on the pluricanonical maps of varieties of intermediate kodaira dimension in this paper we will prove a uniformity result for the iitaka fibration f x rightarrow y provided that the generic fiber has a good minimal model and the prokhorov shokurov conjecture holds in particular the result holds if the variation of f is zero or kappa x text dim x 1

quantization of edge currents for continuous magnetic operators abstract for a magnetic hamiltonian on a half plane given by the sum of the landau operator with dirichlet boundary conditions and a random potential a quantization theorem for the edge currents is proven this shows that the concept of edge channels also makes sense in presence of disorder moreover gaussian bounds on the heat kernel and its covariant derivatives are obtained

convergence of weak kahler ricci flows on minimal models of positive kodaira dimension studying the behavior of the kahler ricci flow on mildly singular varieties one is naturally lead to study weak solutions of degenerate parabolic complex monge ampere equations

the dual complex of calabi yau pairs a log calabi yau pair consists of a proper variety x and a divisor d on it such that k x d is numerically trivial a folklore conjecture predicts that the dual complex of d is homeomorphic to the quotient of a sphere by a finite group the main result of the paper shows that the fundamental group of the dual complex of d is a quotient of the fundamental group of the smooth locus of x hence its pro finite completion is finite this leads to a positive answer in dimension le 4 we also study the dual complex of degenerations of calabi yau varieties the key technical result we prove is that after a volume preserving birational equivalence the transform of d supports an ample divisor

new outlook on the minimal model program ii we prove that the finite generation of adjoint rings implies all the foundational results of the minimal model program the rationality cone and contraction theorems the existence of flips and termination of flips with scaling in the presence of a big boundary

on images of weak fano manifolds we consider a smooth projective morphism between smooth complex projective varieties if the source space is a weak fano or fano manifold then so is the target space our proof is hodge theoretic we do not need mod p reduction arguments we also discuss related topics and questions

stable maps in higher dimensions we formulate a notion of stability for maps between polarised varieties which generalises kontsevich s definition when the domain is a curve and tian donaldson s definition of k stability when the target is a point we give some examples such as kodaira embeddings and fibrations we prove the existence of a projective moduli space of canonically polarised stable maps generalising the kontsevich alexeev moduli space of stable maps in dimensions one and two we also state an analogue of the yau tian donaldson conjecture in this setting relating stability of maps to the existence of certain canonical kahler metrics

the essential skeleton of a degeneration of algebraic varieties in this paper we explore the connections between the minimal model program and the theory of berkovich spaces let k be a field of characteristic zero and let x be a smooth and proper k t variety with semiample canonical divisor we prove that the essential skeleton of x coincides with the skeleton of any minimal dlt model and that it is a strong deformation retract of the berkovich analytification of x as an application we show that the essential skeleton of a calabi yau variety over k t is a pseudo manifold

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

boundary maps for c crossed products with r with an application to the quantum hall effect the boundary map in k theory arising from the wiener hopf extension of a crossed product algebra with r is the connes thom isomorphism in this article the wiener hopf extension is combined with the heisenberg group algebra to provide an elementary construction of a corresponding map on higher traces and cyclic cohomol ogy it then follows directly from a non commutative stokes theorem that this map is dual w r t connes pairing of cyclic cohomology with k theory as an application we prove equality of quantized bulk and edge conductivities for the integer quantum hall effect described by continuous magnetic schrodinger operators 1 motivation and main result in a commonly used approach to study aperiodic solids particles in the bulk of the medium are described by covariant families of one particle schrodinger operators h whereis the probability space of configurations furnished with an ergodic action of space translations crossed product algebras provide a natural framework for such families be86 in particular their bounded functions are represented by elements of a c crossed product the so called bulk algebra the non commutative topology of the c algebra is a useful tool to construct topological invariants resulting from pairings between k group elements and higher traces some of these invariants may be physi cally interpreted as topologically quantised quantities the quantised hall conductivity is such an example the physics near a boundary of the solid can also be described by a c algebra the so called edge algebra the bulk algebra being essentially a crossed product of the edge algebra with r or with z in the tight binding approximation krs02 both algebras are tied together in the wiener hopf extension or respectively the toeplitz extension this extension gives rise to boundary maps between the k groups and the higher traces of the bulk and edge algebra which allow one to equate bulk and edge in variants this topological relation and its physical interpretations is our main objective we discuss one prominent physical example of this the quantum hall effect where the

log canonical singularities are du bois a recurring difficulty in the minimal model program mmp is that while log terminal singularities are quite well behaved for instance they are rational log canonical singularities are much more complicated they need not even be cohenmacaulay the aim of this paper is to prove that as conjectured in kol92 1 13 log canonical singularities are du bois the concept of du bois singularities abbreviated as db was introduced by steenbrink in ste83 as a weakening of rationality it is not clear how to define du bois singularities in positive characteristic so we work over a field of characteristic 0 throughout the paper the precise definition is rather involved see 1 10 but our main applications rely only on the following consequence

extracting non canonical places abstract let x b be a log canonical pair and v be a finite set of divisorial valuations with log discrepancy in 0 1 we prove that there exists a projective birational morphism y x so that the exceptional locus is divisorial the exceptional divisors are q cartier and they correspond to elements of v we study how two such models are related as applications we apply the main theorem to the study of deformations of log canonical singularities furthermore we prove the existence of rationalization for log canonical singularities

poles of maximal order of motivic zeta functions we prove a 1999 conjecture of veys which says that the opposite of the log canonical threshold is the only possible pole of maximal order of denef and loeser s motivic zeta function associated with a germ of a regular function on a smooth variety over a field of characteristic zero we apply similar methods to study the weight function on the berkovich skeleton associated with a degeneration of calabi yau varieties our results suggest that the weight function induces a flow on the non archimedean analytification of the degeneration towards the kontsevich soibelman skeleton

microlocalization percolation and anderson localization for the magnetic schrodinger operator with a random potential abstract we study the spectral properties of the magnetic schrodinger operator with a random potential using results from microlocal analysis and percolation we show that away from the landau levels the spectrum is almost surely pure point with at least exponentially decaying eigenfunctions moreover it is shown that the rate of decay is proportional to b

log canonical models of elliptic surfaces abstract we classify the log canonical models of elliptic surface pairs f x c s f a where f x c is an elliptic fibration s is a section and f a is a weighted sum of marked fibers in particular we show how the log canonical models depend on the choice of the weights we describe a wall and chamber decomposition of the space of weights based on how the log canonical model changes in addition we give a generalized formula for the canonical bundle of an elliptic surface with section and marked fibers this is the first step in constructing compactifications of moduli spaces of elliptic surfaces using the minimal model program

deformations of canonical singularities m reid defined the concept of canonical singularities as those which appear on canonical models of algebraic varieties whose canonical rings are finitely generated from the view point of the classification theory of algebraic varieties it is expected that small deformations of a variety which has only canonical singularities have only canonical singularities the main result of this paper states that this is the case

nef dimension of minimal models we reduce the abundance conjecture in dimension 4 to the following numerical statement if the canonical divisor k is nef and has maximal nef dimension then k is big from this point of view we classify in dimension 2 nef divisors which have maximal nef dimension but which are not big

the singularity of the canonical model of compact kahler manifolds soit x une variete de kahler compacte dont l anneau canonique 7b r m 0 h 0 o x mk x est finement genere alors il existe un q diviseur effectif sur s proj 7b r tel que s est log terminal

landau hamiltonians with random potentials localization and the density of states we prove the existence of localized states at the edges of the bands for the two dimensional landau hamiltonian with a random potential of arbitrary disorder provided that the magnetic field is sufficiently large the corresponding eigenfunctions decay exponentially with the magnetic field and distance we also prove that the integrated density of states is lipschitz continuous away from the landau energies the proof relies on a wegner estimate for the finite area magnetic hamiltonians with random potentials and exponential decay estimates for the finitearea green s functions the proof of the decay estimates for the green s functions uses fundamental results from two dimensional bond percolation theory

dynamical delocalization in random landau hamiltonians we prove the existence of dynamical delocalization for random landau hamiltonians near each landau level since typically there is dynamical localization at the edges of each disordered broadened landau band this implies the existence of at least one dynamical mobility edge at each landau band namely a boundary point between the localization and delocalization regimes which we prove to converge to the corresponding landau level as either the magnetic field goes to infinity or the disorder goes to zero in this article we prove the existence of dynamical delocalization for random landau hamiltonians near each landau level more precisely we prove that for these two dimensional hamiltonians there exists at least one energy e near each landau level such that e 1 where e the local transport exponent introduced in gk5 is a measure of the rate of transport in wave packets with spectral support near e since typically there is dynamical localization at the edges of each disordered broadened landau band this implies the existence of at least one dynamical mobility edge at each landau band namely a boundary point between the localization and delocalization regimes which we prove to converge to the corresponding landau level as either the magnetic field goes to infinity or the disorder goes to zero random landau hamiltonians are the subject of intensive study due to their links with the integer quantum hall e ect kli for which von klitzing received the 1985 nobel prize in physics they describe an electron moving in a very thin flat conductor with impurities under the influence of a constant magnetic field perpendicular to the plane of the conductor and play an important role in the understanding of the quantum hall e ect l aoa t h nt ku be avss bees laughlin s argument l as pointed out by halperin h uses the assumption that under weak disorder and strong magnetic field the energy spectrum consists of bands of extended states separated

subadditivity of kodaira dimensions for fibrations of three folds in positive characteristics abstract in this paper we will prove subadditivity of kodaira dimensions for a fibration with possibly singular geometric generic fiber under certain nefness and relative semi ampleness conditions as an application for a fibration f x y of a smooth projective threefold over an algebraically closed field of characteristic p 5 under the assumption that y is of general type and non uniruled we prove subadditivity of kodaira dimensions when general fibers are smooth or when k x y is relatively big over y

extension of twisted hodge metrics for kahler morphisms let f x y be a holomorphic map of complex manifolds which is proper kahler and surjective with connected fibers and which is smooth over y z the complement of an analytic subset z let e be a nakano semi positive vector bundle on x and consider direct image sheaves f r qf k x y otimes e for q geq 0 in our previous paper we discussed the nakano semi positivity of f with respect to the so called hodge metric when the map f is smooth in this paper we discuss the extension of the induced metric on the tautological line bundle o 1 on the projective space bundle p f over y z as a singular hermitian metric with semi positive curvature over y as a particular consequence if y is projective r qf k x y otimes e is weakly positive over y z in the sense of viehweg

degenerations of rationally connected varieties we prove that a degeneration of rationally connected varieties over a field of characteristic zero always contains a geometrically irreducible subvariety which is rationally connected

cellular neural networks the navier stokes equation and microarray image reconstruction although the last decade has witnessed a great deal of improvements achieved for the microarray technology many major developments in all the main stages of this technology including image processing are still needed some hardware implementations of microarray image processing have been proposed in the literature and proved to be promising alternatives to the currently available software systems however the main drawback of those proposed approaches is the unsuitable addressing of the quantification of the gene spot in a realistic way without any assumption about the image surface our aim in this paper is to present a new image reconstruction algorithm using the cellular neural network that solves the navier stokes equation this algorithm offers a robust method for estimating the background signal within the gene spot region the matcnn toolbox for matlab is used to test the proposed method quantitative comparisons are carried out i e in terms of objective criteria between our approach and some other available methods it is shown that the proposed algorithm gives highly accurate and realistic measurements in a fully automated manner within a remarkably efficient time

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

on isolated singularities with a noninvertible finite endomorphism abstract we prove that if x 0 x 0 is a finite endomorphism of an isolated singularity such that deg 2 and is etale in codimension 1 then x is q gorenstein and log canonical

anderson localization for multi dimensional systems at large disorder or large energy we prove that discrete schrodinger operators on d with a random potential have almost surely only pure point spectrum and exponentially decaying eigenfunctions for large disorder or large energy this is the first proof of localization for multi dimensional anderson models

the quantum hall effect for electrons in a random potential we assume the existence of sufficiently localised states near the edges of each landau band we then prove that the hall conductivity is quantised and the parallel conductivity vanishes when the filling factor stays close to an integer the hall integer is a topological invariant given by the landau band index

simple connectedness of fano log pairs with semi log canonical singularities we show that any union of slc strata of a fano log pair with semi log canonical singularities is simply connected in particular fano log pairs with semi log canonical singularities are simply connected which confirms a conjecture of the first author

constructive proof of localization in the anderson tight binding model we prove that for large disorder or near the band tails the spectrum of the anderson tight binding hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues the corresponding eigenfunctions are exponentially well localized these results hold in arbitrary dimension and with probability one in one dimension we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder our techniques extend to other physical systems which exhibit localization phenomena such as infinite systems of coupled harmonic oscillators or random schrodinger operators in the continuum

on the geometry of thin exceptional sets in manin s conjecture manin s conjecture predicts the rate of growth of rational points of a bounded height after removing those lying on an exceptional set we study whether the exceptional set in manin s conjecture is a thin set using the minimal model program and boundedness of log fano varieties

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

existence of log canonical closures let f x u be a projective morphism of normal varieties and x a dlt pair we prove that if there is an open set u 0 u such that x u u 0 has a good minimal model over u 0 and the images of all the non klt centers intersect u 0 then x has a good minimal model over u as consequences we show the existence of log canonical compactifications for open log canonical pairs and the fact that the moduli functor of stable schemes satisfies the valuative criterion for properness

geometric invariants of the quantum hall effect we study the two dimensional hall effect with a random potential the hall conductivity is identified as a geometric invariant associated with an algebra of observables using the pairing betweenk theory and cyclic cohomology theory we identify this geometric invariant with a topological index thereby giving the hall conductivity a new interpretation

intermixture of extended edge and localized bulk energy levels in macroscopic hall systems we study the spectrum of a random schrodinger operator for an electron subjected to a magnetic field in a finite but macroscopic two dimensional system of linear dimensions equal to l the y direction is periodic and in the x direction the electron is confined by two smooth increasing boundary potentials the eigenvalues of the hamiltonian are classified according to their associated quantum mechanical diamagnetic current in the y direction here we look at an interval of energies inside the first landau band of the random operator for the infinite plane in this energy interval with large probability there exist l eigenvalues with positive or negative currents of 1 between each of these there exist l2 eigenvalues with infinitesimal current e b log l 2 we explain the relevance of this analysis of boundary diamagnetic currents to the integer quantum hall effect

varieties fibred over abelian varieties with fibres of log general type abstract let x b be a complex projective klt pair and let f x z be a surjective morphism onto a normal projective variety with maximal albanese dimension such that k x b is relatively big over z we show that such pairs have good log minimal models

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

infinite degeneracy for a landau hamiltonian with poisson impurities we consider a single band approximation to the random schrodinger operator in an external magnetic field the random potential consists of delta functions of random strengths whose positions have a poisson distribution we prove that if the magnetic field is sufficiently high compared to the density of scatterers then with probability one there exists an infinitely degenerate eigenenergy coinciding with the first landau level in the absence of a random potential

contribution of jumping numbers by exceptional divisors abstract we investigate some necessary and sufficient conditions for an exceptional divisor to contribute jumping numbers of an effective divisor on a variety of arbitrary dimension inspired by the results for curves on surfaces by smith and thompson 17 and tucker 18 in particular we construct an example of an exceptional divisor that is not contracted in the log canonical model and does not contribute any jumping numbers

varieties fibered by good minimal models let f x y be an algebraic fiber space such that the general fiber has a good minimal model we show that if f is the iitaka fibration or if f is the albanese map with relative dimension no more than three then x has a good minimal model

edge and impurity effects on quantization of hall currents we consider the edge hall conductance and show it is invariant under perturbations located in a strip along the edge decaying perturbations far from the edge are also allowed this enables us to prove for the edge conductances a general sum rule relating currents due to the presence of two different media located respectively on the left and on the right half plane as a particular interesting case we put forward a general quantization formula for the difference of edge hall conductances in semi infinite samples with and without a confining wall it implies in particular that the edge hall conductance takes its ideal quantized value under a gap condition for the bulk hamiltonian or under some localization properties for a random bulk hamiltonian provided one first regularizes the conductance we shall discuss this regularization issue our quantization formula also shows that deviations from the ideal value occurs if a semi infinite distribution of impurity potentials is repulsive enough to produce current carrying surface states on its boundary

on absence of diffusion near the bottom of the spectrum for a random schrodinger operator on l 2 bf r nu we consider a random schrodinger operator onl2 v of the form h omega delta v omega v omega x sigma chi c i x q i omega ci being a covering of v with unit cubes around the sites of v and qi i i d random variables with values in 0 1 we assume that theqi s are continuously distributed with bounded densityf q and that 0 p q0 1 2 1 then we show that an ergodic mean of the quantity dx x 2 exp ith x 2 t 1 vanishes provided ge h where is well localized around the origin andge is a positivec function with support in 0 e e e f our estimate ofe f is such that the set x v v x e f may contain with probability one an infinite cluster of cubes ci which are nearest neighbours the proof is based on the technique introduced by frohlich and spencer for the analysis of the anderson model

a canonical bundle formula a higher dimensional analogue of kodaira s canonical bundle formula is obtained as applications we prove that the log canonical ring of a klt pair with 3is finitely generated and that there exists an effectively computable natural number m such that mk x induces the iitaka fibering for every algebraic threefold x with kodaira dimension 1

on fujita s log spectrum conjecture we prove fujita s log spectrum conjecture it follows from the acc of a suitable set of pseudo effective thresholds

jordan property for cremona groups assuming the borisov alexeev borisov conjecture we prove that there is a constant j j n such that for any rationally connected variety x of dimension n and any finite subgroup g subset rm bir x there exists a normal abelian subgroup a subset g of index at most j in particular we obtain that the cremona group rm cr 3 rm bir bbb p 3 enjoys the jordan property

on invariance of plurigenera for foliations on surfaces let x be a smooth algebraic surface a foliation f on x is roughly speaking a subline bundle t f of the tangent bundle of x the dual of t f is called the canonical bundle of the foliation k f in the last few years birational methods have been successfully used in order to study foliations more precisely geometric properties of the foliation are translated into properties of the canonical bundle of the foliation one of the most important invariants describing the properties of a line bundle l is its kodaira dimension kappa l which measures the growth of the global sections of l and its tensor powers the kodaira dimension of a foliation f is defined as the kodaira dimension of its canonical bundle kappa k f in their fundamental works brunella and mcquillan give a classfication of foliations on surfaces on the model of enriques kodaira classification of surfaces the next step is the study of the behaviour of families of foliations brunella proves that for a family of foliations x t f t of dimension one on surfaces satisfying certain hypotheses of regularity the kodaira dimension of the foliation does not depend on t by analogy with siu s invariance of plurigenera it is natural to ask whether for a family of foliations x t f t the dimensions of global sections of the canonical bundle and its powers depend on t in this talk we will discuss to which extent an invariance of plurigenera for foliations is true and under which hypotheses on the family of foliations it holds

subadjunction of log canonical divisors ii we extend a subadjunction formula of log canonical divisors as in kawamata contemp math 207 1997 79 88 to the case when the codimension of the minimal center is arbitrary by using the positivity of the hodge bundles

linear response theory for magnetic schrodinger operators in disordered media abstract we justify the linear response theory for an ergodic schrodinger operator with magnetic field within the noninteracting particle approximation and derive a kubo formula for the electric conductivity tensor to achieve that we construct suitable normed spaces of measurable covariant operators where the liouville equation can be solved uniquely if the fermi level falls into a region of localization we recover the well known kubo str eda formula for the quantum hall conductivity at zero temperature