singular kahler einstein metrics we study degenerate complex monge ampere equations of the form omega dd c f n e t f mu where omega is a big semi kahler form on a compact kahler manifold x of dimension n t in r and mu f omega n is a positive measure with density f in l p x omega n p 1 we prove the existence and unicity of continuous o plurisubharmonic solutions in case x is projective and omega psi omega where psi x to v is a proper birational morphism to a normal projective variety omega in ns r v is an ample class and mu has only algebraic singularities we prove that the solution is smooth in the regular locus of the equation we use these results to construct singular kahler einstein metrics of non positive curvature on projective klt pairs in particular on canonical models of algebraic varieties of general type

k polystability of q fano varieties admitting kahler einstein metrics it is shown that any possibly singular fano variety x admitting a kahler einstein metric is k polystable thus confirming one direction of the yau tian donaldson conjecture in the setting of mathbb q fano varieties equipped with their anti canonical polarization the proof is based on a new formula expressing the donaldson futaki invariants in terms of the slope of the ding functional along a geodesic ray in the space of all bounded positively curved metrics on the anti canonical line bundle of x one consequence is that a toric fano variety x is k polystable iff it is k polystable along toric degenerations iff 0 is the barycenter of the canonical weight polytope p associated to x the results also extend to the logarithmic setting and in particular to the setting of kahler einstein metrics with edge cone singularities applications to geodesic stability bounds on the ricci potential and perelman s lambda entropy functional on k unstable fano manifolds are also given

collapsing limits of the kahler ricci flow and the continuity method we consider the kahler ricci flow on certain calabi yau fibration which is a calabi yau fibration with one dimensional base or a product of two calabi yau fibrations with one dimensional bases assume the kahler ricci flow on total space admits a uniform lower bound for ricci curvature then the flow converges in gromov hausdorff topology to the metric completion of the regular part of generalized kahler einstein current on the base which is a compact length metric space homeomorphic to the base the analogue results for the continuity method on such calabi yau fibrations are also obtained moreover we show the continuity method starting from a suitable kahler metric on the total space of a fano fibration with one dimensional base converges in gromov hausdorff topology to a compact metric on the base during the proof we show the metric completion of the regular part of a generalized kahler einstein current on a riemann surface is compact

ricci flow on kahler einstein surfaces in this paper we prove that if m is a kahler einstein surface with positive scalar curvature if the initial metric has nonnegative sectional curvature and the curvature is positive somewhere then the kahler ricci flow converges to a kahler einstein metric with constant bisectional curvature in a subsequent paper 7 we prove the same result for general kahler einstein manifolds in all dimension this gives an affirmative answer to a long standing problem in kahler ricci flow on a compact kahler einstein manifold does the kahler ricci flow converge to a kahler einstein metric if the initial metric has a positive bisectional curvature our main method is to find a set of new functionals which are essentially decreasing under the kahler ricci flow while they have uniform lower bounds this property gives the crucial estimate we need to tackle this problem

on stability and the convergence of the kahler ricci flow assuming uniform bounds for the curvature the exponential convergence of the kahler ricci flow is established under two con ditions which are a form of stability the mabuchi energy is bound ed from below and the dimension of the space of holomorphic vector fields in an orbit of the diffeomorphism group cannot jump up in the limit

ricci measure for some singular riemannian metrics we define the ricci curvature as a measure for certain singular torsion free connections on the tangent bundle of a manifold the definition uses an integral formula and vector valued half densities we give relevant examples in which the ricci measure can be computed in the time dependent setting we give a weak notion of a ricci flow solution on a manifold

inoue surfaces and the chern ricci flow we investigate the chern ricci flow an evolution equation of hermitian metrics on inoue surfaces these are non kahler compact complex surfaces of type class vii we show that after an initial conformal change the flow always collapses the inoue surface to a circle at infinite time in the sense of gromov hausdorff

maximally genuine multipartite entangled mixed x states of n qubits for every possible spectrum of dimensional density operators we construct an n qubit x state of the same spectrum and maximal genuine multipartite gm concurrence hence characterizing a global unitary transformation that constrained to output x states maximizes the gm concurrence of an arbitrary input mixed state of n qubits we also apply semidefinite programming methods to obtain n qubit x states with maximal gm concurrence for a given purity and to provide an alternative proof of optimality of a recently proposed set of density matrices for the purpose the so called x mems furthermore we introduce a numerical strategy to tailor a quantum operation that converts between any two given density matrices using a relatively small number of kraus operators we apply our strategy to design short operator sum representations for the transformation between any given n qubit mixed state and a corresponding x mems of the same purity

geometry of twisted kahler einstein metrics and collapsing we prove that the twisted kahler einstein metrics that arise on the base of certain holomorphic fiber space with calabi yau fibers have conical type singularities along the discriminant locus these fiber spaces arise naturally when studying the collapsing of ricci flat kahler metrics on calabi yau manifolds and of the kahler ricci flow on compact kahler manifolds with semiample canonical bundle and intermediate kodaira dimension our results allow us to understand their collapsed gromov hausdorff limits when the base is smooth and the discriminant has simple normal crossings

classification of mixed three qubit states we introduce a classification of mixed three qubit states in which we define the classes of separable biseparable w and greenberger horne zeilinger states these classes are successively embedded into each other we show that contrary to pure w type states the mixed w class is not of measure zero we construct witness operators that detect the class of a mixed state we discuss the conjecture that all entangled states with positive partial transpose pptes belong to the w class finally we present a new family of pptes edge states with maximal ranks

the canonical expanding soliton and harnack inequalities for ricci flow we introduce the notion of canonical expanding ricci soliton and use it to derive new harnack inequalities for ricci flow this viewpoint also gives geometric insight into the existing harnack inequalities of hamilton and brendle

coupling of brownian motions and perelman s l functional abstract we show that on a manifold whose riemannian metric evolves under backwards ricci flow two brownian motions can be coupled in such a way that their normalized l distance is a supermartingale as a corollary we obtain the monotonicity of the transportation cost between two solutions of the heat equation in the case that the cost function is the composition of a concave non decreasing function and the normalized l distance in particular it provides a new proof of a recent result of topping p topping l optimal transportation for ricci flow j reine angew math 636 2009 93 122

algebraic characterization of x states in quantum information a class of two qubit states called x states are increasingly being used to discuss entanglement and other quantum correlations in the field of quantum information maximally entangled bell states and werner states are subsets of them apart from being so named because their density matrix looks like the letter x there is not as yet any characterization of them the su 2 su 2 u 1 subalgebra of the full su 4 algebra of two qubits is pointed out as the underlying invariance of this class of states x states are a seven parameter family associated with this subalgebra of seven operators this recognition provides a route to preparing such states and also a convenient algebraic procedure for analytically calculating their properties at the same time it points to other groups of seven parameter states that while not at first sight appearing similar are also invariant under the same subalgebra and it opens the way to analyzing invariant states of other subalgebras in bipartite systems

super ricci flows for metric measure spaces abstract we introduce the notions of super ricci flows and ricci flows for time dependent families of metric measure spaces x d t m t t i the former property is proven to be stable under suitable space time versions of mgh convergence uniformly bounded families of super ricci flows are compact in the spirit of the synthetic lower ricci bounds of lott sturm villani for static metric measure spaces the defining property for super ricci flows is the dynamic convexity of the boltzmann entropy ent m t regarded as a functions on the time dependent geodesic space p x w t t i for ricci flows in addition a nearly dynamic concavity of the boltzmann entropy is requested alternatively super ricci flows will be studied in the framework of the calculus of bakry emery ledoux and equivalence to gradient estimates will be derived for both notions of super ricci flows also enforced versions involving an upper dimension bound n will be presented

the kahler ricci flow on fano bundles we study the behavior of the kahler ricci flow on some fano bundles which is a trivial bundle on one zariski open set we show that if the fiber is mathbb p m blown up at one point or some weighted projective space blown up at the orbifold point and the initial metric is in a suitable kahler class then the fibers collapse in finite time and the metrics converge sub sequentially in gromov hausdorff sense to a metric on the base

super ricci flows and improved gradient and transport estimates we introduce brownian motions on time dependent metric measure spaces proving their existence and uniqueness we prove contraction estimates for their trajectories assuming that the time dependent heat flow satisfies transport estimates with respect to every l p kantorovich distance p in 1 infty these transport estimates turn out to characterize super ricci flows introduced by sturm j funct anal 275 12 3504 3569 2015

the kahler ricci flow with positive bisectional curvature we show that the kahler ricci flow on a manifold with positive first chern class converges to a kahler einstein metric assuming positive bisectional curvature and certain stability conditions

the role of phases in detecting three qubit entanglement we propose separability criteria for three qubit states in terms of diagonal and anti diagonal entries to detect entanglement with positive partial transposes we report that the phases that is the angular parts of anti diagonal entries play a crucial role in determining whether a given three qubit state is separable or entangled and they must obey even an identity for separability in some cases these criteria are strong enough to detect ppt positive partial transpose entanglement with nonzero volume in several cases when all the entries are zero except for diagonal and anti diagonal entries we characterize separability using phases these include the cases when anti diagonal entries of such states share a common magnitude and when ranks are less than or equal to six we also compute the lengths of rank six cases and find three qubit separable states with lengths 8 whose maximum ranks of partial transposes are 7

convergence of calabi yau manifolds abstract in this paper we study the convergence of calabi yau manifolds under kahler degeneration to orbifold singularities and complex degeneration to canonical singularities including the conifold singularities and the collapsing of a family of calabi yau manifolds

the kahler ricci flow and k polystability we consider the k ahler ricci flow on a fano manifold we show that if the curvature remains uniformly bounded along the flow the mabuchi energy is bounded below and the manifold is k polystable then the manifold admits a k ahler einstein metric the main ingredient is a result that says that a sufficiently small perturbation of a csck manifold admits a csck metric if it is k polystable

kahler ricci flow on blowups along submanifolds in this short note we study the behavior of kahler ricci flow on kahler manifolds which contract divisors to smooth submanifolds we show that the kahler potentials are holder continuous and the flow converges sequentially in gromov hausdorff topology to a compact metric space which is homeomorphic to the base manifold

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

the weil petersson geometry of the moduli space of su n 3 calabi yau manifolds i the weil petersson metric is defined on the moduli space of calabi yau manifolds the curvature of this weil petersson metrics is computed and its potential is explicitely defined it is proved that the moduli space of calabi yau manifolds is unobstructed see tian

on the evolution of a hermitian metric by its chern ricci form we consider the evolution of a hermitian metric on a compact complex manifold by its chern ricci form this is an evolution equation first studied by m gill and coincides with the kahler ricci flow if the initial metric is kahler we find the maximal existence time for the flow in terms of the initial data we investigate the behavior of the flow on complex surfaces when the initial metric is gauduchon on complex manifolds with negative first chern class and on some hopf manifolds finally we discuss a new estimate for the complex monge ampere equation on hermitian manifolds

convergence of kahler ricci flow in this paper we prove a theorem on convergence of kahler ricci flow on a compact kahler manifold which admits a kahler ricci soliton

displacement interpolations from a hamiltonian point of view abstract one of the most well known results in the theory of optimal transportation is the equivalence between the convexity of the entropy functional with respect to the riemannian wasserstein metric and the ricci curvature lower bound of the underlying riemannian manifold there are also generalizations of this result to the finsler manifolds and manifolds with a ricci flow background in this paper we study displacement interpolations from the point of view of hamiltonian systems and give a unifying approach to the above mentioned results

ricci flow entropy and optimal transportation let a smooth family of riemannian metrics g tau satisfy the backwards ricer flow equation on a compact oriented n dimensional manifold m suppose two families of normalized n forms omega tau 0 and omega over bar tau 0 satisfy the forwaids in tau heat equation on m generated by the connection laplacian delta g t if these n forms represent two evolving distributions of particles over m the minimum root mean square distance w 2 omega t omega over tilde t t to transport the particles of omega t onto those of omega over tilde t is shown to be non increasing as a function of t without sign conditions on the curvature of mg t moreover this contractivity property is shown to characterize supersolutions to the ricci flow

bounding diameter of singular kahler metric in this paper we investigate the differential geometric and algebro geometric properties of the noncollapsing limit in the continuity method that was introduced by the first two named authors in 19

some discretizations of geometric evolution equations and the ricci iteration on the space of kahler metrics we propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as well as a means to obtain interesting dynamics on certain infinite dimensional spaces we illustrate the fruitfulness of this approach in the context of the ricci flow as well as another flow in kahler geometry we introduce and study dynamical systems related to the ricci operator on the space of kahler metrics that arise as discretizations of these flows we pose some problems regarding their dynamics we point out a number of applications to well studied objects in kahler and conformal geometry such as constant scalar curvature metrics kahler ricci solitons nadel type multiplier ideal sheaves balanced metrics the moser trudinger onofri inequality energy functionals and the geometry and structure of the space of kahler metrics e g we obtain a new sharp inequality strengthening the classical moser trudinger onofri inequality on the two sphere

entanglement criteria for dicke states dicke states are a family of multi qubit quantum states with interesting entanglement properties and they have been observed in many experiments we construct entanglement witnesses for detecting genuine multiparticle entanglement in the vicinity of these states we use the approach of positive partial transpose mixtures to derive the conditions analytically for nearly all cases our criteria are stronger than all conditions previously known

regularity of kahler ricci flows on fano manifolds in this paper we will establish a regularity theory for the kahler ricci flow on fano n manifolds with ricci curvature bounded in l p norm for some p n using this regularity theory we will also solve a long standing conjecture for dimension 3 as an application we give a new proof of the yau tian donaldson conjecture for fano 3 manifolds the results have been announced in 45

ℒ optimal transportation for ricci flow we introduce the notion of l optimal transportation and use it to construct a natural monotonic quantity for ricci flow which includes a selection of other monotonicity results including some key discoveries of perelman 13 both related to entropy and to l length and a recent result of mccann and the author 11

a riemannian interpolation inequality a la borell brascamp and lieb a concavity estimate is derived for interpolations between l 1 m mass densities on a riemannian manifold the inequality sheds new light on the theorems of prekopa leindler borell brascamp and lieb that it generalizes from euclidean space due to the curvature of the manifold the new riemannian versions of these theorems incorporate a volume distortion factor which can however be controlled via lower bounds on ricci curvature the method uses optimal mappings from mass transportation theory along the way several new properties are established for optimal mass transport and interpolating maps on a riemannian manifold

collapsing of the chern ricci flow on elliptic surfaces we investigate the chern ricci flow an evolution equation of hermitian metrics generalizing the kahler ricci flow on elliptic bundles over a riemann surface of genus greater than one we show that starting at any gauduchon metric the flow collapses the elliptic fibers and the metrics converge to the pullback of a kahler einstein metric from the base some of our estimates are new even for the kahler ricci flow a consequence of our result is that on every minimal non kahler surface of kodaira dimension one the chern ricci flow converges in the sense of gromov hausdorff to an orbifold kahler einstein metric on a riemann surface

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

kahler currents and null loci we prove that the non kahler locus of a nef and big class on a compact complex manifold bimeromorphic to a kahler manifold equals its null locus in particular this gives an analytic proof of a theorem of nakamaye and ein lazarsfeld musta nakamaye popa as an application we show that finite time non collapsing singularities of the kahler ricci flow on compact kahler manifolds always form along analytic subvarieties thus answering a question of feldman ilmanen knopf and campana we also extend the second author s results about noncollapsing degenerations of ricci flat kahler metrics on calabi yau manifolds to the nonalgebraic case

the kahler ricci flow through singularities we prove the existence and uniqueness of the weak kahler ricci flow on projective varieties with log terminal singularities we also show that the weak kahler ricci flow can be uniquely continued through divisorial contractions and flips if they exist finally we propose an analytic version of the minimal model program with ricci flow

transport inequalities gradient estimates entropy and ricci curvature we present various characterizations of uniform lower bounds for the ricci curvature of a smooth riemannian manifold m in terms of convexity properties of the entropy considered as a function on the space of probability measures on m as well as in terms of transportation inequalities for volume measures heat kernels and brownian motions and in terms of gradient estimates for the heat semigroup 2004 wiley periodicals inc

the kahler ricci flow on surfaces of positive kodaira dimension 4 estimates 9 4 1 the zeroth order and volume estimates 9 4 2 a partial second order estimate 13 4 3 gradient estimates 14 4 4 the second order estimates 18

bounding scalar curvature for global solutions of the kahler ricci flow we show that the scalar curvature is uniformly bounded for the normalized k ahler ricci flow on a k ahler manifold with semi ample canonical bundle in particular the normalized k ahler ricci flow has long time existence if and only if the scalar curvature is uniformly bounded for k ahler surfaces projective manifolds of complex dimension three and for projective manifolds of all dimensions if assuming the abundance conjecture

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

on the l function and the reduced volume of perelman ii the main purpose of this paper is to present a number of analytic and geometric properties of the l function and the reduced volume of perelman including in particular the monotonicity the upper bound and the rigidities of the reduced volume

device independent witnesses of genuine multipartite entanglement we consider the problem of determining whether genuine multipartite entanglement was produced in an experiment without relying on a characterization of the systems observed or of the measurements performed we present an n partite inequality that is satisfied by all correlations produced by measurements on biseparable quantum states but which can be violated by n partite entangled states such as greenberger horne zeilinger states in contrast to traditional entanglement witnesses the violation of this inequality implies that the state is not biseparable independently of the hilbert space dimension and of the measured operators violation of this inequality does not imply however genuine multipartite nonlocality we show more generically how the problem of identifying genuine tripartite entanglement in a deviceindependent way can be addressed through semidefinite programming

convergence of scalar curvature of kahler ricci flow on manifolds of positive kodaira dimension abstract in this paper we consider kahler ricci flow on n dimensional kahler manifold with semi ample canonical line bundle and 0 m kod x n such manifolds admit a calabi yau fibration over its canonical model we prove that the scalar curvature of the kahler metrics along the normalized kahler ricci flow converge to m outside the singular set of this fibration

non explosion of diffusion processes on manifolds with time dependent metric we study the problem of non explosion of diffusion processes on a manifold with time dependent riemannian metric in particular we obtain that brownian motion cannot explode in finite time if the metric evolves under backwards ricci flow our result makes it possible to remove the assumption of non explosion in the pathwise contraction result established by arnaudon et al arxiv 0904 2762 to appear in sem prob as an important tool which is of independent interest we derive an ito formula for the distance from a fixed reference point generalising a result of kendall ann prob 15 1491 1500 1987

ricci flow on kahler einstein manifolds this is the continuation of our earlier article 10 for any kahler einstein surfaces with positive scalar curvature if the initial metric has positive bisectional curvature then we have proved see 10 that the kahler ricci flow converges exponentially to a unique kahler einstein metric in the end this partially answers a long standing problem in ricci flow on a compact kahler einstein manifold does the kahler ricci flow converge to a kahler einstein metric if the initial metric has positive bisectional curvature in this article we give a complete affirmative answer to this problem

a criterion for flatness of hodge bundles over curves and geometric applications for the definition of the kodaira dimension as well as for a discussion of the conjecture cn m i refer to the bourbaki talk by esnault 7 here i only want to say that c n and cn 1 are true and that for applications to albanese mappings a stronger version of cn m is needed which in 7 is called conjecture c m if m 1 it merely says that in addition to the validity of cn in the special case when y is an elliptic curve and k x f 0 there should exist a surjective morphism y y such that the fibre product x x y is birational to a product y x f for u n 2 m 1 this last requirement follows from a result of arakelov 1 theorem 1 1 which implies that deg f tox r 0 and equality holds if and only if f is isotrivial i e after a finite covering of y the fibre product is birational to a product here cox r is the relative dualizing sheaf see sect 5 1 kawamata s proof of c as presented in 13 depends on a result of fujita 11 about line bundle quotients of f t o x r d i m y l very roughly fujita s result says that arakelov s inequality generalizes to the case when dimx is arbitrary but still dim y 1 here i want to look more closely to what happens if deg f o2x r 0 the natural framework for this question is griffith s theory of variations of hodge structure our main result is the following

three qubit entanglement witnesses with the full spanning properties we construct three qubit entanglement witnesses with the full spanning property by which we mean that all the partial transposes have the spanning properties these witnesses determine faces for separable states whose interior lies in the interior of positive partial transposes ppt states and so they detect sets of ppt entangled states with nonempty interiors these witnesses also detect fully bi separable states as well as all kinds of bi separable pure product states our construction depends on the observation that multi partite entanglement witnesses correspond to positive multi linear maps

weak solutions to degenerate complex monge ampere flows ii abstract studying the long term behavior of the kahler ricci flow on mildly singular varieties one is naturally led to study weak solutions of degenerate parabolic complex monge ampere equations the purpose of this article the second of a series on this subject is to develop a viscosity theory for degenerate complex monge ampere flows on compact kahler manifolds our general theory allows in particular to define and study the normalized kahler ricci flow on varieties with canonical singularities generalizing results of song and tian

contracting exceptional divisors by the kahler ricci flow we give a criterion under which a solution g t of the kahler ricci flow contracts exceptional divisors on a compact manifold and can be uniquely continued on a new manifold as t tends to the singular time t from each direction we prove the convergence of g t in the sense of gromov hausdorff and smooth convergence away from the exceptional divisors we call this behavior for the kahler ricci flow a canonical surgical contraction in particular our results show that the kahler ricci flow on a projective algebraic surface will perform a sequence of canonical surgical contractions until in finite time either the minimal model is obtained or the volume of the manifold tends to zero

on the convex cones arising from classifications of partial entanglement in the three qubit system in order to classify partial entanglement of multi partite states it is natural to consider the convex hulls intersections and differences of basic convex cones obtained from partially separable states with respect to partitions of systems in this paper we consider convex cones consisting of x shaped three qubit states arising in this way the class of x shaped states includes important classes like greenberger horne zeilinger diagonal states we find all the extreme rays of those convex cones to exhibit corresponding partially separable states we also give characterizations for those cones which give rise to necessary criteria in terms of diagonal and anti diagonal entries for general three qubit states

scalar v soliton equation and kahler ricci flow on symplectic quotients abstract in this paper we consider the v soliton equation which is a degenerate fully nonlinear equation introduced by la nave and tian in their work on kahler ricci flow on symplectic quotients one can apply the interpretation to study finite time singularities of the kahler ricci flow as in the case of kahler einstein metrics we can also reduce the v soliton equation to a scalar equation on kahler potentials which is of monge ampere type we formulate some preliminary estimates for such a scalar equation on a compact kahler manifold m

construction of multi qubit optimal genuine entanglement witnesses we interpret multi partite genuine entanglement witnesses as simultaneous positivity of various maps arising from them we apply this result to multi qubit x shaped hermitian matrices and characterize the conditions for them to be genuine entanglement witnesses in terms of entries furthermore we find all optimal entries among them they turn out to have spanning properties and so they detect a non zero volume set of multi qubit genuine entanglement we also characterize decomposability for x shaped entanglement witnesses

collapsing of abelian fibered calabi yau manifolds we study the collapsing behavior of ricci flat kahler metrics on a projective calabi yau manifold which admits an abelian fibration when the volume of the fibers approaches zero we show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature and along the fibers the rescaled metrics become flat in the limit the limit metric on the base minus the critical locus is locally isometric to an open dense subset of any gromov hausdorff limit space of the ricci flat metrics we then apply these results to study metric degenerations of families of polarized hyperkahler manifolds in the large complex structure limit in this setting we prove an analogue of a result of gross and wilson for k3 surfaces which is motivated by the strominger yau zaslow picture of mirror symmetry