blowing up kahler manifolds with constant scalar curvature ii in this paper we prove the existence of kahler metrics of constant scalar curvature on the blow up at finitely many points of a compact manifold that already carries a constant scalar curvature kahler metric in the case where the manifold has nontrivial holomorphic vector fields with zeros we give necessary conditions on the number and locations of the blow up points for the blow up to carry constant scalar curvature kahler metrics

scalar curvature and projective embeddings i we prove that a metric of constant scalar curvature on a polarised kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings satisfying a condition introduced by h luo this gives as a corollary the uniqueness of constant scalar curvature kahler metrics in a given rational cohomology class the proof uses results in the literature on the asymptotics of the bergman kernel the arguments are presented in a general framework involving moment maps for two different group actions

extremal metrics on blowups in this paper we provide conditions that are sufficient to guarantee the existence of extremal metrics on blowups at finitely many points of kahler manifolds which already carry an extremal metric as a particular case we construct extremal metrics on mathbb p 2 blown up k points in general position with k lt m 2

k stability and parabolic stability parabolic structures with rational weights encode certain iterated blowups of geometrically ruled surfaces in this paper we show that the three notions of parabolic polystability k polystability and existence of constant scalar curvature kahler metrics on the iterated blowup are equivalent for certain polarizations close to the boundary of the kahler cone

on the second solution to a critical growth robin problem abstract we investigate the existence of the second mountain pass solution to a robin problem where the equation is at critical growth and depends on a positive parameter more precisely we determine existence and nonexistence regions for this type of solutions depending both on and on the parameter in the boundary conditions

laumon spaces and the calogero sutherland integrable system this paper contains a proof of a conjecture of braverman concerning laumon quasiflag spaces we consider the generating function z m whose coefficients are the integrals of the equivariant chern polynomial with variable m of the tangent bundles of the laumon spaces we prove braverman s conjecture which states that z m coincides with the eigenfunction of the calogero sutherland hamiltonian up to a simple factor which we specify this conjecture was inspired by the work of nekrasov in the affine widehat mathfrak sl n setting where a similar conjecture is still open

moduli spaces of critical riemannian metrics in dimension four abstract we obtain a compactness result for various classes of riemannian metrics in dimension four in particular our method applies to anti self dual metrics kahler metrics with constant scalar curvature and metrics with harmonic curvature with certain geometric non collapsing assumptions the moduli space can be compactified by adding metrics with orbifold like singularities similar results were obtained for einstein metrics in j amer math soc 2 3 1989 455 invent math 97 2 1989 313 invent math 101 1 1990 101 but our analysis differs substantially from the einstein case in that we do not assume any pointwise ricci curvature bound

normal moveout revisited inhomogeneous media and curved interfaces the equation of normal moveout t2 t02 x2 va2 is valid for a reflection from the base of a single homogeneous and isotropic bed but is only an approximation in the real world of multilayered inhomogeneous media and curved interfaces using the theory of geometrical optics we can find another second order equation which represents hyperbolas that are also symmetrical about the time axis however the centers of these hyperbolas do not coincide with the center of coordinates but are shifted along the time axis the equation describing this second type of hyperbola is t tp t0 2 tp2 x2 v12 where tp is the time of focusing depth and v1 the velocity of the input medium this equation is not only more accurate than the usual normal moveout but its use is more economical on a vector computer because the traditional dynamic correction is a static correction in the tp analysis this procedure makes it possible to compute velocities for all the samples of all the stacked traces and produces a velocity secti

generalized moveout approximation for qp and qsv waves in a homogeneous transversely isotropic medium the moveout approximations can be used in kinematic modeling velocity analysis and time migration the generalized moveout approximation involves five approximation parameters and has several known approximations as special cases a method is demonstrated for determining parameters of the generalized nonhyperbolic moveout approximation for qp and qsv waves in a homogeneous transversely isotropic medium with vertical symmetry axis vti medium the additional parameters for the generalized approximation are computed from the hyperbolic asymptote at infinite offset comparison with a few well known moveout approximations for higher order terms in the taylor series and asymptotic behavior shows that the generalized moveout approximation is superior to other nonhyperbolic approximations a few numerical examples for qp and qsv waves in a vti medium also indicate that the generalized approximation performs the best

generalized nonhyperbolic moveout approximation reflection moveout approximations are commonly used for velocity analysis stacking and time migration a novel functional form for approximating the moveout of reflection traveltimes at large offsets is introduced in comparison with the classic hyperbolic approximation which uses only two parameters zero offset time and moveout velocity this form involves five parameters that can be determined in a known medium from zero offset computations and from tracing one nonzero offset ray it is called a generalized approximation because it reduces to some known three parameter forms with a particular choice of coefficients by testing the accuracy of the proposed approximation with analytical and numerical examples the new approximation is shown to bring an improvement in accuracy of several orders of magnitude compared to known analytical approximations which makes it as good as exact for many practical purposes

blowing up and desingularizing constant scalar curvature kahler manifolds this paper is concerned with the existence of constant scalar curvature kahler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature kahler metrics we also consider the desingularization of isolated quotient singularities of compact orbifolds which carry constant scalar curvature kahler metrics

interaction between the geometry of the boundary and positive solutions of a semilinear neumann problem with critical nonlinearity abstract we consider the problem u u un 2 n 2 u 0 in u v 0 on where is a bounded smooth domain in r n n 3 we show that for large least energy solutions of the above problem have a unique maximum point p on and the limit points of p as are contained in the set of the points of maximum mean curvature we also prove that if has k peaks then the equation has at least k solutions for large

parabolic sheaves on logarithmic schemes abstract we show how the natural context for the definition of parabolic sheaves on a scheme is that of logarithmic geometry the key point is a reformulation of the concept of logarithmic structure in the language of symmetric monoidal categories which might be of independent interest our main result states that parabolic sheaves can be interpreted as quasi coherent sheaves on certain stacks of roots

counter examples to the generalized positive action conjecture we give examples of complete locally asymptotically flat riemannian 4 manifolds with zero scalar curvature and negative mass the generalized positive action conjecture of hawking and pope 5 is therefore false

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

traveltime approximations for a layered transversely isotropic medium we consider multiple transmitted reflected and converted qp qsv waves or multiple transmitted and reflected sh waves in a horizontally layered medium that is transversely isotropic with a vertical symmetry axis vti traveltime and offset horizontal distance between a source and receiver not necessarily in the same layer are expressed as functions of horizontal slowness these functions are given in terms of a taylor series in slowness in exactly the same form as for a layered isotropic medium the coefficients depend on the parameters of the anisotropic layers through which the wave has passed and there is no weak anisotropy assumption using classical formulas the traveltime or traveltime squared can then be expressed as a taylor series in even powers of offset these taylor series give rise to a shifted hyperbola traveltime approximation and a new continued fraction approximation described by four parameters that match the taylor series up to the sixth power in offset further approximations give several simplified continued fraction approximations all of which depend on three parameters zero offset traveltime nmo velocity and a heterogeneity coefficient the approximations break down when there is a cusp in the group velocity for the qsv wave numerical studies indicate that approximations of traveltime squared are generally better than those for traveltime a new continued fraction approximation that depends on three parameters is more accurate than the commonly used continued fraction approximation and the shifted hyperbola

nonhyperbolic moveout analysis in vti media using rational interpolation anisotropic velocity analysis using qp waves in transversely isotropic media with a vertical symmetry axis vti usually is done by inferring the anellipticity parameter and the normal moveout velocity vnmo from the nonhyperbolic character of the moveout several approximations explicit in these parameters exist with varying degrees of accuracy here we present a rational interpolation approach to nonhyperbolic moveout analysis in the t x domain this method has no additional computational overhead compared to using expressions explicit in and vnmo the lack of such overhead stems from the observation that for fixed and zero offset two way traveltime t0 the moveout curve for different values of vnmo can be calculated by simple stretching of the offset axis this observation is based on the assumptions that the traveltimes of qp waves in transversely isotropic media mainly depend on and vnmo and that the shear wave velocity along the symmetry axis has a negligibleinfluence on these traveltim

an acoustic wave equation for anisotropic media a wave equation derived using the acoustic medium assumption for p waves in transversely isotropic ti media with a vertical symmetry axis vti media yields a good kinematic approximation to the familiar elastic wave equation for vti media the wavefield solutions obtained using this vti acoustic wave equation are free of shear waves which significantly reduces the computation time compared to the elastic wavefield solutions for exploding reflector type applications from this vti acoustic wave equation the eikonal and transport equations that describe the ray theoretical aspects of wave propagation in a ti medium are derived these equations based on the acoustic assumption shear wave velocity 0 are much simpler than their elastic counterparts yet they yield an accurate description of traveltimes and geometrical amplitudes numerical examples prove the usefulness of this acoustic equation in simulating the kinematic aspects of wave propagation in complex ti models

inversion of reflection traveltimes for transverse isotropy in anisotropic media the short spread stacking ve locity is generally different from the root mean square vertical velocity the influence of anisotropy makes it impossible to recover the vertical velocity or the reflector depth using hyperbolic moveout analysis on short spread common midpoint cmp gathers even if both p and s waves are recorded hence we examine the feasibility of inverting long spread nonhyperbolic reflection moveouts for param eters of transversely isotropic media with a vertical symmetry axis one possible solution is to recover the quartic term of the taylor series expansion for t 2 x 2 curves for p and sv waves and to use it to determine the anisotropy however this procedure turns out to be unstable because of the ambiguity in the joint inversion of intermediate spread i e spreads of about 1 5 times the reflector depth p and sv move

critical metrics on connected sums of einstein four manifolds abstract we develop a gluing procedure designed to obtain canonical metrics on connected sums of einstein four manifolds the main application is an existence result using two well known einstein manifolds as building blocks the fubini study metric on cp 2 and the product metric on s 2 s 2 using these metrics in various gluing configurations toric invariant critical metrics are found on connected sums for a specific riemannian functional which depends on the global geometry of the factors furthermore using certain quotients of s 2 s 2 as one of the gluing factors critical metrics on several non simply connected manifolds are also obtained

group valued implosion and parabolic structures the purpose of this paper is twofold first we extend the notion of symplectic implosion to the category of quasi hamiltonian k manifolds where k is a simply connected compact lie group the imploded cross section of the double k k turns out to be universal in a suitable sense it is a singular space but some of its strata have a nonsingular closure this observation leads to interesting new examples of quasi hamiltonian k manifolds such as the spinning 2 n sphere for k su n secondly we construct a universal master moduli space of parabolic bundles with structure group k over a marked riemann surface the master moduli space carries a natural action of a maximal torus of k and a torus invariant stratification into manifolds each of which has a symplectic structure an essential ingredient in the construction is the universal implosion paradoxically although the universal implosion has no complex structure it is the four sphere for k su 2 the master moduli space turns out to be a complex algebraic variety

further study on the effect of boundary conditions abstract this paper concerns the least energy solutions to a semilinear elliptic equation the role of geometry of the boundary is studied when a parameter in the boundary condition tends to a critical index as well as to infinity

a relation between the parabolic chern characters of the de rham bundles in this paper we consider the weight i de rham gauss manin bundles on a smooth variety arising from a smooth projective morphism f x u longrightarrow u for i geq 0 we associate to each weight i de rham bundle a certain parabolic bundle on s and consider their parabolic chern characters in the rational chow groups for a good compactification s of u we show the triviality of the alternating sum of these parabolic bundles in the positive degree rational chow groups this removes the hypothesis of semistable reduction in the original result of this kind due to esnault and viehweg

quantum generalization of the horn conjecture recall that the lie algebra of the special unitary group su n is isomorphic to the real vector space of traceless hermitian matrices as representations of su n and hence the terminology additive eigenvalue problem for su n in 1962 horn 17 gave a conjectural solution to a problem equivalent to the above additive eigenvalue problem for su n by a recursively determined system of inequalities in 19 klyachko gave a solution to the additive eigenvalue problem for su n in terms of a certain system of inequalities to write down this system we need to know which structure constants in the cohomology of gr r n written in the schubert basis 0 2 be a positive integer characterize the possible eigenvalues a1 a2 as of matrices a a 2 a e su n which satisfy a a a 1

long offset moveout for vti using pade approximation abstractthe approximation of normal moveout is essential for estimating the anisotropy parameters of the transversally isotropic media with vertical symmetry axis vti we have approximated the long offset moveout using the pade approximation based on the higher order taylor series coefficients for vti media for a given anellipticity parameter we have the best accuracy when the numerator is one order higher than the denominator i e l l 1 thus we suggest using 4 3 and 7 6 orders for practical applications a 7 6 pade approximation can handle a much larger offset and stronger anellipticity parameter we have further compared the relative traveltime errors between the pade approximation and several approximations our method shows great superiority to most existing methods over a wide range of offset normalized offset up to 2 or offset to depth ratio up to 4 and anellipticity parameter 0 0 5 the pade approximation provides us with an attractive high accuracy scheme with an error that

blowing up extremal kahler manifolds ii this is a continuation of the work of arezzo pacard singer and the author on blowups of extremal kahler manifolds we prove the conjecture stated in szekelyhidi duke math j 161 8 1411 1453 2012 and we relate this result to the k stability of blown up manifolds as an application we prove that if a kahler manifold m of dimension 2 admits a constant scalar curvature csck metric then the blowup of m at a point admits a csck metric if and only if it is k stable as long as the exceptional divisor is sufficiently small

multiple positive solutions for robin problem involving critical weighted hardy sobolev exponents with boundary singularities abstract this paper deals with the existence of positive solutions for robin elliptic problems involving critical weighted hardy sobolev exponents with boundary singularities using the caffarelli kohn nirenberg inequalities and variational methods we prove the existence and multiplicity of positive solutions

the mathbb a _ q t algebra and parabolic flag hilbert schemes the earlier work of the first and the third name authors introduced the algebra mathbb a q t and its polynomial representation in this paper we construct an action of this algebra on the equivariant k theory of certain smooth strata in the flag hilbert scheme of points on the plane in this presentation the fixed points of the torus action correspond to generalized macdonald polynomials and the matrix elements of the operators have an explicit presentation

the construction of ale spaces as hyper kahler quotients on decrit la construction d une famille particuliere de 4 varietes hyper kahler les espaces asymptotiquement localement euclidiens ale on decrit une 4 variete de riemann avec juste une extremite qui a l infini ressemble au quotient r 4 de l espace euclidien r 4 par un groupe fini d identifications

nonhyperbolic reflection moveout in anisotropic media the standard hyperbolic approximation for reflection moveouts in layered media is accurate only for relatively short spreads even if the layers are isotropic velocity anisotropy may significantly enhance deviations from hyperbolic moveout nonhyperbolic analysis in anisotropic media is also important because conventional hyperbolic moveout processing on short spreads is insufficient to recover the true vertical velocity hence the depth we present analytic and numerical analysis of the combined influence of vertical transverse isotropy and layering on long spread reflection moveouts qualitative description of nonhyperbolic moveout on intermediate spreads offset to depth ratio x z 1 7 2 is given in terms of the exact fourth order taylor series expansion for p sv and p sv traveltime curves valid for multilayered transversely isotropic media with arbitrary strength of anisotropy we use this expansion to provide an analytic explanation for deviations from hyperbolic moveout such as the strong

rigidity and stability of einstein metrics for quadratic curvature functionals we investigate rigidity and stability properties of critical points of quadratic curvature functionals on the space of riemannian metrics we show it is possible to gauge the euler lagrange equations in a self adjoint fashion to become elliptic fredholm theory may then be used to describe local properties of the moduli space of critical metrics we show a number of compact examples are infinitesimally rigid and consequently are isolated critical points in the space of unit volume riemannian metrics we then give examples of critical metrics which are strict local minimizers up to diffeomorphism and scaling a corollary is a local reverse bishop s inequality for such metrics in particular any metric g in a c neighborhood of the round metric s gs satisfying ric g ric gs has volume v ol g v ol gs with equality holding if and only if g is isometric to gs

conformal duality and compact complex surfaces on etudie les structures demi conformement plates sur une variete de riemann a 4 dimensions m g avec l hypothese supplementaire que m est complexe

smoothing singular constant scalar curvature kahler surfaces and minimal lagrangians abstract given a complex surface x with singularities of class t and no nontrivial holomorphic vector field endowed with a kahler class 0 we consider smoothings m t t where t is a kahler class on m t degenerating to 0 under an hypothesis of nondegeneracy of the smoothing at each singular point we prove that if x admits a constant scalar curvature kahler metric in 0 then m t admits a constant scalar curvature kahler metric in t for small t in addition we construct small lagrangian stationary spheres which represent lagrangian vanishing cycles when t is small

parabolic refined invariants and macdonald polynomials a string theoretic derivation is given for the conjecture of hausel letellier and rodriguez villegas on the cohomology of character varieties with marked points their formula is identified with a refined bps expansion in the stable pair theory of a local root stack generalizing previous work of the first two authors in collaboration with pan haiman s geometric construction for macdonald polynomials is shown to emerge naturally in this context via geometric engineering in particular this yields a new conjectural relation between macdonald polynomials and refined local orbifold curve counting invariants the string theoretic approach also leads to a new spectral cover construction for parabolic higgs bundles in terms of holomorphic symplectic orbifolds

stability of vector bundles and extremal metrics on donne des exemples de surfaces reglees m qui n ont pas de champs holomorphes vectoriels non triviaux et qui n admettent pas de metrique de kahler extremale dans une classe de kahler specifique donnee

global structure of solutions for equations of brezis nirenberg type on the unit ball the brezis nirenberg equation and the scalar field equation on the three dimensional unit ball are studied under the robin condition we show the existence and uniqueness of radial solutions in a unified way in particular it is shown that the global structure of solutions changes qualitatively when a parameter in the boundary condition exceeds a certain critical value

parabolic sheaves with real weights as sheaves on the kato nakayama space abstract we define quasi coherent parabolic sheaves with real weights on a fine saturated log analytic space and explain how to interpret them as quasi coherent sheaves of modules on its kato nakayama space this recovers the description as sheaves on root stacks of 5 and 27 for rational weights but also includes the case of arbitrary real weights

einstein metrics and complex singularities this paper is concerned with the construction of special metrics on non compact 4 manifolds which arise as resolutions of complex orbifold singularities our study is close in spirit to the construction of the hyperkahler gravitational instantons but we focus on a different class of singularities we show that any resolution x of an isolated cyclic quotient singularity admits a complete scalar flat kahler metric which is hyperkahler if and only if k x is trivial and that if k x is strictly nef then x also admits a complete non kahler self dual einstein metric of negative scalar curvature in particular complete self dual einstein metrics are constructed on simply connected non compact 4 manifolds with arbitrary second betti number

a theory of normal moveout three geophysical principles are shown to be sufficient to determine the most general practical normal moveout nmo equation the principles are reciprocity in a common midpoint cmp gather finite slowness and exact constant velocity limit the resulting equation is the shifted hyperbola nmo equation that has three parameters comparisons at both near and far offsets between the shifted hyperbola nmo equation and the results for layered media assign geophysical meaning to the parameters two of the parameters zero offset time and nmo velocity are constants and control the very near offset behavior the third parameter is dimensionless and controls the far offset behavior of the nmo curve but it may be a function of offset so as to exactly fit any traveltime curve the parameters may be found by a linear least squares fit to data the theory applies to all offsets for nonturning wave reflections in an isotropic earth for both p waves and converted p sv waves

prescribing curvature on compact surfaces with conical singularities we study the berger nirenberg problem on surfaces with conical singularities i e we discuss conditions under which a function on a riemann surface is the gaussian curvature of some conformal metric with a prescribed set of singularities of conical types

non minimal scalar flat kahler surfaces and parabolic stability a new construction is presented of scalar flat kahler metrics on non minimal ruled surfaces the method is based on the resolution of singularities of orbifold ruled surfaces which are closely related to rank 2 parabolically stable holomorphic bundles this rather general construction is shown also to give new examples of low genus in particular it is shown that mathbb cp 2 blown up at 10 suitably chosen points admits a scalar flat kahler metric this answers a question raised by claude lebrun in 1986 in connection with the classification of compact self dual 4 manifolds

holomorphic vector bundles on sasakian threefolds and parabolic vector bundles a bijective correspondence is constructed between the parabolic vector bundles with rational parabolic weights on a riemann surface x and the holomorphic vector bundles on quasiregular sasakian threefolds over x this bijective correspondence preserves the stable objects

global optimization of generalized nonhyperbolic moveout approximation for long offset normal moveout the approximation of normal moveout is essential for estimating the anisotropy parameters of anisotropic media the generalized nonhyperbolic moveout approximation gma brings a considerable improvement in accuracy compared to known analytical approximations however this is still prone to relatively large errors in the presence of relatively long offsets and large anisotropic parameters which would degrade the inversion accuracy due to error accumulation in velocity analysis we optimize the constant coefficients for all possible groupings of the anellipticity parameter and the offset to depth ratio within practical ranges theoretical analyses and numerical experiments show that the traditional optimization scheme using 2 norm solved by the least squares method could not provide an error constrained result in addition a direct optimization without constant coefficient extension would not lead to a satisfactory accuracy improvement we construct the objective function using the maximum norm and so

existence and nonexistence of least energy solutions of the neumann problem for a semilinear elliptic equation with critical sobolev exponent and a critical lower order perturbation let be a smooth bounded domain in rn with n 5 a 0 0 and 2 2nn 2 we show that the exponent q 2 n 1 n 2 plays a critical role regarding the existence of least energy or ground state solutions of the neumann problem u au u2 1 uq 1in u 0in u 0on namely we prove that when q 2 n 1 n 2 there exists an 0 0 such that the problem has a least energy solution if 0