplets we present a new construction of hyperkahler metrics that derives from the 3 dimensionaln 4 nonlinear supermultiplet further we give a detailed description of the nonlinear multiplet inn 2 and 4 superspace

birational geometry of singular moduli spaces of o grady type abstract following bayer and macri we study the birational geometry of singular moduli spaces m of sheaves on a k3 surface x which admit symplectic resolutions more precisely we use the bayer macri map from the space of bridgeland stability conditions stab x to the cone of movable divisors on m to relate wall crossing in stab x to birational transformations of m we give a complete classification of walls in stab x and show that every minimal birational model of m in the sense of the log minimal model program appears as a moduli space of bridgeland semistable objects on x an essential ingredient of our proof is an isometry between the orthogonal complement of a mukai vector inside the algebraic mukai lattice of x and the neron severi lattice of m which generalises results of yoshioka as well as perego and rapagnetta moreover this allows us to conclude that the symplectic resolution of m is deformation equivalent to the 10 dimensional irreducible holomorphic symplectic manifold found by o grady

subsolution theorem and the dirichlet problem for the quaternionic monge ampere equation in this paper we obtain the existence of solution to the dirichlet problem for quaternionic monge ampere equation after showing a stability theorem we prove the subsolution theorem for quaternionic monge ampere equation by combining our stability theorem and kolodziej s approach for the complex case

fano varieties of cubic fourfolds containing a plane we prove that the fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a k3 surface on the other hand we show that the fano varieties are always birational to moduli spaces of twisted stable coherent sheaves on a k3 surface the moduli spaces of complexes and of sheaves are related by wall crossing in the derived category of twisted sheaves on the corresponding k3 surface

twisted cubics on singular cubic fourfolds on starr s fibration we show that the hilbert scheme compactification of the total space of starr s fibration on the space of twisted cubics on a cubic hypersurface in mathbb p 5 not containing a plane admits a contraction to a singular projective symplectic variety of dimension eight which has a crepant resolution deformation equivalent to the symplectic eightfold constructed from twisted cubics on a smooth cubic fourfold this yields another proof that the symplectic eightfold and the hilbert scheme of four points on a k3 surface are deformation equivalent as a byproduct we obtain similar results for the variety of lines on a singular cubic fourfold

on the kodaira dimension of orthogonal modular varieties we prove that up to scaling there are only finitely many integral lattices l of signature 2 n with n ge 21 or n 17 such that the modular variety defined by the orthogonal group of l is not of general type in particular when n ge 108 every modular variety defined by an arithmetic group for a rational quadratic form of signature 2 n is of general type we also obtain similar finiteness in n ge 9 for the stable orthogonal groups as a byproduct we derive finiteness of lattices admitting reflective modular form of bounded vanishing order which proves a conjecture of gritsenko and nikulin

choosing selection pressure for wide gap problems abstract to exploit an evolutionary algorithm s performance to the full extent the selection scheme should be chosen carefully empirically it is commonly acknowledged that low selection pressure can prevent an evolutionary algorithm from premature convergence and is thereby more suitable for wide gap problems however there are few theoretical time complexity studies that actually give the conditions under which a high or a low selection pressure is better in this paper we provide a rigorous time complexity analysis showing that low selection pressure is better for the wide gap problems with two optima

hyperkahler quotient construction of bps monopole moduli spaces we use the hyperkahler quotient of flat space to obtain some monopole moduli space metrics in explicit form using this new description we discuss their topology completeness and isometries we construct the moduli space metrics in the limit when some monopoles become massless which corresponds to non maximal symmetry breaking of the gauge group we also introduce a new family of hyperkahler metrics which depending on the mass parameter being positive or negative tend to either the asymptotic metric on the moduli space of many su 2 monopoles or to previously unknown metrics these new metrics are singular or complete depending on the particular choice of the level set of the moment map the singular metrics are of relevance to the moduli spaces of vacua of three dimensional gauge theories for higher rank gauge groups finally we make a few comments concerning the existence of closed or bound orbits on some of these manifolds and the integrability of the geodesic flow

drift analysis and average time complexity of evolutionary algorithms the computational time complexity is an important topic in the theory of evolutionary algorithms eas this paper reports some new results on the average time complexity of eas based on drift analysis some useful drift conditions for deriving the time complexity of eas are studied including conditions under which an ea will take no more than polynomial time in problem size to solve a problem and conditions under which an ea will take at least exponential time in problem size to solve a problem the paper first presents the general results and then uses several problems as examples to illustrate how these general results can be applied to concrete problems in analyzing the average time complexity of eas while previous work only considered 1 1 eas without any crossover the eas considered in this paper are fairly general which use a finite population crossover mutation and selection 2001 elsevier science b v all rights reserved

numerical characterization of the kahler cone of a compact kahler manifold the goal of this work is to give a precise numerical description of the kahler cone of a compact kahler manifold our main result states that the kahler cone depends only on the intersection form of the cohomology ring the hodge structure and the homology classes of analytic cycles if x is a compact kahler manifold the kahler cone k of x is one of the connected components of the set p of real 1 1 cohomology classes which are numerically positive on analytic cycles i e y p 0 for every irreducible analytic set y in x p dim y this result is new even in the case of projective manifolds where it can be seen as a generalization of the well known nakai moishezon criterion and it also extends previous results by campana peternell and eyssidieux the principal technical step is to show that every nef class which has positive highest self intersection numberx n 0 contains a kahler current this is done by using the calabi yau theorem and a mass concentration technique for monge ampere equations the main result admits a number of variants and corollaries including a description of the cone of numerically effective 1 1 classes and their dual cone another important consequence is the fact that for an arbitrary deformation x s of compact kahler manifolds the kahler cone of a very general fibre xt is independent of t i e invariant by parallel transport under the 1 1 component of the gauss manin connection

a lie algebra attached to a projective variety each choice of a kahler class on a compact complex manifold defines an action of the lie algebra sl 2 on its total complex cohomology if a nonempty set of such kahler classes is given then we prove that the corresponding sl 2 copies generate a semisimple lie algebra we investigate the formal properties of the resulting representation and we work things out explicitly in the case of complex tori hyperkahler manifolds and flag varieties we pay special attention to the cases where this leads to a jordan algebra structure or a graded frobenius algebra

teichmuller spaces and torelli theorems for hyperkahler manifolds kreck and yang su recently gave counterexamples to a version of the torelli theorem for hyperkahlerian manifolds as stated by verbitsky we extract the correct statement and give a short proof of it we also revisit a few of its consequences some of which are given new shorter proofs and ask some questions

n 2 super yang mills theory in projective superspace we constructn 2 yang mills theory in projective superspace by exploiting the analogy to ward s twistor construction of self dual yang mills fields

integrating over higgs branches we develop some useful techniques for integrating over higgs branches in supersymmetric theories with 4 and 8 supercharges in particular we define a regularized volume for hyperkahler quotients we evaluate this volume for certain ale and alf spaces in terms of the hyperkahler periods we also reduce these volumes for a large class of hyperkahler quotients to simpler integrals these quotients include complex coadjoint orbits instanton moduli spaces on 4 and ale manifolds hitchin spaces and moduli spaces of parabolic higgs bundles on riemann surfaces in the case of hitchin spaces the evaluation of the volume reduces to a summation over solutions of bethe ansatz equations for the non linear schrodinger system we discuss some applications of our results

reflection functors for quiver varieties and weyl group actions we introduce reflectionfunctors on quiver varieties they are hyper kahler isometries between quiver varieties with different parameters related by elements in the weyl group the definition is motivated by the origial reflection functor given by bernstein gelfand ponomarev 1 but they behave much nicely they are isomorphisms and satisfy the weyl group relations as an application we define weyl group representations of homology groups of quiver varieties they are analogues of slodowy s construction of springer representations of the weyl group

a large population size can be unhelpful in evolutionary algorithms the utilization of populations is one of the most important features of evolutionary algorithms eas there have been many studies analyzing the impact of different population sizes on the performance of eas however most of such studies are based on computational experiments except for a few cases the common wisdom so far appears to be that a large population would increase the population diversity and thus help an ea indeed increasing the population size has been a commonly used strategy in tuning an ea when it did not perform as well as expected for a given problem he and yao 2002 8 showed theoretically that for some problem instance classes a population can help to reduce the runtime of an ea from exponential to polynomial time this paper analyzes the role of population further in eas and shows rigorously that large populations may not always be useful conditions under which large populations can be harmful are discussed in this paper although the theoretical analysis was carried out on one multimodal problem using a specific type of eas it has much wider implications the analysis has revealed certain problem characteristics which can be either the problem considered here or other problems that lead to the disadvantages of large population sizes the analytical approach developed in this paper can also be applied to analyzing eas on other problems

mapping class group and a global torelli theorem for hyperkahler manifolds a mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies we compute a mapping class group of a hyperkahler manifold m showing that it is commensurable to an arithmetic lattice in so 3 b2 3 a teichmuller space of m is a space of complex structures on m up to isotopies we define a birational teichmuller space by identifying certain points corresponding to bimeromorphically equivalent manifolds we show that the period map gives the isomorphism between connected components of the birational teichmuller space and the corresponding period space so b2 3 3 so 2 so b2 3 1 we use this result to obtain a torelli theorem identifying each connected component of the birational moduli space with a quotient of a period space by an arithmetic group when m is a hilbert scheme of n points on a k3 surface with n 1 a prime power our torelli theorem implies the usual hodge theoretic birational torelli theorem for other examples of hyperkahler manifolds the hodge theoretic torelli theorem is known to be false

addendum to effective metastability of halpern iterates in cat 0 spaces adv math 231 5 2012 2526 2556 abstract we show that one proposition in our original paper allows for a much simpler proof resulting in certain improvements of the bounds in the main theorems

the use of tail inequalities on the probable computational time of randomized search heuristics for the purpose of analyzing the time cost of evolutionary algorithms eas or other types of randomized search heuristics rshs with certain requirements on the probability of obtaining a target solution this paper proposes a new index called the probable computational time pct which complements expected running time analysis using simple tail inequalities such as markov s inequality and chebyshev s inequality we also provide basic properties of pct explicitly exhibiting the general relationships between the expected running time and the pct to present deeper estimations of the pct for specific rshs and problems we demonstrate a new inequality that is based on the general form of the chernoff inequality and previous methods such as fitness based partitions and potential functions which have been used to analyze the expected running time of rshs the precondition of the new inequality is that the total running time can be described as the sum of a linear combination of some independent geometrically distributed variables and a constant term the new inequality always provides meaningful upper bounds for the pct under such circumstances some applications of the new inequality for simple eas ant colony optimization aco and particle swarm optimization pso algorithms on simple pseudo boolean functions are illustrated in this paper

effective metastability of halpern iterates in cat 0 spaces abstract this paper provides an effective uniform rate of metastability in the sense of tao on the strong convergence of halpern iterations of nonexpansive mappings in cat 0 spaces the extraction of this rate from an ineffective proof due to saejung is an instance of the general proof mining program which uses tools from mathematical logic to uncover hidden computational content from proofs this methodology is applied here for the first time to a proof that uses banach limits and hence makes a substantial reference to the axiom of choice

hyperkahler structures and infinite dimensional grassmannians abstract in this paper we describe an example of a hyperkahler quotient of a banach manifold by a banach lie group although the initial manifold is not diffeomorphic to a hilbert manifold not even to a manifold modelled on a reflexive banach space the quotient space obtained is a hilbert manifold which can be furthermore identified either with the cotangent space of a connected component gr res j j z of the restricted grassmannian or with a natural complexification of this connected component thus proving that these two manifolds are isomorphic hyperkahler manifolds moreover kahler potentials associated with the natural complex structure of the cotangent space of gr res j and with the natural complex structure of the complexification of gr res j are computed using kostant souriau s theory of prequantization

a construction of non kahler calabi yau manifolds and new solutions to the strominger system abstract we propose a new construction of compact non kahler calabi yau manifolds with balanced metrics and study the strominger system on them in particular we obtain explicit solutions to the strominger system with degeneracies on g t 4 where g is an immersed minimal surface of genus g 3 in flat t 3

tangentially positive isometric actions and conjugate points let m g be a complete riemannian manifold with no conjugate points and f m g b g b a principal g bundle where g is a lie group acting by isometries and b the smooth quotient with g b the riemannian submersion metric we obtain a characterization of conjugate point free quotients b g b in terms of symplectic reduction and a canonical pseudo riemannian metric on the tangent bundle tm from which we then derive necessary conditions involving g and m for the quotient metric to be conjugate point free particularly for m a reducible riemannian manifold let g tm with the lie algebra of g be the moment map of the tangential g action on tm and let gp be the canonical pseudo riemannian metric on tm defined by the symplectic form d and the map f tm m x m f z exp z exp z first we prove a theorem stating that if gp is not positive definite on the action vector fields for the tangential action along g 1 0 then b g b acquires conjugate points we proved the converse result in 2005 then we characterize self parallel vector fields on m in terms of the positivity of the gp length of their tangential lifts along certain canonical subsets of tm we use this to derive some necessary conditions on g and m for actions to be tangentially positive on relevant subsets of tm which we then apply to isometric actions on complete conjugate point free reducible riemannian manifolds when one of the irreducible factors satisfies certain curvature conditions

linear perturbations of quaternionic metrics we extend the twistor methods developed in our earlier work on linear deformations of hyperkahler manifolds 1 to the case of quaternionic kahler manifolds via swann s construction deformations of a 4d dimensional quaternionic kahler manifold mathcal m are in one to one correspondence with deformations of its 4d 4 dimensional hyperkahler cone mathcal s the latter can be encoded in variations of the complex symplectomorphisms which relate different locally flat patches of the twistor space mathcal z mathcal s with a suitable homogeneity condition that ensures that the hyperkahler cone property is preserved equivalently we show that the deformations of mathcal m can be encoded in variations of the complex contact transformations which relate different locally flat patches of the twistor space mathcal z mathcal m of mathcal m by passing the swann bundle and its twistor space we specialize these general results to the case of quaternionic kahler metrics with d 1 commuting isometries obtainable by the legendre transform method and linear deformations thereof we illustrate our methods for the hypermultiplet moduli space in string theory compactifications at tree and one loop level

quantitative results for halpern iterations of nonexpansive mappings abstract we give a rate of metastability for halpern s iteration relative to a rate of metastability for the resolvent for nonexpansive mappings in uniformly smooth banach spaces extracted from a proof due to xu in hilbert space the latter is known so we get an explicit rate of metastability we also extract a rate of asymptotic regularity for general normed spaces such rates have already been extracted by kohlenbach and leustean for different and incomparable conditions on n the proof analyzed in this paper is also more effective than the proofs treated by kohlenbach and leustean in that it does not use banach limits or weak compactness which makes the extraction particularly efficient moreover we also give an equivalent axiomatization of uniformly smooth banach spaces this paper is part of an ongoing case study of proof mining in nonlinear fixed point theory

nahm s equations and hyperkahler geometry the geometry of certain moduli spaces of solutions to nahm s equations is studied and a family of gravitational instationsj is shown to arise as a deformation of theh atiyah hitchin maniford

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

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

the kahler cone of a compact hyperkahler manifold this paper is a sequel to 11 we study a number of questions only touched upon in 11 in more detail in particular what is the relation between two birational compact hyperkahler manifolds what is the shape of the cone of all kahler classes on such a manifold how can the birational kahler cone be described most of the results are motivated by either the well established two dimensional theory i e the theory of k3 surfaces or the theory of calabi yau threefolds and string theory

curvature and characteristic numbers of hyper kahler manifolds it is perhaps surprising to see this norm expressed solely in terms of the volume andpontryagin classes in fact the context for this formula is the work of rozansky andwitten who showed in 10 how one could associate to each compact hyperk ahlermanifold a three manifold invariant in more abstract terms what they show is that

energy of sections of the deligne hitchin twistor space we study a natural functional on the space of holomorphic sections of the deligne hitchin moduli space of a compact riemann surface generalizing the energy of equivariant harmonic maps corresponding to twistor lines we show that the energy is the residue of the pull back along the section of a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by hitchin as a byproduct we show the existence of a hyper kahler potentials for new components of real holomorphic sections of twistor spaces of hyper kahler manifolds with rotating s 1 action additionally we prove that for a certain class of real holomorphic sections of the deligne hitchin moduli space the energy functional is basically given by the willmore energy of corresponding equivariant conformal map to the 3 sphere as an application we use the functional to distinguish new components of real holomorphic sections of the deligne hitchin moduli space from the space of twistor lines

remarks on kawamata s effective non vanishing conjecture for manifolds with trivial first chern classes kawamata proposed a conjecture predicting that every nef and big line bundle on a smooth projective variety with trivial first chern class has nontrivial global sections we verify this conjecture for several cases including i all hyperkahler varieties of dimension le 6 ii all known hyperkahler varieties except for o grady s 10 dimensional example iii general complete intersection calabi yau varieties in certain fano manifolds e g toric ones moreover we investigate the effectivity of todd classes of hyperkahler varieties and calabi yau varieties we prove that the fourth todd classes are quasi effective for all hyperkahler varieties and general complete intersection calabi yau varieties in products of projective spaces

a uniform estimate for general quaternionic calabi problem with appendix by daniel barlet abstract we prove a c 0 a priori estimate on a solution of the quaternionic calabi problem on an arbitrary compact connected hkt manifold this generalizes earlier results 9 and 7 where this result was proven under certain extra assumptions on the manifold

conification of kahler and hyper kahler manifolds given a kahler manifold m endowed with a hamiltonian killing vector field z we construct a conical kahler manifold hat m such that m is recovered as a kahler quotient of hat m similarly given a hyper kahler manifold m g j1 j2 j3 endowed with a killing vector field z hamiltonian with respect to the kahler form of j1 and satisfying mathcal l zj 2 2j 3 we construct a hyper kahler cone hat m such that m is a certain hyper kahler quotient of hat m in this way we recover a theorem by haydys our work is motivated by the problem of relating the supergravity c map to the rigid c map we show that any hyper kahler manifold in the image of the c map admits a killing vector field with the above properties therefore it gives rise to a hyper kahler cone which in turn defines a quaternionic kahler manifold our results for the signature of the metric and the sign of the scalar curvature are consistent with what we know about the supergravity c map

local stability of ergodic averages we consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages it is not difficult to construct an example of a computable lebesgue measure preserving transformation of 0 1 and a characteristic function f xa such that the ergodic averages a n f do not converge to a computable element of l 2 0 1 in particular there is no computable bound on the rate of convergence for that sequence on the other hand we show that for any nonexpansive linear operator t on a separable hilbert space and any element f it is possible to compute a bound on the rate of convergence of a n f from t f and the norm f of the limit in particular if t is the koopman operator arising from a computable ergodic measure preserving transformation of a probability space x and f is any computable element of l 2 x then there is a computable bound on the rate of convergence of the sequence anf the mean ergodic theorem is equivalent to the assertion that for every function k n and every e 0 there is an n with the property that the ergodic averages a m f are stable to within e on the interval n k n even in situations where the sequence a n f does not have a computable limit one can give explicit bounds on such n in terms of k and f e this tells us how far one has to search to find an n so that the ergodic averages are locally stable on a large interval we use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem and we show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to bishop and ivanov finally we explain how our positive results can be viewed as an application of a body of general proof theoretic methods falling under the heading of proof mining

kuga satake construction and cohomology of hyperkahler manifolds abstract let m be a simple hyperkahler manifold kuga satake construction gives an embedding of h 2 m c into the second cohomology of a torus compatible with the hodge structure we construct a torus t and an embedding of the graded cohomology space h m c h l t c for some l which is compatible with the hodge structures and the poincare pairing moreover this embedding is compatible with an action of the lie algebra generated by all lefschetz sl 2 triples on m