on the homotopy types of compact kahler and complex projective manifolds the celebrated kodaira theorem 6 says that a compact complex manifold is projective if and only if it admits a kahler form whose cohomology class is integral this suggests that kahler geometry is an extension of projective geometry obtained by relaxing the integrality condition on a kahler class this point of view together with the many restrictive conditions on the topology of kahler manifolds provided by hodge theory the strongest one being the formality theorem 4 would indicate that compact kahler manifolds and complex projective ones cannot be distinguished by topological invariants this is supported by the results known for kahler surfaces for which a much stronger statement is known as a consequence of kodaira s classification recall first that two compact complex manifolds x and x are said to be deformation equivalent if there exist a family x b where b is a connected analytic space and is smooth and proper and two points b b b such that xb x xb x

the borel conjecture for hyperbolic and cat 0 groups we prove the borel conjecture for a class of groups containing word hyperbolic groups and groups acting properly isometrically and cocompactly on a finite dimensional cat 0 space

minimal models for kahler threefolds let x be a compact kahler threefold that is not uniruled we prove that x has a minimal model

the noether inequality for algebraic 3 folds we establish the noether inequality for projective 3 folds and specifically we prove that the inequality vol x 43pg x 103 holds for all projective 3 folds x of general type with either pg x 4 or pg x 21 where pg x is the geometric genus and vol x is the canonical volume this inequality is optimal due to known examples found by kobayashi in 1992

the moduli space of smooth ample hypersurfaces in abelian varieties abstract we give a characterization of smooth ample hypersurfaces in abelian varieties and also describe the connected component of the moduli space containing such hypersurfaces of a given polarization type we show that this component is irreducible and that it consists of these hypersurfaces plus the smooth iterated univariate coverings of normal type of the same polarization type the above manifolds yield also a connected component of the open set of teichmuller space consisting of kahler complex structures

locality in distributed graph algorithms this paper concerns a number of algorithmic problems on graphs and how they may be solved in a distributed fashion the computational model is such that each node of the graph is occupied by a processor which has its own id processors are restricted to collecting data from others which are at a distance at most t away from them in t time units but are otherwise computationally unbounded this model focuses on the issue of locality in distributed processing namely to what extent a global solution to a computational problem can be obtained from locally available data three results are proved within this model a 3 coloring of an n cycle requires time omega log n this bound is tight by previous work of cole and vishkin any algorithm for coloring the d regular tree of radius r which runs for time at most 2r 3 requires at least omega sqrt d colors in an n vertex graph of largest degree delta an o delta 2 coloring may be found in time o log n

global spectra polytopes and stacky invariants given a convex polytope we define its geometric spectrum a stacky version of batyrev s stringy e functions and we prove a stacky version of a formula of libgober and wood about the e polynomial of a smooth projective variety as an application we get a closed formula for the variance of the geometric spectrum and a noether s formula for two dimensional fano polytopes polytopes whose vertices are primitive lattice points a fano polytope is not necessarily smooth we also show that this geometric spectrum is equal to the algebraic spectrum the spectrum at infinity of a tame laurent polynomial whose newton polytope is the polytope alluded to this gives an explanation and some positive answers to hertling s conjecture about the variance of the spectrum of tame regular functions

on the diffeomorphism types of certain algebraic surfaces i l invariant de donaldson la geometrie de certaines surfaces rationnelles surfaces dolgachev generalites sur des formes de type 1 n la classe canonique d une chambre et un coin le cone de kahler d une bonne surface generique une etude de p cellules les super p cellules

improved distributed algorithms for coloring interval graphs with application to multicoloring trees we give a distributed 1 epsilon approximation algorithm for the minimum vertex coloring problem on interval graphs which runs in the mathcal local model and operates in mathrm o frac 1 epsilon log n rounds if nodes are aware of their interval representations then the algorithm can be adapted to the mathcal congest model using the same number of rounds prior to this work only constant factor approximations using mathrm o log n rounds were known 12 linial s ring coloring lower bound implies that the dependency on log n cannot be improved we further prove that the dependency on frac 1 epsilon is also optimal

lessons from the congested clique applied to mapreduce the main result of this paper is a simulation algorithm which under quite general constraints transforms algorithms running in the congested clique model into algorithms running in the mapreduce model as a case study of applying this simulation algorithm we first present a distributed o coloring algorithm running on the congested clique in o 1 rounds if log 2 n and o log log log n rounds otherwise applying the simulation theorem to this congested clique o coloring algorithm yields an o 1 round o coloring algorithm in the mapreduce model we apply our simulation algorithm to other congested clique algorithms including the 2 ruling set and metric facility location algorithms of berns et al icalp 2012 our simulation algorithm illustrates a natural correspondence between per node bandwidth in the congested clique model and memory per machine in the mapreduce model in the congested clique and more generally any network in the congest model the major impediment to constructing fast algorithms is the o log n restriction on message sizes similarly in the mapreduce model the combined restrictions on memory per machine and total system memory have a dominant effect on algorithm design in showing a fairly general simulation algorithm we highlight the similarities and differences between these models

new approximation algorithms for graph coloring the problem of coloring a graph with the minimum number of colorsis well known to be np hard even restricted to k colorable graphs for constant k 3 this paper explores theapproximation problem of coloring k colorable graphs with as fewadditional colors as possible in polynomial time with special focus onthe case of k 3 the previous best upper bound on the number of colors needed forcoloring 3 colorable n vertex graphsin polynomial time was o n log n colors by berger and rompel improving a bound of o n colors by wigderson this paper presents an algorithmto color any 3 colorable graph with o n3 8 polylog n colors thus breaking an o n 1 2 o 1 barrier the algorithm given here is based on examiningsecond order neighborhoods of vertices rather than just immediateneighborhoods of vertices as in previous approaches we extend ourresults to improve the worst case bounds for coloring k colorable graphs for constant k 3 as well

invariance of plurigenera in this paper we give a proof of the following long conjectured result on the invariance of the plurigenera main theorem let p x d be a smooth projective family of compact complex manifolds parametrized by the open unit 1 disk d assume that the bers xt py1 t t 2 d are of general type then for every positive integer m the plurigenus dimc c xt mkxt is independent of t 2 d where kxt is the canonical line bundle of xt

elliptic genera of toric varieties and applications to mirror symmetry the paper contains a proof that elliptic genus of a calabi yau manifold is a jacobi form finds in which dimensions the elliptic genus is determined by the hodge numbers and shows that elliptic genera of a calabi yau hypersurface in a toric variety and its mirror coincide up to sign the proof of the mirror property is based on the extension of elliptic genus to calabi yau hypersurfaces in toric varieties with gorenstein singularities

12 24 and beyond abstract we generalize the well known 12 and 24 theorems for reflexive polytopes of dimension 2 and 3 to any smooth reflexive polytope our methods apply to a wider category of objects here called reflexive gkm graphs that are associated with certain monotone symplectic manifolds which do not necessarily admit a toric action as an application we provide bounds on the betti numbers for certain monotone hamiltonian spaces which depend on the minimal chern number of the manifold

the genus and a regularization of an s1 equivariant euler class we show that a new multiplicative genus in the sense of hirzebruch can be obtained by generalizing a calculation due to atiyah and witten we introduce this as the genus compute its value for some examples and highlight some of its interesting properties we also indicate a connection with the study of multiple zeta values which gives an algebraic interpretation for our proposed regularization procedure

convergence of the forward euler method for nonconvex differential inclusions the convergence of reachable sets for nonconvex differential inclusions is considered when the right hand side in the differential inclusion is a compact valued lipschitz continuous set valued function it is shown that the convergence in hausdorff distance of reachable sets for a forward euler discretization is linear in the time step

a fast network decomposition algorithm and its applications to constant time distributed computation abstract a partition c 1 c 2 c q of g v e into clusters of strong respectively weak diameter d such that the supergraph obtained by contracting each c i is l colorable is called a strong resp weak d l network decomposition network decompositions were introduced in a seminal paper by awerbuch goldberg luby and plotkin in 1989 awerbuch et al showed that strong d l network decompositions with d l exp o log n log log n can be computed in distributed deterministic time o d even more importantly they demonstrated that network decompositions can be used for a great variety of applications in the message passing model of distributed computing the result of awerbuch et al was improved by panconesi and srinivasan in 1992 in the latter result d l exp o log n and the running time is o d as well in another remarkable breakthrough linial and saks in 1992 showed that weak o log n o log n network decompositions can be computed in distributed randomized time o log 2 n much more recently barenboim 2012 devised a distributed randomized constant time algorithm for computing strong network decompositions with d o 1 however the parameter l in his result is o n 1 2 in this paper we drastically improve the result of barenboim and devise a distributed randomized constant time algorithm for computing strong o 1 o n network decompositions as a corollary we derive a constant time randomized o n approximation algorithm for the distributed minimum coloring problem improving the previously best known o n 1 2 approximation guarantee we also derive other improved distributed algorithms for a variety of problems most notably for the extremely well studied distributed minimum dominating set problem currently there is no known deterministic polylogarithmic time algorithm we devise a deterministic polylogarithmic time approximation algorithm for this problem addressing an open problem of lenzen and wattenhofer 2010

chern numbers for singular varieties and elliptic homology a fundamental goal of algebraic geometry is to do for singular varieties whatever we can do for smooth ones intersection homology for example directly produces groups associated to any variety which have almost all the properties of the usual homology groups of a smooth variety minimal model theory suggests the possibility of working more indirectly by relating any singular variety to a variety which is smooth or nearly so here we use ideas from minimal model theory to define some characteristic numbers for singular varieties generalizing the chern numbers of a smooth variety this was suggested by goresky and macpherson as a next natural problem after the definition of intersection homology 11 we find that only a subspace of the chern numbers can be defined for singular varieties a convenient way to describe this subspace is to say that a smooth variety has a fundamental class in complex bordism whereas a singular variety can at most have a fundamental class in a weaker homology theory elliptic homology we use this idea to give an algebro geometric definition of elliptic homology complex bordism modulo flops equals elliptic homology this paper was inspired by some questions asked by jack morava the descriptions of elliptic homology given by gerald hohn 13 were also an important influence thanks to dave bayer mike stillman john stembridge sheldon katz and stein stromme for their computer algebra programs macaulay sf and schubert which helped in guessing the right answer

the locality of distributed symmetry breaking symmetry breaking problems are among the most well studied in the field of distributed computing and yet the most fundamental questions about their complexity remain open in this article we work in the local model where the input graph and underlying distributed network are identical and study the randomized complexity of four fundamental symmetry breaking problems on graphs computing miss maximal independent sets maximal matchings vertex colorings and ruling sets a small sample of our results includes the following an mis algorithm running in o log2 2o log log n time where is the maximum degree this is the first mis algorithm to improve on the 1986 algorithms of luby and alon babai and itai when log n l l 2 log n and comes close to the log log log lower bound of kuhn moscibroda and wattenhofer a maximal matching algorithm running in o log log 4log n time this is the first significant improvement to the 1986 algorithm of israeli and itai moreover its dependence on is nearly optimal a 1 coloring algorithm requiring o log 2o log log n time improving on an o log log n time algorithm of schneider and wattenhofer a method for reducing symmetry breaking problems in low arboricity degeneracy graphs to low degree graphs roughly speaking the arboricity or degeneracy of a graph bounds the density of any subgraph corollaries of this reduction include an o log n time maximal matching algorithm for graphs with arboricity up to 2 log n and an o log 2 3n time mis algorithm for graphs with arboricity up to 2 log n 1 3 each of our algorithms is based on a simple but powerful technique for reducing a randomized symmetry breaking task to a corresponding deterministic one on a poly log n size graph

second order runge kutta approximations in control constrained optimal control in this paper we analyze second order runge kutta approximations to a nonlinear optimal control problem with control constraints if the optimal control has a derivative of bounded variation and a coercivity condition holds we show that for a special class of runge kutta schemes the error in the discrete approximating control is o h2 where h is the mesh spacing

deterministic δ 1 coloring in sublinear in δ time in static dynamic and faulty networks we study the distributed 1 vertex coloring and 2 1 edge coloring problems these problems are among the most important and intensively studied problems in distributed computing despite very intensive research in the last 30 years no deterministic algorithms for these problems with sublinear in time have been known so far moreover for more restricted scenarios and some related problems there are lower bounds of goos et al 2014 hirvonen and suomela 2012 kuhn and wattenhofer 2006 szegedy and vishwanathan 1993 the question of the possibility to devise algorithms that overcome this challenging barrier is one of the most fundamental questions in distributed symmetry breaking barenboim and elkin 2009 2011 goos et al 2014 hirvonen and suomela 2012 kuhn 2009 panconesi and rizzi 2001 in this article we settle this question for 1 vertex coloring and 2 1 edge coloring by devising deterministic algorithms that require o 3s4log log n time in the static dynamic and faulty settings the term log n is unavoidable in view of the lower bound of linial 1987 moreover for 1 o 1 vertex coloring and 2 o 1 edge coloring we devise algorithms with o s log n deterministic time this is roughly a quadratic improvement comparing to the state of the art that requires o log n time barenboim and elkin 2009 kuhn 2009 panconesi and rizzi 2001 our results are actually more general than that since they apply also to a variant of the list coloring problem that generalizes ordinary coloring our results are obtained using a novel technique for coloring partially colored graphs also known as fixing we partition the uncolored parts into a small number of subgraphs with certain helpful properties then we color these subgraphs gradually using a technique that employs constructions of polynomials in a novel way our construction is inspired by the algorithm of linial 1987 for ordinary o 2 coloring however it is a more sophisticated construction that differs from that of linial 1987 in several important respects these new insights in using systems of polynomials allow us to significantly speed up the o coloring algorithms moreover they allow us to devise algorithms with the same running time also in the more complicated settings of dynamic and faulty networks

kahler structures on spin 6 manifolds we show that many spin 6 manifolds have the homotopy type but not the homeomorphism type of a kahler manifold moreover for given betti numbers there are only finitely many deformation types and hence topological types of smooth complex projective spin threefolds of general type finally on a fixed spin 6 manifold the chern numbers take on only finitely many values on all possible kahler structures

on the discretization of switched linear systems abstract a bilinear single control system which can be viewed as a control formulation of a linear switched system is considered the control is restricted to take values either i in 0 1 switched system or ii in 0 1 relaxed system in more practical considerations the control is often allowed to change only at the points of a given time net with a step length h the paper investigates what is the approximation error in terms of the reachable set in the two cases i and ii the error estimates that follow directly from known results are of order h and h respectively in the present paper estimations of order h and h 1 5 are proved in a constructive way the second one makes use of the effect of non accumulation of errors established earlier by the second author

a simple parallel algorithm for the maximal independent set problem two basic design strategies are used to develop a very simple and fast parallel algorithms for the maximal independent set mis problem the first strategy consists of assigning identical copies o

boundedness of pluricanonical maps of varieties of general type using the techniques of 20 and 10 we prove that certain log forms may be lifted from a divisor to the ambient variety as a consequence of this result following 22 we show that for any positive integer n there exists an integer rn such that if x is a smooth projective variety of general type and dimension n then phi rk x colon x dasharrow mathbb p h 0 mathcal o x rk x is birational for all r rn

second order discrete approximation to linear differential inclusions an approximation to a linear differential inclusion by means of n stage single step discrete inclusions is presented which is of second order accuracy with respect to n approximations of this type with higher order of accuracy are shown not to exist in general the result is applied for a second order discretization of control constrained optimal control problems and approximate feedback design making use of dynamic programming

topologically invariant chern numbers of projective varieties abstract we prove that a rational linear combination of chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the euler and pontryagin numbers in dimension at least three we prove that only multiples of the top chern number which is the euler characteristic are invariant under diffeomorphisms that are not necessarily orientation preserving these results solve a long standing problem of hirzebruch s we also determine the linear combinations of chern numbers that can be bounded in terms of betti numbers

characteristic numbers of algebraic varieties a rational linear combination of chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the euler and pontryagin numbers in dimension at least 3 only multiples of the top chern number which is the euler characteristic are invariant under diffeomorphisms that are not necessarily orientation preserving in the space of chern numbers there are 2 distinguished subspaces one spanned by the euler and pontryagin numbers and the other spanned by the hirzebruch todd numbers their intersection is the span of the euler number and the signature

symmetry breaking depending on the chromatic number or the neighborhood growth we deterministically compute a d 1 coloring and a maximal independent set mis in time o d 1 2 q 1 h log n for d 1 email protected email protected 1 i h where d j is defined as the maximal number of nodes within distance j for a node and d email protected 1 our greedy coloring and mis algorithms improve the state of the art algorithms running in o d log n for a large class of graphs i e graphs of moderate neighborhood growth with h 36 we also state and analyze a randomized coloring algorithm in terms of the chromatic number the run time and the used colors our algorithm runs in time o email protected log n for email protected email protected log 1 1 l o g nn and email protected o d log 1 1 l o g nn for graphs of polylogarithmic chromatic number the analysis reveals an exponential gap compared to the fastest d 1 coloring algorithm running in time o email protected logn the algorithm works without knowledge of g and uses less than d colors i e 1 1 o g d with high probability to the best of our knowledge this is the first distributed algorithm for such general graphs taking the chromatic number g into account we also improve on the state of the art deterministic computation of 2 c ruling sets