constructions in sasakian geometry we first generalize the join construction described previously by the first two authors 4 for quasi regular sasakian einstein orbifolds to the general quasi regular sasakian case this allows for the further construction of specific types of sasakian structures that are preserved under the join operation such as positive negative or null sasakian structures as well as sasakian einstein structures in particular we show that there are families of sasakian einstein structures on certain 7 manifolds homeomorphic to s 2 s 5 we next show how the join construction emerges as a special case of lerman s contact fibre bundle construction 32 in particular when both the base and the fiber of the contact fiber bundle are toric we show that the construction yields a new toric sasakian manifold finally we study toric sasakian manifolds in dimension 5 and show that any simply connected compact oriented 5 manifold with vanishing torsion admits regular toric sasakian structures this is accomplished by explicitly constructing circle bundles over the equivariant blow ups of hirzebruch surfaces

sasaki einstein manifolds and volume minimisation we study a variational problem whose critical point determines the reeb vector field for a sasaki einstein manifold this extends our previous work on sasakian geometry by lifting the condition that the manifolds are toric we show that the einstein hilbert action restricted to a space of sasakian metrics on a link l in a calabi yau cone x is the volume functional which in fact is a function on the space of reeb vector fields we relate this function both to the duistermaat heckman formula and also to a limit of a certain equivariant index on x that counts holomorphic functions both formulae may be evaluated by localisation this leads to a general formula for the volume function in terms of topological fixed point data as a result we prove that the volume of a sasaki einstein manifold relative to that of the round sphere is always an algebraic number in complex dimension n 3 these results provide via ads cft the geometric counterpart of a maximisation in four dimensional superconformal field theories we also show that our variational problem dynamically sets to zero the futaki invariant of the transverse space the latter being an obstruction to the existence of a kahler einstein metric

some families of mordell curves associated to cubic fields we introduce some mordell curves of two different natures both of which are associated to cubic fields one set of them consists of those elliptic curves whose rational points over the rational number field are described by or closely related to cubic fields the other is a one parameter family of mordell curves which gives all cyclic cubic twists and all quadratic twists of the fermat curve x3 y3 z3 0

energy properness and sasakian einstein metrics in this paper we show that the existence of sasakian einstein metrics is closely related to the properness of corresponding energy functionals under the condition of admitting no nontrivial hamiltonian holomorphic vector field we prove that the existence of sasakian einstein metric implies a moser trudinger type inequality at the end of this paper we also obtain a miyaoka yau type inequality in sasakian geometry

the cr geometry of weighted extremal kahler and sasaki metrics we establish an equivalence between conformally einstein maxwell kahler 4 manifolds recently studied in apostolov et al j reine angew math 721 109 147 2016 apostolov and maschler j eur math soc 21 1319 1360 2019 futaki and ono j math soc jpn 70 1493 1521 2018 koca et al ann glob anal geom 50 29 46 2016 lebrun einstein maxwell equations extremal kahler metrics and seiberg witten theory in the many facets of geometry a tribute to nigel hitchin oxford university press oxford pp 17 33 2009 lebrun j geom phys 91 171 2015 and lebrun commun math phys 344 621 653 2016 and extremal kahler 4 manifolds in the sense of calabi extremal kahler metrics seminar on differential geometry annals of mathematics studies vol 102 pp 259 290 princeton university press princeton 1982 with nowhere vanishing scalar curvature the corresponding pairs of kahler metrics arise as transversal kahler structures of sasaki metrics compatible with the same cr structure and having commuting sasaki reeb vector fields this correspondence extends to higher dimensions using the notion of a weighted extremal kahler metric apostolov et al in levi kahler reduction of cr structures product of spheres and toric geometry arxiv 1708 05253 apostolov et al in weighted extremal kahler metrics and the einstein maxwell geometry of projective bundles arxiv 1808 02813 lahdili in j geom anal 29 542 568 2019 lahdili in int math res not arxiv 1710 00235 lahdili in proc lond math soc 119 1065 1114 2019 illuminating and uniting several explicit constructions in kahler and sasaki geometry it also leads to new existence and non existence results for extremal sasaki metrics suggesting a link between the notions of relative weighted k stability of a polarized variety introduced in apostolov et al weighted extremal kahler metrics and the einstein maxwell geometry of projective bundles arxiv 1808 02813 and lahdili proc lond math soc 119 1065 1114 2019 and relative k stability of the kahler cone corresponding to a sasaki polarization studied in boyer and van coevering math res lett 25 1 19 2018 and collins and szekelyhidi j differ geom 109 81 109 2018

on the homology of the hilbert scheme of points in the plane geir ellingsrud 1 and stein arild stromme 2 i matematisk institutt universitetet i oslo blindern n oslo 3 norway 2 matematisk institutt universitetet i bergen n 5014 bergen norway although several authors have been interested in the hilbert scheme hilba ip z parametrizing finite subschemes of length d in the projective plane i1 i2 f 1 f2 br among others not much is known about the topological properties of this space the picard group has been calculated i f2 and the homology groups of hilb3 ip 2 have been computed hi in this paper we give a precise description of the additive structure of the homology of hilba ip2 applying the results of birula bialynicki b1 b2 on the cellular decompositions de fined by a torus action to the natural action of a maximal torus of sl 3 on hilbe ipz a rather easy consequence of the fact that this action has finitely many fixpoints is that the cycle maps between the chow groups and the homology groups are isomorphisms in particular there is no odd homology and the homology groups are all free the main objective of this work is to compute their ranks the betti numbers of hilba ipz as a byproduct of our method we get similar results on the homology of the punctual hilbert scheme and of the hilbert scheme of points in the affine plane it seems natural to generalize our results to any toric smooth surface however we give the results only for the rational ruled surfaces if with an indication of the necessary changes in the proofs for simplicity we work over the field of complex numbers but with an appropriate interpretation of the word homology our results remain valid over any base field

extremal k contact metrics extending a result of he to the non integrable case of k contact manifolds it is shown that the transverse hermitian scalar curvature may be interpreted as a moment map for the strict contactomorphism group as a consequence we may generalize the sasaki futaki invariant to k contact geometry and establish a number of elementary properties moreover we prove that in dimension 5 certain deformation theoretic results can be established also under weaker integrability conditions by exploiting the relationship between j anti invariant and self dual 2 forms

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

regularity of the geodesic equation in the space of sasakian metrics abstract in this paper we study a geodesic equation in the space of sasakian metrics h the equation leads to the dirichlet problem of a complex monge ampere type equation on the kahler cone this equation differs from the standard complex monge ampere equation in a significant way with gradient terms involved in the 1 1 symmetric tensor of the operator we establish appropriate regularity estimates for this complex monge ampere type equation as geometric application we show that the space of sasakian metrics h is a metric space and the constant transversal scalar curvature metric realizes the global minimum of k energy if the first basic chern class c 1 b 0 we also prove that the constant transversal scalar curvature metric is unique in each basic kahler class if the first basic chern class is either strictly negative or zero

ricci flat kahler metrics on crepant resolutions of kahler cones we prove that a crepant resolution y x of a ricci flat kahler cone x admits a complete ricci flat kahler metric asymptotic to the cone metric in every kahler class in h 2 c y mathbb r a kahler cone x bar g is a metric cone over a sasaki manifold s g i e x c s s times mathbb r 0 with bar g dr 2 r 2 g and x bar g is ricci flat precisely when s g einstein of positive scalar curvature this result contains as a subset the existence of ale ricci flat kahler metrics on crepant resolutions pi y rightarrow x mathbb c n gamma with gamma subset sl n mathbb c due to p kronheimer n 2 and d joyce n 2 we then consider the case when x c s is toric it is a result of a futaki h ono and g wang that any gorenstein toric kahler cone admits a ricci flat kahler cone metric it follows that if a toric kahler cone x c s admits a crepant resolution y x then y admits a tn invariant ricci flat kahler metric asymptotic to the cone metric x bar g in every kahler class in h 2 c y mathbb r a crepant resolution in this context is a simplicial fan refining the convex polyhedral cone defining x we then list some examples which are easy to construct using toric geometry

on the modularity of elliptic curves over q wild 3 adic exercises in this paper building on work of wiles wi and of wiles and one of us r t tw we will prove the following two theorems see 2 2 theorem a if e q is an elliptic curve then e is modular theorem b if gal q q gl2 f5 is an irreducible continuous representation with cyclotomic determinant then is modular we will first remind the reader of the content of these results and then briefly outline the method of proof if n is a positive integer then we let 1 n denote the subgroup of sl2 z consisting of matrices that modulo n are of the form 1 0 1

a cochlear frequency position function for several species 29 years later accurate cochlear frequency position functions based on physiological data would facilitate the interpretation of physiological and psychoacoustic data within and across species such functions might aid in developing cochlear models and cochlear coordinates could provide potentially useful spectral transforms of speech and other acoustic signals in 1961 an almost exponential function was developed greenwood 1961b 1974 by integrating an exponential function fitted to a subset of frequency resolution integration estimates critical bandwidths the resulting frequency position function was found to fit cochlear observations on human cadaver ears quite well and with changes of constants those on elephant cow guinea pig rat mouse and chicken bekesy 1960 as well as in vivo behavioral anatomical data on cats schucknecht 1953 since 1961 new mechanical and other physiological data have appeared on the human cat guinea pig chinchilla monkey and gerbil it is shown here that the newer extended data on human cadaver ears and from living animal preparations are quite well fit by the same basic function the function essentially requires only empirical adjustment of a single parameter to set an upper frequency limit while a slope parameter can be left constant if cochlear partition length is normalized to 1 or scaled if distance is specified in physical units constancy of slope and form in dead and living ears and across species increases the probability that the function fitting human cadaver data may apply as well to the living human ear this prospect increases the function s value in plotting auditory data and in modeling concerned with speech and other bioacoustic signals since it fits the available physiological data well and consequently if those data are correct remains independent of and an appropriate means to examine psychoacoustic data and assumptions

the moduli space of asymptotically cylindrical calabi yau manifolds we prove that the deformation theory of compactifiable asymptotically cylindrical calabi yau manifolds is unobstructed this relies on a detailed study of the dolbeault hodge theory and its description in terms of the cohomology of the compactification we also show that these calabi yau metrics admit a polyhomogeneous expansion at infinity a result that we extend to asymptotically conical calabi yau metrics as well we then study the moduli space of calabi yau deformations that fix the complex structure at infinity there is a weil petersson metric on this space which we show is kahler by proving a local families l 2 index theorem we exhibit its kahler form as a multiple of the curvature of a certain determinant line bundle

toric geometry sasaki einstein manifolds and a new infinite class of ads cft duals recently an infinite family of explicit sasaki einstein metrics yp q on s2 s3 has been discovered where p and q are two coprime positive integers with q p these give rise to a corresponding family of calabi yau cones which moreover are toric aided by several recent results in toric geometry we show that these are kahler quotients open image in new window namely the vacua of gauged linear sigma models with charges p p p q p q thereby generalising the conifold which is p 1 q 0 we present the corresponding toric diagrams and show that these may be embedded in the toric diagram for the orbifold open image in new window for all q p with fixed p we hence find that the yp q manifolds are ads cft dual to an infinite class of open image in new window superconformal field theories arising as ir fixed points of toric quiver gauge theories with gauge group su n 2p as a non trivial example we show that y2 1 is an explicit irregular sasaki einstein metric on the horizon of the complex cone over the first del pezzo surface the dual quiver gauge theory has already been constructed for this case and hence we can predict the exact central charge of this theory at its ir fixed point using the ads cft correspondence the value we obtain is a quadratic irrational number and remarkably agrees with a recent purely field theoretic calculation using a maximisation

from sheaves on p 2 to a generalization of the rademacher expansion moduli spaces of stable coherent sheaves on a surface are of much interest for both mathe matics and physics yoshioka computed generating functions of poincare polynomials of such moduli spaces if the surface is p 2 and the rank of the sheaves is 2 motivated by physical arguments this pa per investigates the modular properties of these generating functions it is shown that these functions can be written in terms of the lerch sum and theta function based on this we prove a conjecture by vafa and witten which expresses the generating functions of euler numbers as a mixed mock mod ular form moreover we derive an exact formula of rademacher type for the fourier coefficients of this function this formula requires a generalization of the classical circle method this is the first example of an exact formula for the fourier coefficients of mixed mock modular forms which is of independent mathematical interest 1 introduction and statement of results in the past interactions between physics and mathematics have led to many interesting results motivated by strong weak coupling duality or s duality in physics this article considers various gen erating functions which appear in the study of moduli spaces of stable coherent sheaves on the projective plane p 2 we express the generating functions of poincare polynomials of moduli spaces of rank 2 sheaves in terms of the lerch sum and theta function which we will recall later using these expressions we prove a conjec ture by vafa and witten 45 for the generating functions of euler numbers these functions appear to be related to ramanujan s mock theta functions and therefore transform almost as weakly holomorphic modular forms i e meromorphic mod ular forms whose poles if there are any may only lie in cusps our second main result is an exact formula for the fourier coefficients of these generating functions that formally resembles the rademacher expansion for the coefficients of weakly holomorphic modular forms moduli spaces of coherent sheaves on a complex surface s receive much attention see for example 28 for an extensive work on such moduli spaces more specifically one is interested in the moduli space m r c1 c 2 of semi stable sheaves of rank r with first chern class c1 and second chern class c2 w e will consider topological invariants of m in particular the poincare polynomial p m s 2d imc m i 0 bi m s i and the euler number m p m 1 w here

elliptic genera of symmetric products and second quantized strings in this note we prove an identity that equates the elliptic genus partition function of a supersymmetric sigma model on the n fold symmetric product m n sn of a manifold m to the partition function of a second quantized string theory on the space m s 1 the generating function of these elliptic genera is shown to be almost an automorphic form for o 3 2 z in the context of d brane dynamics this result gives a precise computation of the free energy of a gas of d strings inside a higher dimensional brane

toric generalized kahler ricci solitons with hamiltonian 2 form we show that the generalized kahler ricci soliton equation on 4 dimensional toric kahler orbifolds reduces to odes assuming there is a hamiltonian 2 form this leads to an explicit resolution of this equation on labelled triangles and convex labelled quadrilaterals in particular we give the explicit expression of the kahler ricci solitons of weighted projective planes as well as new examples

kahler ricci solitons on toric manifolds with positive first chern class abstract in this paper we prove there exists a kahler ricci soliton unique up to holomorphic automorphisms on any toric kahler manifold with positive first chern class and the kahler ricci soliton is a kahler einstein metric if and only if the futaki invariant vanishes

deformations of special legendrian submanifolds in sasaki einstein manifolds in this paper we study the deformation theory of submanifolds characterized by a system of differential forms and provide a criterion for deformations of such submanifolds to be unobstructed we apply this deformation theory to special legendrian submanifolds in sasaki einstein manifolds in general special legendrian deformations have the obstruction however we show that the deformation space of special legendrian submanifolds is the intersection of two larger smooth deformation spaces of different types we also prove that any special legendrian submanifold admits smooth deformations which are not special legendrian deformations given by harmonic 1 forms

s duality and refined bps indices whenever available refined bps indices provide considerably more information on the spectrum of bps states than their unrefined version extending earlier work on the modularity of generalized donaldson thomas invariants counting d4 d2 d0 brane bound states in type iia strings on a calabi yau threefold mathfrak y we construct the modular completion of generating functions of refined bps indices supported on a divisor class although for compact mathfrak y the refined indices are not protected switching on the refinement considerably simplifies the construction of the modular completion furthermore it leads to a non commutative analogue of the tba equations which suggests a quantization of the moduli space consistent with s duality in contrast for a local cy threefold given by the canonical bundle over a complex surface s refined dt invariants are well defined and equal to vafa witten invariants of s our construction provides a modular completion of the generating function of these refined invariants for arbitrary rank in cases where all reducible components of the divisor class are collinear which occurs e g when b 2 mathfrak y 1 or in the local case we show that the holomorphic anomaly equation satisfied by the completed generating function truncates at quadratic order in the local case it agrees with an earlier proposal by minahan et al for unrefined invariants and extends it to the refined level using the afore mentioned non commutative structure finally we show that these general predictions reproduce known results for u 2 and u 3 vafa witten theory on mathrm p 2 and make them explicit for u 4

hilbert series and obstructions to asymptotic semistability abstract given a polarized manifold there are obstructions for asymptotic chow semistability described as integral invariants which can be regarded as characters of the lie algebra of holomorphic vector fields in this paper we show that on toric fano manifolds the linear span of those lie algebra characters coincides with the derivatives of the laurent series of the hilbert series

s duality for surfaces with a_n type singularities we show that the generating series of euler characteristics of hilbert schemes of points on any algebraic surface with at worst a n type singularities is described by the theta series determined by integer valued positive definite quadratic forms and the dedekind eta function in particular it is a fourier development of a meromorphic modular form with possibly half integer weight the key ingredient is to apply the flop transformation formula of donaldson thomas type invariants counting two dimensional torsion sheaves on threefolds proved in the author s previous paper

integral operators and integral cohomology classes of hilbert schemes the methods of integral operators on the cohomology of hilbert schemes of points on surfaces are developed they are used to establish integral bases for the cohomology groups of hilbert schemes of points on a class of surfaces and conjecturally for all simply connected surfaces

invariance principles for cochlear mechanics hearing phases a functional model of the cochlea is devised on the basis of the results from classical experiments the basilar membrane filter is investigated in detail its phase is close to linear in the region around the peak of the amplification on one side this has consequences for the time analysis and on the other side this has led to a prediction on phase perception for very simple combinations of tones a prediction which is now confirmed by experiments equivariance under the dilation group permits one to describe the model by a wavelet transform daubechies ten lectures on wavelets siam philadelphia 1992 the wavelet is discussed in reference to the phase analysis of the basilar membrane filter

frankel conjecture and sasaki geometry abstract we classify simply connected compact sasaki manifolds of dimension 2 n 1 with positive transverse bisectional curvature in particular the kahler cone corresponding to such manifolds must be bi holomorphic to c n 1 0 as an application we recover the theorem of mori and siu yau on the frankel conjecture and extend it to certain orbifold version the main idea is to deform such sasaki manifolds to the standard round sphere in two steps both fixing the complex structure on the kahler cone first we deform the metric along the sasaki ricci flow and obtain a limit sasaki ricci soliton with positive transverse bisectional curvature then by varying the reeb vector field which essentially decreases the volume functional we deform the sasaki ricci soliton to a sasaki einstein metric with positive transverse bisectional curvature i e a round sphere the second deformation is only possible when one treats simultaneously regular and irregular sasaki manifolds even if the manifold one starts with is regular quasi regular i e kahler manifolds orbifolds

vertex operators grassmannians and hilbert schemes we approximate the infinite grassmannian by finite dimensional cutoffs and define a family of fermionic vertex operators as the limit of geometric correspondences on the equivariant cohomology groups with respect to a one dimensional torus action we prove that in the localization basis these are the well known fermionic vertex operators on the infinite wedge representation furthermore the boson fermion correspondence locality and intertwining properties with the virasoro algebra are the limits of relations on the finite dimensional cutoff spaces which are true for geometric reasons we then show that these operators are also almost by definition the vertex operators defined by okounkov and the author in carlsson and okounkov http arxiv org abs 0801 2565v2 math ag 2009 on the equivariant cohomology groups of the hilbert scheme of points on mathbb c 2 with respect to a special torus action

genus zero gopakumar vafa type invariants for calabi yau 4 folds abstract in analogy with the gopakumar vafa conjecture on cy 3 folds klemm and pandharipande defined gv type invariants on calabi yau 4 folds using gromov witten theory and conjectured their integrality in this paper we propose a sheaf theoretic interpretation of their genus zero invariants using donaldson thomas theory on cy 4 folds more specifically we conjecture genus zero gv type invariants are d t 4 invariants for one dimensional stable sheaves on cy 4 folds some examples are computed for both compact and non compact cy 4 folds to support our conjectures we also propose an equivariant version of the conjectures for local curves and verify them in certain cases

comparison of the roex and gammachirp filters as representations of the auditory filter although the rounded exponential roex filter has been successfully used to represent the magnitude response of the auditory filter recent studies with the roex p w t filter reveal two serious problems the fits to notched noise masking data are somewhat unstable unless the filter is reduced to a physically unrealizable form and there is no time domain version of the roex p w t filter to support modeling of the perception of complex sounds this paper describes a compressive gammachirp cgc filter with the same architecture as the roex p w t which can be implemented in the time domain the gain and asymmetry of this parallel cgc filter are shown to be comparable to those of the roex p w t filter but the fits to masking data are still somewhat unstable the roex p w t and parallel cgc filters were also compared with the cascade cgc filter patterson et al j acoust soc am 114 1529 1542 2003 which was found to provide an equivalent fit with 25 fewer coefficients moreover the fits were stable the advantage of the cascade cgc filter appears to derive from its parsimonious representation of the high frequency side of the filter it is concluded that cgc filters offer better prospects than roex filters for the representation of the auditory filter

the cochlear frequency map for the cat labeling auditory nerve fibers of known characteristic frequency iontophoresis of horseradish peroxidase was used to label single auditory nerve fibers after determination of threshold tuning curves and rates of spontaneous discharge the relation between characteristic frequency cf and cochlear longitudinal location is reconstructed from 52 labeled neurons in 16 cochleas the length of the organ of corti alotted to an octave of stimulus frequency increases steadily from low to high frequencies thus there is not a simple linear distance to log frequency conversion when comparing cochleas of different total length the best predictor of cf at a given location is the distance from base or apex expressed as a percentage of the total length the cochlear frequency map derived from these single neuron labeling experiments is compared to maps derived by a number of different physiological and psychophysical techniques and the significance of the similarities and differences is discussed

the geometric dual of a maximisation for toric sasaki einstein manifolds we show that the reeb vector and hence in particular the volume of a sasaki einstein metric on the base of a toric calabi yau cone of complex dimension n may be computed by minimising a function z on mathbb r n which depends only on the toric data that defines the singularity in this way one can extract certain geometric information for a toric sasaki einstein manifold without finding the metric explicitly for complex dimension n 3 the reeb vector and the volume correspond to the r symmetry and the a central charge of the ads cft dual superconformal field theory respectively we therefore interpret this extremal problem as the geometric dual of a maximisation we illustrate our results with some examples including the yp q singularities and the complex cone over the second del pezzo surface

calabi yau volumes and reflexive polytopes we study various geometrical quantities for calabi yau varieties realized as cones over gorenstein fano varieties obtained as toric varieties from reflexive polytopes in various dimensions focus is made on reflexive polytopes up to dimension 4 and the minimized volumes of the sasaki einstein base of the corresponding calabi yau cone are calculated by doing so we conjecture new bounds for the sasaki einstein volume with respect to various topological quantities of the corresponding toric varieties we give interpretations about these volume bounds in the context of associated field theories via the ads cft correspondence

a note of sasakian metrics with constant scalar curvature in this paper we consider compact sasakian manifolds with constant scalar curvature under some positive curvature assumption we prove that such sasakian metrics must be einstein metrics

uniqueness and examples of compact toric sasaki einstein metrics in 11 it was proved that given a compact toric sasaki manifold with positive basic first chern class and trivial first chern class of the contact bundle one can find a deformed sasaki structure on which a sasaki einstein metric exists in the present paper we first prove the uniqueness of such einstein metrics on compact toric sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure and secondly remark that the result of 11 implies the existence of compatible einstein metrics on all compact sasaki manifolds obtained from the toric diagrams with any height or equivalently on all compact toric sasaki manifolds whose cones have flat canonical bundle we further show that there exists an infinite family of inequivalent toric sasaki einstein metrics on s 5 sharp k s 2 times s 3 for each positive integer k

conformally einstein maxwell kahler metrics and structure of the automorphism group let m g be a compact kahler manifold and f a positive smooth function such that its hamiltonian vector field k j mathrm grad g f for the kahler form omega g is a holomorphic killing vector field we say that the pair g f is conformally einstein maxwell kahler metric if the conformal metric tilde g f 2 g has constant scalar curvature in this paper we prove a reductiveness result of the reduced lie algebra of holomorphic vector fields for conformally einstein maxwell kahler manifolds extending the lichnerowicz matsushima theorem for constant scalar curvature kahler manifolds more generally we consider extensions of calabi functional and extremal kahler metrics and prove an extension of calabi s theorem on the structure of the lie algebra of holomorphic vector fields for extremal kahler manifolds the proof uses a hessian formula for the calabi functional under the set up of donaldson fujiki picture

on eta einstein sasakian geometry a compact quasi regular sasakian manifold m is foliated by one dimensional leaves and the transverse space of this characteristic foliation is necessarily a compact kahler orbifold in the case when the transverse space is also einstein the corresponding sasakian manifold m is said to be sasakian einstein in this article we study einstein geometry as a class of distinguished riemannian metrics on contact metric manifolds in particular we use a previous solution of the calabi problem in the context of sasakian geometry to prove the existence of einstein structures on many different compact manifolds including exotic spheres we also relate these results to the existence of einstein weyl and lorenzian sasakian einstein structures

counting rational curves on text k3 surfaces the aim of these notes is to explain the remarkable formula found by yau and zaslow y z to express the number of rational curves on a k3 surface projective k3 surfaces fall into countably many families fg g 1 a surface in fg admits a gdimensional linear system of curves of genus g a naive count of constants suggests that such a system will contain a positive number say n g of rational highly singular curves the formula is

on the classification of reflexive polyhedra reflexive polyhedra encode the combinatorial data for mirror pairs of calabi yau hypersurfaces in toric varieties we investigate the geometrical structures of circumscribed polytopes with a minimal number of facets and of inscribed polytopes with a minimal number of vertices these objects which constrain reflexive pairs of polyhedra from the interior and the exterior can be described in terms of certain non negative integral matrices a major tool in the classification of these matrices is the existence of a pair of weight systems indicating a relation to weighted projective spaces this is the cornerstone for an algorithm for the construction of all dual pairs of reflexive polyhedra that we expect to be efficient enough for an enumerative classification in up to 4 dimensions which is the relevant case for calabi yau compactifications in string theory

hard lefschetz theorem for sasakian manifolds we prove that on a compact sasakian manifold m g of dimension 2n 1 for any 0 p n the wedge product with d defines an isomorphism between the spaces of harmonic forms n p m and n p 1 m therefore it induces an isomorphism between the de rham cohomology spaces h m and h m such isomorphism is proven to be independent of the choice of a compatible sasakian metric on a given contact manifold as a consequence an obstruction for a contact manifold to admit sasakian structures is found

transverse fully nonlinear equations on sasakian manifolds and applications abstract we prove a priori estimates for a class of transverse fully nonlinear equations on sasakian manifolds and give some geometric applications such as the transversion calabi yau theorem for transverse balanced and strongly gauduchon metrics we also explain that similar results hold on compact oriented taut transverse hermitian foliated manifold of complex co dimension n and give some geometric applications such as the transverse calabi yau theorems for transverse hermitian and strongly gauduchon metrics