the bogomolov conjecture for totally degenerate abelian varieties we prove the bogomolov conjecture for a totally degenerate abelian variety a over a function field we adapt zhang s proof of the number field case replacing the complex analytic tools by tropical analytic geometry a key step is the tropical equidistribution theorem for a at the totally degenerate place v as a corollary we obtain finiteness of torsion points with coordinates in the maximal unramified algebraic extension over v

omega admissible theory ii deligne pairings over moduli spaces of punctured riemann surfaces in part i deligne riemann roch isometry is generalized for punctured riemann surfaces equipped with quasi hyperbolic metrics this is achieved by proving the mean value lemmas which explicitly explain how metrized deligne pairings for omega admissible metrized line bundles depend on omega in part ii we first introduce several line bundles over knudsen deligne mumford compactification of the moduli space or rather the algebraic stack of stable n pointed algebraic curves of genus g which are rather natural and include weil petersson takhtajan zograf and logarithmic mumford line bundles then we use deligne riemann roch isomorphism and its metrized version proved in part i to establish some fundamental relations among these line bundles finally we compute first chern forms of the metrized weil petersson takhtajan zograf and logarithmic mumford line bundles by using results of wolpert and takhtajan zograf and show that the so called takhtajan zograf metric on the moduli space is algebraic

riemann roch and abel jacobi theory on a finite graph abstract it is well known that a finite graph can be viewed in many respects as a discrete analogue of a riemann surface in this paper we pursue this analogy further in the context of linear equivalence of divisors in particular we formulate and prove a graph theoretic analogue of the classical riemann roch theorem we also prove several results analogous to classical facts about riemann surfaces concerning the abel jacobi map from a graph to its jacobian as an application of our results we characterize the existence or non existence of a winning strategy for a certain chip firing game played on the vertices of a graph

comparison of two notions of subharmonicity on non archimedean curves we show that a continuous function on the analytification of a smooth proper algebraic curve over a non archimedean field is subharmonic in the sense of thuillier if and only if it is psh i e subharmonic in the sense of chambert loir and ducros this equivalence implies that the property psh for continuous functions is stable under pullback with respect to morphisms of curves furthermore we prove an analogue of the monotone regularization theorem on the analytification of mathbb p 1 and mumford curves using this equivalence

moduli spaces of framed instanton bundles on cp 3 and twistor sections of moduli spaces of instantons on c 2 abstract we show that the moduli space m of framed instanton bundles on cp 3 is isomorphic as a complex manifold to a subvariety in the moduli of rational curves of the twistor space of the moduli space of framed instantons on r 4 we then use this characterization to prove that m is equipped with a torsion free affine connection with holonomy in sp 2 n c

semistable sheaves on projective varieties and their restriction to curves let x be a nonsingular projective variety of dimension n over an algebraically closed field k let h be a very ample line bundle on x if v is a torsion free coherent sheaf on x we define deg v to be cl v c h and v deg v rk v we call v sernistable resp stable if for all proper subsheaves w of v we have w v resp w v cf 7 14 in this paper we prove that if v is semistable on x then its restriction to a general complete intersection curve of sufficiently high degree is semistable theorem 6 1 to give an idea of the proof assume x is a surface and v a vector bundle of rank 2 the restriction of v to a general curve c of degree m is not semistable if and only if it is not semistable on the generic curve ym defined over the function field of iph h let s be the line bundle on y contradicting the semistability of vi y cf sects 4 1 and 4 2 first we show that l extends uniquely to a line bundle l m on x proposition 2 1 if we can get l as a subbundle of v we are through for then l would contradict the semistability of v so we would like the restriction map h hom lm v h hom l m v to be surjective now for fixed l it follows from the lemma of enriques severi proposition 3 2 6 corollary 7 8 that h horn l v h horn l v is surjective for large m therefore it is enough if the l remain the same line bundle l for infinitely many m to prove that l l we construct a degenerating family of curves d f s x x s 3 d p x such that the generic fibre is a curve ct 1 of degree 2 1 and the special fibre is a reduced curve with two nonsingular components ci of degree 2 cf sect 5 let m denote 2 extending the subbundle lt ct to a subsheaf of p v on d and restricting the extension to ci gives a lower bound for the maximal degree of a line subbundle of v cim in terms of that for v ctm i proposition 4 3 this implies that degl is bounded lemma 6 5 1 so that for an infinite subsequence of m degl is constant if degltm r deglt by refining the above argument with the degenerating family one can prove that lt ci

imaging geometric graphs using internal measurements abstract this article presents two topologically motivated algorithms for reconstructing a metric graph from solutions to a wave equation on it our algorithms rely on narrowband and visibility measurements and are therefore of considerable value to urban sensing applications this is an advancement over the wideband methods traditionally used to study metric graphs our methodology exposes and separates the impact of the graph topology and geometry and is driven by a complete cohomological characterization of the space of wave equation solutions

mesures et equidistribution sur les espaces de berkovich the proof by ullmo and zhang of bogomolov s conjecture about points of small height in abelian varieties made a crucial use of an equidistribution property for small points in the associated complex abelian variety we study the analogous equidistribution property at p adic places our results can be conveniently stated within the framework of the analytic spaces defined by berkovich the first one is valid in any dimension but is restricted to algebraic metrics the second one is valid for curves but allows for more general metrics in particular to the normalized heights with respect to dynamical systems

severi inequality for varieties of maximal albanese dimension in this paper we prove the general severi inequality for varieties of maximal albanese dimension suppose that x is an n dimensional projective normal minimal and mathbb q gorenstein variety of general type in characteristic zero if x is of maximal albanese dimension then k n x ge 2 n chi omega x

second variation of zhang s λ invariant on the moduli space of curves we compute the second variation of the invariant recently introduced by s zhang on the complex moduli space mg of curves of genus g 2 using work of n kawazumi as a result we prove that 8g 4 is equal up to a constant to the invariant introduced some time ago by r hain and d reed we deduce some consequences for example we calculate the invariant for each hyperelliptic curve expressing it in terms of the petersson norm of the discriminant modular form 1 introduction recently independently s zhang 27 and n kawazumi 16 introduced a new interesting real valued function on the moduli space mg of complex curves of genus g 2 its value at a curve x m g is given as follows let h 0 x x be the space of holomorphic differentials on x equipped with the

positive line bundles on arithmetic varieties p 5 u integers is defined to count the volume of the lattice i l n of integral sections in the space j7 lon of real sections with supremum norm we want to prove that the leading term of this hilbert function as n f oo is given in terms of the height of x in as 1 this is known as a theorem of gillet and soule gs2 if x has a regular generic fiber beside this known result our proof uses hironaka s theorem on resolutions of singularities and minkowski s theorem on successive minima by hironaka s theorem we may construct

harder narasimhan reduction for principal bundles over a compact kahler manifold generalizing the harder narasimhan filtration of a vector bundle it is shown that a principal g bundle e g over a compact kahler manifold admits a canonical reduction of its structure group to a parabolic subgroup of g here g is a complex connected reductive algebraic group in the special case where g rm gl n mathbb c this reduction is the harder narasimhan filtration of the vector bundle associated to e g by the standard representation of rm gl n mathbb c the reduction of e g in question is determined by two conditions if p denotes the parabolic subgroup l its levi factor and e p subset e g the canonical reduction then the first condition says that the principal l bundle obtained by extending the structure group of the p bundle e p using the natural projection of p to l is semistable denoting by mathfrak u the lie algebra of the unipotent radical of p the second condition says that for any irreducible p module v occurring in mathfrak u mathfrak u mathfrak u the associated vector bundle e p times p v is of positive degree here mathfrak u mathfrak u mathfrak u is considered as a p module using the adjoint action the second condition has an equivalent reformulation which says that for any nontrivial character chi of p which can be expressed as a nonnegative integral combination of simple roots with respect to any borel subgroup contained in p the line bundle associated to e p for chi is of positive degree the equivalence of these two conditions is a consequence of a representation theoretic result proved here

symmetric roots and admissible pairing using the discriminant modular form and the noether formula it is possible to write the admissible self intersection of the relative dualising sheaf of a semistable hyperelliptic curve over a number field or function field as a sum over all places of a certain adelic invariant we provide a simple geometric interpretation for this invariant based on the arithmetic of symmetric roots we propose the conjecture that the invariant coincides with the invariant introduced in a recent paper by s w zhang this conjecture is true in the genus 2 case and we obtain a new proof of the bogomolov conjecture for curves of genus 2 over number fields

fibered algebraic surfaces with low slope the background of this paper is the following conjecture on complex surfaces of general type let s be a smooth projective minimal surface of general type over ti with c 2 throughout the paper

accelerated landweber iterations for the solution of ill posed equations in this paper the potentials of so calledlinear semiiterative methods are considered for the approximate solution of linear ill posed problems and ill conditioned matrix equations several efficient two step methods are presented most of which have been introduced earlier in the literature stipulating certain conditions concerning the smoothness of the solution a notion of optimal speed of convergence may be formulated various direct and converse results are derived to illustrate the properties of this concept if the problem s right hand side data are contaminated by noise semiiterative methods may be used asregularization methods assuming optimal rate of convergence of the iteration for the unperturbed problem the regularized approximations will be of order optimal accuracy to derive these results specific properties of polynomials are used in connection with the basic theory of solving ill posed problems rather recent results onfast decreasing polynomials are applied to answer an open question of brakhage numerical examples are given including a comparison to the method of conjugate gradients

towards the ample cone of mg n in this paper we study the ample cone of the moduli space m g n of stable n pointed curves of genus g our motivating conjecture is that a divisor on m g n is ample iff it has positive intersection with all 1 dimensional strata the components of the locus of curves with at least 3g n 2 nodes this translates into a simple conjectural description of the cone by linear inequalities and as all the 1 strata are rational includes the conjecture that the mori cone is polyhedral and generated by rational curves our main result is that the conjecture holds iff it holds for g 0 more precisely there is a natural finite map r m0 2g n m g n whose image is the locus rg n of curves with all components rational any 1 strata either lies in rg n or is numerically equivalent to a family e of elliptic tails and we show that a divisor d is nef iff d e 0 and r d is nef we also give results on contractions i e morphisms with connected fibers to projective varieties of m g n for g 1 showing that any fibration factors through a tautological one given by forgetting points and that the exceptional locus of any birational contraction is contained in the boundary

zhang s conjecture and the effective bogomolov conjecture over function fields we prove the effective bogomolov conjecture and so the bogomolov conjecture over a function field of characteristic 0 by proving zhang s conjecture about certain invariants of metrized graphs in the function field case these conjectures were previously known to be true only for curves of good reduction for curves of genus at most 4 and a few other special cases we also either verify or improve the previous results we relate the invariants involved in zhang s conjecture to the tau constant of metrized graphs then we use and extend our previous results on the tau constant by proving another conjecture of zhang we obtain a new proof of the slope inequality for faltings heights on moduli space of curves

points de petite hauteur sur les courbes modulaires x0 n let n be a square free integer prime to 6 let the imbeding of x 0 n in its jacobian relative to the point we show that the set is finite and that is infinite this explicit form of the bogomolov conjecture is obtained by an estimation of the self intersection of the dualizing sheaf in the sense of arakelov theory of modular curves this result is obtained by estimating several quantities attached to the arakelov metric on x 0 n starting with petersson s trace formula

stability of special instanton bundles on ℙ 2n 1 we prove that the special instanton bundles of rank 2n on p2n 1 c with a symplectic structure studied by spindler and trautmann are stable in the sense of mumford takemoto this implies that the generic special instanton bundle is stable moreover all instanton bundles on p5 are stable we get also the stability of other related vector bundles

relative noether inequality on fibered surfaces abstract we prove effective upper bounds on the global sections of nef line bundles of small generic degree over a fibered surface over a field of any characteristic it can be viewed as a relative version of the classical noether inequality for surfaces as a consequence we give a new proof of the slope inequality for fibered surface without using any stability method the treatment is essentially different from those of xiao cornalba harris and moriwaki we also study the geography problem of surfaces in positive characteristics and show that the severi inequality is true for surfaces of general type in positive characteristic whose albanese map is generically finite moreover the geography of surfaces with albanese fibrations is studied

uniform convergence of regularization methods for linear ill posed problems abstract after a general discussion about convergence and convergence rates for regularization methods in banach spaces we present a general method that can be used to modify regularization methods in l2 in such a way that uniform convergence which is often preferred in concrete applications to just l2 convergence is obtained we prove results about convergence rates in the uniform norm and discuss questions of parameter choice the theoretical results will be supported by numerical examples which indicate that although the results are asymptotic in character they are of some relevance also for actual computations

the theorem of grauert mulich spindler the boundedness of semi stable sheaves on p is a simple corollary to the above see 13 if one looks into the proof of this theorem by spindler and the techniques by van de ven 14 and barth 1 he can generalize immediately the theorem in the following way let e be a torsion free coherent sheaf on a non singular projective variety x over an algebraically closed field k and ox 1 a very ample invertible sheaf on x if g l y n dimx are sufficiently general member of a very ample linear subsystem l of iox 1 l then y yl i1 is a non singular subvariety of codimension n in x and elf is torsion free

on abelian automorphism group of a surface of general type lets be a minimal surface of general type over k the canonical divisor ofs letg be an abelian automorphism group ofs ifk 2 140 then the order ofg is at most 52k 2 32 examples are also provided with an abelian automorphism group of order 12k 2 96 the automorphism groups for a complex algebraic curve of genusg 2 have been thoroughly studied by many authors including many recent ones in particular various bounds have been established for the order of such groups for example the order of the total automorphism group is 84 g 1 hu that of an abelian subgroup is 4g 4 n while the order of any automorphism is 4g 2 w see also ha it is an intriguing problem to generalise these bounds to higher dimensions for example for surfaces of general type it is well known that the automorphism groups are finite and the bound of the orders of these groups depends only on the chern numbers of the surface a in the attempts to such generalisations the order of abelian subgroups has a special importance due to jordan s theorem on group representations and its followers a bound on the order of abelian subgroups induces a bound on that of the whole automorphism group although bounds thus obtained are generally far from satisfactory in h s it is shown that for surfaces of general type the order of such an abelian subgroup is bounded by the square of the chern numbers times a constant the purpose of this article is to give a further analysis to the abelian case for surfaces of general type in proving that the order is bounded linearly by the chern numbers of the surface in good analogy with the case of curves more precisely our main result is the following

canonical measures on metric graphs and a kazhdan s theorem we extend the notion of canonical measures to all possibly non compact metric graphs this will allow us to introduce a notion of hyperbolic measures on universal covers of metric graphs kazhdan s theorem for riemann surfaces describes the limiting behavior of canonical arakelov measures on finite covers in relation to the hyperbolic measure we will prove a generalized version of this theorem for metric graphs allowing any infinite galois cover to replace the universal cover we will show all such limiting measures satisfy a version of gauss bonnet formula which using the theory of von neumann dimensions can be interpreted as a trace formula in the special case where the infinite cover is the universal cover we will provide explicit methods to compute the corresponding limiting hyperbolic measure our ideas are motivated by non archimedean analytic and tropical geometry

lower bounds on the arithmetic self intersection number of the relative dualizing sheaf on arithmetic surfaces we give an explicitly computable lower bound for the arithmetic self intersection number of the dualizing sheaf on a large class of arithmetic surfaces if some technical conditions are satisfied then this lower bound is positive in particular these technical conditions are always satisfied for minimal arithmetic surfaces with simple multiplicities and at least one reducible fiber but we have also used our techniques to obtain lower bounds for some arithmetic surfaces with non reduced fibers

calculus on arithmetic surfaces in a2 and a3 arakelov introduces an intersection calculus for arithmetic surfaces that is for stable models of curves over a number field in this paper we intend to show that his intersection product has a lot of useful properties more precisely we show that the following properties from the theory of algebraic surfaces have an analogue in our situation

restriction theorems for principal bundles let e be a semistable or stable principal bundle over a smooth complex projective variety x and let d x be a complete intersection we study the semi stability of the restriction e d some of the results known for vector bundles such as grauert mulich flenner and mehta ramanathan theorems are generalized to principal bundles

linear series on metrized complexes of algebraic curves a metrized complex of algebraic curves over an algebraically closed field kappa is roughly speaking a finite metric graph gamma together with a collection of marked complete nonsingular algebraic curves c v over kappa one for each vertex v of gamma the marked points on c v are in bijection with the edges of gamma incident to v we define linear equivalence of divisors and establish a riemann roch theorem for metrized complexes of curves which combines the classical riemann roch theorem over kappa with its graph theoretic and tropical analogues from amini and caporaso adv math 240 1 23 2013 baker and norine adv math 215 2 766 788 2007 gathmann and kerber math z 259 1 217 230 2008 and mikhalkin and zharkov tropical curves their jacobians and theta functions contemporary mathematics 203 231 2007 providing a common generalization of all of these results for a complete nonsingular curve x defined over a non archimedean field mathbb k together with a strongly semistable model mathfrak x for x over the valuation ring r of mathbb k we define a corresponding metrized complex mathfrak c mathfrak x of curves over the residue field kappa of mathbb k and a canonical specialization map tau mathfrak c mathfrak x from divisors on x to divisors on mathfrak c mathfrak x which preserves degrees and linear equivalence we then establish generalizations of the specialization lemma from baker algebra number theory 2 6 613 653 2008 and its weighted graph analogue from amini and caporaso adv math 240 1 23 2013 showing that the rank of a divisor cannot go down under specialization from x to mathfrak c mathfrak x as an application we establish a concrete link between specialization of divisors from curves to metrized complexes and the theory of limit linear series due to eisenbud and harris invent math 85 337 371 1986 using this link we formulate a generalization of the notion of limit linear series to curves which are not necessarily of compact type and prove among other things that any degeneration of a mathfrak g r d in a regular family of semistable curves is a limit mathfrak g r d on the special fiber

log minimal model program for the moduli space of stable curves the first flip we give a geometric invariant theory git construction of the log canonical model m g a of the pairs m g ad for a 7 10 7 10 for small q we show that m g 7 10 is isomorphic to the git quotient of the chow variety of bicanonical curves m g 7 10 is isomorphic to the git quotient of the asymptotically linearized hilbert scheme of bicanonical curves in each case we completely classify the semi stable curves and their orbit closures chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic tails hilbert semistable curves satisfy further conditions e g they do not contain elliptic chains we show that there is a small contraction m g 7 10 m g 7 10 that contracts the locus of elliptic bridges moreover by using the git interpretation of the log canonical models we construct a small contraction m g 7 10 m g 7 10 that is the mori flip of

arithmetic height functions over finitely generated fields in this paper we propose a new height function for a variety defined over a finitely generated field over for this height function we prove northcott s theorem and bogomolov s conjecture so that we can recover the original raynaud s theorem manin mumford s conjecture

on the discrepancy principle for iterative and parametric methods to solve linear ill posed equations for the numerical solution of non necessarily well posed linear equations in banach spaces we consider a class of iterative methods which contains well known methods like the richardson iteration if the associated resolvent operator fulfils a condition with respect to a sector it is the purpose of this paper to show that for given noisy right hand side the discrepancy principle being a stopping rule for the iteration methods belonging to the mentioned class defines a regularization method and convergence rates are proved under additional smoothness conditions on the initial error this extends similar results obtained for positive semidefinite problems in hilbert spaces then we consider a class of parametric methods which under the same resolvent condition contains the method of the abstract cauchy problem and under a weaker resolvent condition the iterated method of lavrentiev a modified discrepancy principle is formulated for them and finally numerical illustrations are presented

a scheme for stable numerical differentiation a method for stable numerical differentiation of noisy data is proposed the method requires solving a volterra integral equation of the second kind this equation is solved analytically in the examples considered its solution is computed analytically some numerical results of its application are presented these examples show that the proposed method for stable numerical differentiation is numerically more efficient than some other methods in particular than variational regularization

the uniformization of the moduli space of principally polarized abelian 6 folds author s alexeev valery donagi ron farkas gavril izadi elham ortega angela abstract abstractstarting from a beautiful idea of kanev we construct a uniformization of the moduli space mathcal a 6 of principally polarized abelian 6 folds in terms of curves and monodromy data we show that the general principally polarized abelian variety of dimension 6 is a prym tyurin variety corresponding to a degree 27 cover of the projective line having monodromy the weyl group of the e 6 lattice along the way we establish numerous facts concerning the geometry of the hurwitz space of such e 6 covers including 1 a proof that the canonical class of the hurwitz space is big 2 a concrete geometric description of the hodge hurwitz eigenbundles with respect to the kanev correspondence and 3 a description of the ramification divisor of the prym tyurin map from the hurwitz space to mathcal a 6 in the terms of syzygies of the abel prym tyurin curve

bosonization on higher genus riemann surfaces we prove the equivalence between certain fermionic and bosonic theories in two spacetime dimensions the theories have fields of arbitrary spin on compact surfaces with any number of handles global considerations require that we add new topological terms to the bosonic action the proof that our prescription is correct relies on methods of complex algebraic geometry

algebras with cycle finite galois coverings we prove that the finite dimensional algebras over an algebraically closed field which admit cycle finite galois coverings with torsion free galois groups are of tame representation type and derive some consequences

effective bounds of linear series on algebraic varieties and arithmetic varieties in this paper we prove effective upper bounds for effective sections of line bundles on projective varieties and hermitian line bundles on arithmetic varieties in terms of the volumes they are effective versions of the hilbert samuel formula and the arithmetic hilbert samuel formula the treatments are high dimensional generalizations of duke math j 162 2013 1723 1770 and relative noether inequality on fibered surfaces preprint 2013 similar results are obtained independently by huayi chen majorations explicites de fonctions de hilbert samuel geometrique et arithmetique preprint 2014 with less explicit error terms

a riemann roch theorem in tropical geometry recently baker and norine have proven a riemann roch theorem for finite graphs we extend their results to metric graphs and thus establish a riemann roch theorem for divisors on abstract tropical curves

modular compactifications of the space of marked trigonal curves abstract we construct a sequence of modular compactifications of the space of marked trigonal curves by allowing the branch points to coincide to a given extent beginning with the standard admissible cover compactification the sequence first proceeds through contractions of the boundary divisors and then through flips of the so called maroni strata culminating in a fano model for even genera and a fano fibration for odd genera while the sequence of divisorial contractions arises from a more general construction the sequence of flips uses the particular geometry of triple covers we explicitly describe the mori chamber decomposition given by this sequence of flips

on the regularizing properties of the gmres method the gmres method is a popular iterative method for the solution of large linear systems of equations with a nonsymmetric nonsingular matrix however little is known about the behavior of this method when it is applied to the solution of nonsymmetric linear ill posed problems with a right hand side that is contaminated by errors we show that when the associated error free right hand side lies in a finite dimensional krylov subspace the gmres method is a regularization method the iterations are terminated by a stopping rule based on the discrepancy principle

relative bogomolov s inequality and the cone of positive divisors on the moduli space of stable curves an interesting point of the above theorem is that even if the weak positivity of disx y e at y is a global property on y it can be derived from the local assumption the goodness of xg and the semistability of eg this gives a great advantage to our applications in order to understand the intuition underlying the theorem let us consider a simulated case namely we suppose that f x y is a smooth surface fibred over a curve and the fiber is general bogomolov s instability theorem 1 says that if e is semistable then the codimension two cycle 2rc2 e r 1 cl e 2 has nonnegative degree so if we push it down to a codimension one cycle on y then one can rephrase bogomolov s theorem as saying that the semistability of ey implies the non negativity of disx y e

on a minimax equality for seminorms abstract for seminorms 0 and 1 defined on a real or complex vector space x and induced by positive semidefinite hermitian forms we present two different proofs of the equality sup x 1 max 1 min 0 t 1 sup x 1 t 1 where max max 0 1 and 2 t 1 t 2 0 t 2 1 t 0 1 during the course of the first proof results on the geometry of the joint numerical range of two real valued quadratic forms are given for spaces equipped with a semidefinite hermitian form which may be of independent interest in the second proof using a more direct approach the minimax equality is first proved for finite dimensional x and norms generated by inner products and this result is then extended to the general case