negative holomorphic curvature and positive canonical bundle in this note we show that if a projective manifold admits a kahler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample this confirms a conjecture of the second author

algebraic deformations of compact kahler surfaces ii in the first version of this paper a short proof was given of kodaira s result that every compact kahler surface is a deformation of an algebraic surface under the extra assumption that the infinitesimal deformations of the surface were unobstructed in this paper the extra assumption is removed

a potential theory for the k curvature equation abstract in this paper we introduce a potential theory for the k curvature equation which can also be seen as a pde approach to curvature measures we assign a measure to a bounded upper semicontinuous function which is strictly subharmonic with respect to the k curvature operator and establish the weak continuity of the measure

reduction of manifolds with semi negative holomorphic sectional curvature in this note we continue the investigation of a projective kahler manifold m of semi negative holomorphic sectional curvature h we introduce a new differential geometric numerical rank invariant which measures the number of linearly independent truly flat directions of h in the tangent spaces we prove that this invariant is bounded above by the nef dimension and bounded below by the numerical kodaira dimension of m we also prove a splitting theorem for m in terms of the nef dimension and under some additional hypotheses in terms of the new rank invariant

on kahler manifolds with positive orthogonal bisectional curvature abstract for any irreducible kahler manifold which admits positive orthogonal bisectional curvature and c 1 0 if this positivity condition is preserved under the flow then the underlying manifold is biholomorphic to c p n

a new positivity condition for the curvature of hermitian manifolds in this note we introduce a new type of positivity condition for the curvature of a hermitian manifold which generalizes the notion of nonnegative quadratic orthogonal bisectional curvature to the non kahler case we derive a bochner formula for closed 1 1 forms from which this condition appears naturally and prove that if a hermitian manifold satisfies our positivity condition then any class alpha in h 1 1 bc x can be represented by a closed 1 1 form which is parallel with respect to the bismut connection lastly we show that such a curvature positivity condition holds on certain generalized hopf manifolds and on certain vaisman manifolds

scheduled relaxation jacobi method elliptic partial differential equations epdes appear in a wide variety of areas of mathematics physics and engineering typically epdes must be solved numerically which sets an ever growing demand for efficient and highly parallel algorithms to tackle their computational solution the scheduled relaxation jacobi srj is a promising class of methods atypical for combining simplicity and efficiency that has been recently introduced for solving linear poisson like epdes the srj methodology relies on computing the appropriate parameters of a multilevel approach with the goal of minimizing the number of iterations needed to cut down the residuals below specified tolerances the efficiency in the reduction of the residual increases with the number of levels employed in the algorithm applying the original methodology to compute the algorithm parameters with more than 5 levels notably hinders obtaining optimal srj schemes as the mixed non linear algebraic differential system of equations from which they result becomes notably stiff here we present a new methodology for obtaining the parameters of srj schemes that overcomes the limitations of the original algorithm and provide parameters for srj schemes with up to 15 levels and resolutions of up to 215 points per dimension allowing for acceleration factors larger than several hundreds with respect to the jacobi method for typical resolutions and in some high resolution cases close to 1000 most of the success in finding srj optimal schemes with more than 10 levels is based on an analytic reduction of the complexity of the previously mentioned system of equations furthermore we extend the original algorithm to apply it to certain systems of non linear epdes we compute new optimal parameters of the scheduled relaxation jacobi method the new parameters are calculated for srj schemes with p 6 to p 15 levels we reduce the stiffness in the computation of optimal srj parameters analytically we provide a grid of optimal parameters for different p and numerical resolutions we benchmark srj methods against other algorithms to solve linear systems

acceleration of the jacobi iterative method by factors exceeding 100 using scheduled relaxation we present a methodology that accelerates the classical jacobi iterative method by factors exceeding 100 when applied to the finite difference approximation of elliptic equations on large grids the method is based on a schedule of over and under relaxations that preserves the essential simplicity of the jacobi method mathematical conditions that maximize the convergence rate are derived and optimal schemes identified the convergence rate predicted from the analysis is validated via numerical experiments the substantial acceleration of the jacobi method enabled by the current method has the potential to significantly accelerate large scale simulations in computational mechanics as well as other areas where elliptic equations are prominent

hermitian metrics of positive holomorphic sectional curvature on fibrations the main result of this note essentially is that if the base and fibers of a compact fibration carry hermitian metrics of positive holomorphic sectional curvature then so does the total space of the fibration the proof is based on the use of a warped product metric as in the work by cheung in case of negative holomorphic sectional curvature but differs in certain key aspects e g in that it does not use the subadditivity property for holomorphic sectional curvature due to grauert reckziegel and wu

monge ampere equations in big cohomology classes we define non pluripolar products of an arbitrary number of closed positive 1 1 currents on a compact kahler manifold x given a big 1 1 cohomology class on x i e a class that can be represented by a strictly positive current and a positive measure on x of total mass equal to the volume of and putting no mass on pluripolar sets we show that can be written in a unique way as the top degree self intersection in the non pluripolar sense of a closed positive current in we then extend kolodziedj s approach to sup norm estimates to show that the solution has minimal singularities in the sense of demailly if has l 1 e density with respect to lebesgue measure if is smooth and positive everywhere we prove that t is smooth on the ample locus of provided is nef using a fixed point theorem we finally explain how to construct singular kahler einstein volume forms with minimal singularities on varieties of general type

a numerical method for the incompressible navier stokes equations based on an approximate projection in this method we present a fractional step discretization of the time dependent incompressible navier stokes equations the method is based on a projection formulation in which we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields our treatment of the diffusion convection step uses a specialized second order upwind method for differencing the nonlinear convective terms that provides a robust treatment of these terms at a high reynolds number in contrast to conventional projection type discretizations that impose a discrete form of the divergence free constraint we only approximately impose the constraint i e the velocity field we compute is not exactly divergence free the approximate projection is computed using a conventional discretization of the laplacian and the resulting linear system is solved using conventional multigrid methods numerical examples are presented to validate the second order convergence of the method for euler finite reynolds number and stokes flow a second example illustrating the behavior of the algorithm on an unstable shear layer is also presented

determinants of dirac operators with local boundary conditions we study functional determinants for dirac operators on manifolds with boundary we give for local boundary conditions an explicit formula relating these determinants to the corresponding green s functions we finally apply this result to the case of a bidimensional disk under baglike conditions

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

analysis of the sor iteration for the 9 point laplacian the sor iteration for solving linear systems of equations depends upon an overrelaxation factor omega a theory for determining omega was given by young iterative methods for solving partial differential equations of elliptic types trans amer math soc 76 1954 pp 92 111 for consistently ordered matrices here we determine the optimal omega for the 9 point stencil for the model problem of laplace s equation on a square we consider several orderings of the equations including the natural rowwise and multicolor orderings all of which lead to nonconsistently ordered matrices and find two equivalence classes of orderings with different convergence behavior and optimal omega s we compare our results for the natural rowwise ordering to those of garabedian estimation of the relaxation factor for small mesh size math comp 10 1956 pp 183 185 and explain why both results are in a sense correct even though they differ we also analyze a pseudo sor method for the model p

fermionic determinants for chiral bag like two dimensional systems this paper evaluates the determinant of a dirac like operator for a system of fermions confined to a region of two dimensional euclidean space and subject to bag like boundary conditions in the framework of the function regularization method use is made of seeley s developments for the resolvent of such operators a relation is derived between the green s function and the finite part of the diagonal element of the kernel of the power z of the operator for z 1 which amounts to having information about the whole seeley s series

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

conjugate gradient like algorithms for solving nonsymmetric linear systems this paper presents a unified formulation of a class of the conjugate gradient like algorithms for solving nonsymmetric linear systems the common framework is the petrov galerkin method on krylov subspaces we discuss some practical points concerning the methods and point out some of the interrelations between them 1 introduction in the recent few years a large number of generalizations of the conjugate gradient and conjugate residual methods which are very successful in solving symmetric positive definite linear systems have been proposed for solving nonsymmetric linear systems 3 6 5 7 12 in this paper we present an abstract framework which includes most of these methods and many new ones our goal is to understand the relationships among the methods and to synthesize consider the general linear system

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

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

richardson s iteration for nonsymmetric matrices abstract to solve the linear n n system 1 ax a for any nonsingular matrix a richardson s iteration 2 x j 1 x j j ax j a j 1 2 n which is applied in a cyclic manner with cycle length n is investigated where the j are free parameters the objective is to minimize the error x n 1 x where x is the solution of 1 if the spectrum of a is known to lie in a compact set s one is led to the chebyshev type approximation problem 3 min p 1 v n max z s p z where v n is the linear span of z z 2 z n if p solves 3 then the reciprocals of the zeros of p are optimal iteration parameters j it is shown that for a real problem 1 the iteration 2 can be carried out with real arithmetic alone even when there are complex j the stationary case n 1 is solved completely i e for all compact sets s the problem 3 is solved explicitly as a consequence the converging stationary iteration processes can be characterized for arbitrary closed disks s the problem 3 can be solved for all n n and a simple proof is provided the lemniscates associated with s are introduced they appear as an important tool for studying the stability of the iteration 2

fully nonlinear elliptic equations for conformal deformations of chern ricci forms abstract we study a general class of fully nonlinear elliptic equations of second order on hermitian manifolds we derive a priori estimates and prove some existence results for these equations in particular we prove interior estimates for the complex hessian which in general do not hold for other types of equations we apply our results to draw geometric conclusions on conformal deformations of the mixed chern ricci forms a notion we introduced in the article our work is motivated by the close connections of these equations to problems in non kahler geometry and the fact that there have been increasing interests in fully nonlinear pde s beyond the complex monge ampere equation from complex geometry

ricci flow on quasi projective manifolds we consider the kahler ricci flow on complete finite volume metrics that live on the complement of a divisor in a compact kahler manifold x assuming certain spatial asymptotics on the initial metric we compute the singularity time in terms of cohomological data on x we also give a sufficient condition for the singularity if there is one to be type ii

the complex monge ampere type equation on compact hermitian manifolds and applications abstract we prove the existence and uniqueness of continuous solutions to the complex monge ampere type equation with the right hand side in l p p 1 on compact hermitian manifolds next we generalise results of eyssidieux guedj and zeriahi 17 18 to compact hermitian manifolds which a priori are not in the fujiki class these generalisations lead to a number of applications we obtain partial results on a conjecture of tosatti and weinkove 40 and on a weak form of a conjecture of demailly and paun 11

holomorphic sectional curvature nefness and miyaoka yau type inequality on a compact kahler manifold we introduce a notion of almost nonpositivity for the holomorphic sectional curvature which by definition is weaker than the existence of a kahler metric with semi negative holomorphic sectional curvature we prove that a compact kahler manifold of almost nonpositive holomorphic sectional curvature has a nef canonical line bundle contains no rational curves and satisfies some miyaoka yau type inequalities in the course of the discussions we attach a real value to any fixed kahler class which up to a constant factor depending only on the dimension of manifold turns out to be an upper bound for the nef threshold

acceleration of the scheduled relaxation jacobi method promising strategies for solving large sparse linear systems abstract the main aim of this paper is to develop two algorithms based on the scheduled relaxation jacobi srj method yang and mittal 2014 7 for solving problems arising from the finite difference discretization of elliptic partial differential equations on large grids these two algorithms are the alternating anderson scheduled relaxation jacobi aasrj method by utilizing anderson mixing after each srj iteration cycle and the minimal residual scheduled relaxation jacobi mrsrj method by minimizing residual after each srj iteration cycle respectively through numerical experiments we show that aasrj is competitive with the optimal version of the srj method adsuara et al 2017 9 in most problems we considered here and mrsrj outperforms srj in all cases the properties of aasrj and mrsrj are demonstrated both of them are promising strategies for solving large sparse linear systems while maintaining the simplicity of the jacobi method

families of lagrangian fibrations on hyperkahler manifolds abstract a holomorphic lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with lagrangian fibers it is known due to huybrechts that a given compact manifold admits only finitely many holomorphic symplectic structures up to deformation we prove that a given compact simple hyperkahler manifold with b 2 7 admits only finitely many deformation types of holomorphic lagrangian fibrations we also prove that all known hyperkahler manifolds are never kobayashi hyperbolic

local curvature estimates of long time solutions to the kahler ricci flow abstract we study the local curvature estimates of long time solutions to the normalized kahler ricci flow on compact kahler manifolds with semi ample canonical line bundle using these estimates we prove that on such a manifold the set of singular fibers of the semi ample fibration on which the riemann curvature blows up at time infinity is independent of the choice of the initial kahler metric moreover when a regular fiber of the semi ample fibration is not a finite quotient of a torus we determine the exact curvature blow up rate of the kahler ricci flow near the regular fiber

an extension of a theorem of wu yau we show that a compact kahler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle if the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample confirming a conjecture of yau the key ingredient is the recent solution of this conjecture in the projective case by wu yau

delayed over relaxation for iterative methods we propose a variant of the relaxation step used in the most widespread iterative methods e g jacobi over relaxation successive over relaxation which combines the iteration at the predicted step namely n 1 with the iteration at step n 1 we provide a theoretical analysis of the proposed algorithm by applying such a delayed relaxation step to a generic convergent iterative scheme we prove that under proper assumptions this significantly improves the convergence rate of the initial iterative method as a relevant example we apply the proposed algorithm to the solution of the poisson equation highlighting the advantages in comparison with classical iterative models we propose a variant of the relaxation step for iterative solvers this variant improves the convergence for matrices with real eigenvalues the proposed scheme profitably applies to elliptic problems

on the equivalence between the scheduled relaxation jacobi method and richardson s non stationary method abstract the scheduled relaxation jacobi srj method is an extension of the classical jacobi iterative method to solve linear systems of equations a u b associated with elliptic problems it inherits its robustness and accelerates its convergence rate computing a set of p relaxation factors that result from a minimization problem in a typical srj scheme the former set of factors is employed in cycles of m consecutive iterations until a prescribed tolerance is reached we present the analytic form for the optimal set of relaxation factors for the case in which all of them are strictly different and find that the resulting algorithm is equivalent to a non stationary generalized richardson s method where the matrix of the system of equations is preconditioned multiplying it by d diag a our method to estimate the weights has the advantage that the explicit computation of the maximum and minimum eigenvalues of the matrix a or the corresponding iteration matrix of the underlying weighted jacobi scheme is replaced by the much easier calculation of the maximum and minimum frequencies derived from a von neumann analysis of the continuous elliptic operator this set of weights is also the optimal one for the general problem resulting in the fastest convergence of all possible srj schemes for a given grid structure the amplification factor of the method can be found analytically and allows for the exact estimation of the number of iterations needed to achieve a desired tolerance we also show that with the set of weights computed for the optimal srj scheme for a fixed cycle size it is possible to estimate numerically the optimal value of the parameter in the successive overrelaxation sor method in some cases finally we demonstrate with practical examples that our method also works very well for poisson like problems in which a high order discretization of the laplacian operator is employed e g a 9 or 17 points discretization this is of interest since the former discretizations do not yield consistently ordered a matrices and hence the theory of young cannot be used to predict the optimal value of the sor parameter furthermore the optimal srj schemes deduced here are advantageous over existing sor implementations for high order discretizations of the laplacian operator in as much as they do not need to resort to multi coloring schemes for their parallel implementation

functional determinants and geometry in this paper we study the zeta function determinant in the context of elliptic boundary value problems our main technique is to relate the determinant of an operator or a ratio of determinants to the boundary values of the solutions of the operator this has the advantage of restricting attention to the solutions of the operator which do not depend on the boundary conditions and can often be written down explicitly rather than the eigenvalues which are usually difficult to work with in addition the problem is reduced to a calculation over the boundary of the manifold which is a closed manifold of dimension one less than the original manifold this has special significance in the case that the manifold is a finite interval in this case the boundary is a pair of points and the determinant of an ordinary differential operator is expressed in terms of the determinant of a finite matrix

on the use of higher order projection methods for incompressible turbulent flow an important issue in the development of higher order methods for incompressible flow is how they perform when the flow is turbulent a useful diagnostic of a method for turbulent flow is the minimum resolution that is required to adequately resolve the turbulent energy cascade at a given reynolds number in this paper we present careful numerical experiments to assess the utility of higher order numerical methods based on this metric we first introduce a numerical method for the incompressible navier stokes equations based on fourth order discretizations in both space and time the method is based on an auxiliary variable formulation and combines fourth order finite volume differencing with a semi implicit spectral deferred correction temporal integration scheme we also introduce for comparison purposes versions based on second order spatial and or temporal discretizations we demonstrate that for smooth problems each of the methods exhibits the expected order of convergence in time and space we