global smoothing of calabi yau threefolds yoshinori namikawa i j h m steenbrink 2 i department of mathematics sophia university kioi cho tokyo 102 japan 2 mathematical institute university of nijmegen toernooiveld 6525 ed nijmegen the netherlands oblatum 10 x 1994 13 vi 1995 introduction friedman fr has studied the relationship between local and global deforma tions of a threefold z with isolated hypersurface singularities which admits small resolutions one of his main results is as follows let z be a moishezon threefold with only ordinary double points pl p assume that the canon ical line bundle

n 2 super riemann surfaces and algebraic geometry the geometric framework for n 2 superconformal field theories are described by studying susy2 curves a nickname for n 2 super riemann surfaces it is proved that single susy2 curves are actually split supermanifolds and their local model is a serre self dual locally free sheaf of rank two over a smooth algebraic curve superconformal structures on these sheaves are then examined by setting up deformation theory as a first step in studying moduli problems

general gauge and conditional gauge theorems general gauge and conditional gauge theorems are established for a large class of not necessarily symmetric strong markov processes including brownian motions with singular drifts and symmetric stable processes furthermore new classes of functions are introduced under which the general gauge and conditional gauge theorems hold these classes are larger than the classical kato class when the process is brownian motion in a bounded c 1 1 domain

projective embeddings of complex supermanifolds we generalize the kodaira embedding theorem and chow s theorem to the context of families of complex supermanifolds in particular we show that every family of super riemann surfaces is a family of projective superalgebraic varieties

super riemann surfaces uniformization and teichmuller theory teichmuller theory for super riemann surfaces is rigorously developed using the supermanifold theory of rogers in the case of trivial topology in the soul directions relevant for superstring applications the following results are proven the super teichmuller space is a complex super orbifold whose body is the ordinary teichmuller space of the associated riemann surfaces with spin structure for genusg 1 it has 3g 3 complex even and 2g 2 complex odd dimensions the super modular group which reduces super teichmuller space to super moduli space is the ordinary modular group there are no new discrete modular transformations in the odd directions the boundary of super teichmuller space contains not only super riemann surfaces with pinched bodies but rogers supermanifolds having nontrivial topology in the odd dimensions as well we also prove the uniformization theorem for super riemann surfaces and discuss their representation by discrete supergroups of fuchsian and schottky type and by beltrami differentials finally we present partial results for the more difficult problem of classifying super riemann surfaces of arbitrary topology

relative fatou s theorem for δ α 2 harmonic functions in bounded κ fat open sets recently it was shown in p kim fatou s theorem for censored stable processes stochastic process appl 108 1 2003 63 92 that fatou s theorem for transient censored stable processes in a bounded c1 1 open set is true here we give a probabilistic proof of relative fatou s theorem for 2 harmonic functions equivalently for symmetric stable processes in bounded fat open set where 0 2 that is if u is positive 2 harmonic function in a bounded fat open set d and h is singular positive 2 harmonic function in d then nontangential limits of u h exist almost everywhere with respect to the martin representing measure of h this extends the result of bogdan and dyda k bogdan b dyda relative fatou theorem for harmonic functions of rotation invariant stable processes in smooth domain studia math 157 1 2003 83 96 it is also shown that under the gaugeability assumption relative fatou s theorem is true for operators obtained from the generator of the killed stable process in bounded fat open set d through nonlocal feynman kac transforms as an application relative fatou s theorem for relativistic stable processes is also true if d is bounded c1 1 open set

estimates on green functions and poisson kernels for symmetric stable processes one of the most basic and most important subfamily of levy processes is symmetric stable processes a symmetric stable process x on rn is a levy process whose transition density p t x y relative to the lebesgue measure is uniquely determined by its fourier transform rn e ix p t x dx e t here must be in the interval 0 2 when 2 we get a brownian motion running with a time clock twice as fast as the standard one brownian motion plays a central role in modern probability theory and has numerous important applications in other scientific areas as well as in many other branches of mathematics thus it has been intensively studied in this paper symmetric stable processes are referred to the case when 0 2 unless otherwise specified in the last few years there has been an explosive growth in the study of physical and economic systems that can be successfully modeled with the use of stable processes stable processes are now widely used in physics operations research queuing theory mathematical finance and risk estimation in some physics literatures symmetric stable processes are called levy flights and they have been applied to a wide range of very complex physics issues such as turbulent diffusion vortex dynamics anomalous diffusion in rotating flows and molecular spectral fluctuations in mathematical finance stable processes can be used to model stock returns in incomplete market for these and more applications of stable processes please see the interesting book 14 by janicki and weron and the references therein and the recent article 15 by klafter shlesinger and zuomofen in order to make precise predictions about natural phenomena and to better cope with these

calabi yau manifolds motivations and constructions the possible ways of compacitification of thee8 e8 superstring theory to four dimensions are reviewed the phenomenological need forn 1 supersymmetry is argued on quite general grounds to favour the choice of a calabi yau manifold for the compact internal manifold the massless spectrum after compactification is derived in full detail revealing beside the usual particles others that may have great phenomenological impact the technical aspects of the construction of such manifolds are examined and the methods of calculation of the relevant topological properties are given a big family of such constructions giving rise to many new calabi yau manifolds is presented and its relevance to the search of a phenomenologically acceptable solution is discussed

line bundles on super riemann surfaces we give the elements of a theory of line bundles their classification and their connections on super riemann surfaces there are several salient departures from the classical case for example the dimension of the picard group is not constant and there is no natural hermitian form on pic furthermore the bundles with vanishing chern number aren t necessarily flat nor can every such bundle be represented by an antiholomorphic connection on the trivial bundle nevertheless the latter representation is still useful in investigating questions of holomorphic factorization we also define a subclass of all connections those which are compatible with the superconformal structure the compatibility conditions turn out to be constraints on the curvature 2 form

thickening and supersymmetric extensions of complex manifolds en partant d une variete complexe arbitraire on developpe une machinerie cohomologique pour etudier des epaississements et des extensions supersymetriques de cette variete

on a new kato class and positive solutions of dirichlet problems for the fractional laplacian in bounded domains abstract the purpose of this paper is to extend some results of the potential theory of an elliptic operator to the fractional laplacian 2 0 2 in a bounded c 1 1 domain d in r n in particular we introduce a new kato class k d and we exploit the properties of this class to study the existence of positive solutions of some dirichlet problems for the fractional laplacian

blow up boundary solutions of some semilinear fractional equations in the unit ball for 0 we are interested in blow up solutions u c b of the fractional problem in the unit ball b we distinguish particularly two orders of singularity at the boundary solutions exploding at the same rate than 2 1 denotes the euclidean distance and those higher singular than 2 1 as a consequence it will be shown that the classical keller osserman condition cannot be readopted in the fractional setting

supermanifolds from feynman graphs we generalize the computation of feynman integrals of log divergent graphs in terms of the kirchhoff polynomial to the case of graphs with both fermionic and bosonic edges to which we assign a set of ordinary and grassmann variables this procedure gives a computation of the feynman integrals in terms of a period on a supermanifold for graphs admitting a basis of the first homology satisfying a condition generalizing the log divergence in this context the analog in this setting of the graph hypersurfaces is a graph supermanifold given by the divisor of zeros and poles of the berezinian of a matrix associated with the graph inside a superprojective space we introduce a grothendieck group for supermanifolds and identify the subgroup generated by the graph supermanifolds this can be seen as a general procedure for constructing interesting classes of supermanifolds with associated periods

minimizing effective resistance of a graph the effective resistance between two nodes of a weighted graph is the electrical resistance seen between the nodes of a resistor network with branch conductances given by the edge weights the effective resistance comes up in many applications and fields in addition to electrical network analysis including for example markov chains and continuous time averaging networks in this paper we study the problem of allocating edge weights on a given graph in order to minimize the total effective resistance i e the sum of the resistances between all pairs of nodes we show that this is a convex optimization problem and can be solved efficiently either numerically or in some cases analytically we show that optimal allocation of the edge weights can reduce the total effective resistance of the graph compared to uniform weights by a factor that grows unboundedly with the size of the graph we show that among all graphs with n nodes the path has the largest value of optimal total effective resistance and the complete graph has the least

connecting moduli spaces of calabi yau threefolds we demonstrate that many families of calabi yau threefolds consist generically of small resolutions of nodal forms in other families and in fact that a large class of families is connected by this relation our result resonates with a conjecture of reid that calabi yau threefolds may have a universal moduli space even though they are of different homotopy types such ideas tie quite naturally to alluring prospects of unifying super string models

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

holomorphic coordinates for supermoduli space two dimensional supergravity provides complex coordinates for supermoduli space compatible with its natural complex structure such coordinates are useful for investigating questions of holomorphic factorization of superdeterminants they also demonstrate explicitly that the complex structure on supermoduli space is indeed integrable

a note on the global structure of supermoduli spaces we recall some deformation theory of susy curves and study obstructions to projectedness of supermoduli spaces both from a general standpoint and by means of the local coordinate charts most commonly used in the physical literature we prove that these give rise to a projected atlas for genusg 2 only

holomorphic supercurves and supersymmetric sigma models we introduce a natural generalisation of holomorphic curves to morphisms of supermanifolds referred to as holomorphic supercurves more precisely supercurves are morphisms from a riemann surface endowed with the structure of a supermanifold which is induced by a holomorphic line bundle to an ordinary almost complex manifold they are called holomorphic if a generalised cauchy riemann condition is satisfied we show by means of an action identity that holomorphic supercurves are special extrema of a supersymmetric action functional

the geometry of super riemann surfaces we define super riemann surfaces as smooth 2 2 dimensional supermanifolds equipped with a reduction of their structure group to the group of invertible upper triangular 2 2 complex matrices the integrability conditions for such a reduction turn out to be most of the torsion constraints of 2d supergravity we show that they are both necessary and sufficient for a frame to admit local superconformal coordinates the other torsion constraints are merely conditions to fix some of the gauge freedom in this description or to specify a particular connection on such a manifold analogous to the levi civita connection in riemannian geometry unlike ordinary riemann surfaces a super riemann surface cannot be regarded as having only one complex dimension nevertheless in certain important aspects super riemann surfaces behave as nicely as if they had only one dimension in particular they posses an analog widehat partial of the cauchy riemann operator on ordinary riemann surfaces a differential operator taking values in the bundle of half volume forms this operator furnishes a short resolution of the structure sheaf making possible a quillen theory of determinant line bundles finally we show that the moduli space of super riemann surfaces is embedded in the larger space of complex curves of dimension 1 1

complete csck metrics on the local models of the conifold transition in this paper we construct complete constant scalar curvature kahler csck metrics on the complement of the zero section in the total space of mathcal o 1 oplus2 over mathbb p 1 which is biholomorphic to the smooth part of the cone c 0 in mathbb c 4 defined by equation sigma i 1 4 w i 2 0 on its small resolution and its deformation we also consider complete csck metrics and find that if the csck metrics are homogeneous then they must be ricci flat

super beltrami differentials superconformal structures on a given 2 2 dim supermanifold give rise to the notion of super riemann surfaces srs s we investigate super beltrami coefficients which parametrize almost superconformal structures on the supermanifold the integrability condition of the structure reduces to a simple relation among the coefficients taking this into account we can write the super beltrami equations in a transparent form then an analysis of these equations enables us to prove the possibility of the special gauge choice of the wess zumino type for the super beltrami differentials this gauge choice simplifies the description of the deformations of srs s considerably and its existence will afford a better understanding of the structure of the super teichmuller space

green function for schrodinger operator and conditioned feynman kac gauge consider the schrodinger operator h 2 q in a bounded c1 1 domain d with q klocd d q satisfies the condition sup spec 2 q d v x y g x y e y x exp 0 t q x t d t x y d where xt is the conditioned brownian motion starting at x and ending at y with the life time t and v g can be extended to a continuous and positive function on d d specifically we give the comparison theorem there exists a constant c c d q 0 such that 1 c g x y v x y c g x y x y d another result is that v nz x z x d z d exists and represents the poisson kernel for the dirichlet problem corresponding to schrodinger operator h

the complete matter sector in a three generation compactification we consider a calabi yau compactification paradigm with three light generations and anr symmetry from a special form of the tian yau manifold we also construct a new three generation model with markedly different phenomenology thecomplete spectrum of all light matter fields is obtained in a universal way and moreover in a physically suitable basis allowing a straightforward analysis of all their couplings here we discuss all the renormalizable yukawa couplings this computation can equally well be repeated for all compactification models based on calabi yau complete intersections in products of homogeneous spaces

the moore penrose inverse of matrices with an acyclic bipartite graph abstract the moore penrose inverse of a real matrix having no square submatrix with two or more diagonals is described in terms of bipartite graphs for such a matrix the sign of every entry of the moore penrose inverse is shown to be determined uniquely by the signs of the matrix entries i e the matrix has a signed generalized inverse necessary and sufficient conditions on an acyclic bipartite graph are given so that each nonnegative matrix with this graph has a nonnegative moore penrose inverse nearly reducible matrices are proved to contain no submatrix having two or more diagonals implying that a nearly reducible matrix has a signed generalized inverse furthermore it is proved that the term rank and rank are equal for each submatrix of a nearly reducible matrix

torsion constraints in supergeometry we derive the torsion constraints for superspace versions of supergravity theories by means of the theory ofg structures we also discuss superconformal geometry and superkahler geometry

polynomial deformations and cohomology of calabi yau manifolds we study the method of polynomial deformations that is used in the physics literature to determine the hodge numbers of calabi yau manifolds as well as the related yukawa couplings we show that the argument generally presented in the literature in support of these computations is seriously misleading give a correct proof which applies to all the cases we found in the literature and present examples which show that the method is not universally valid we present a general analysis which applies to all calabi yau manifolds embedded as complete intersections in products of complex projective spaces yields sufficient conditions for the validity of the polynomial deformation method and provides an alternative computation of all the hodge numbers in many cases in which the polynomial method fails

the moore penrose inverse of the normalized graph laplacian abstract we prove a formula that relates the moore penrose inverses of two matrices a b such that a n 1 b m 1 and discuss some applications in particular to the representation of the moore penrose inverse of the normalized laplacian of a graph the laplacian matrix of an undirected graph is symmetric and is strictly related to its connectivity properties however our formula applies to asymmetric matrices so that we can generalize our results for asymmetric laplacians whose importance for the study of directed graphs is increasing

existence and estimates of solutions for singular nonlinear elliptic problems for both bounded and unbounded domain in r d 3 with smooth boundary the function 0 0 is measurable and satisfies some appropriate hypotheses detailed below the problem 1 has been studied extensively for the special nonlinearity x t p x t 0 indeed in 7 edelson proved the existence of a positive solution u c2 loc r decaying as x 2 n as x provided that p c loc r 0 0 for x 0 and satisfying the growth condition

moduli of super riemann surfaces the basic properties of super riemann surfaces are presented and their supermoduli spaces are constructed in a manner suitable for the application of algebro geometric techniques to string theory

on the semigroup nature of superconformal symmetry a semigroup of n 1 superconformal transformations is introduced and analyzed noninvertible ones can describe transitions from body to soul and form a proper ideal containing a set of nilpotent transformations the projective superspace is also considered transformations twisting the parity of a tangent space are brought in they can be a nonsuperconformal square root of noninvertible superconformal transformations and the analogs of the poincare metric and conformal invariance are suggested for them

deformations of super riemann surfaces two different approaches to kostant leites super riemann surfaces are investigated in the local approach i e glueing open superdomains by superconformal transition functions deformations of the superconformal structure are discussed on the other hand the representation of compact super riemann surfaces of genus greater than one as a fundamental domain in the poincare upper half plane provides a simple description of super laplace operators acting on automorphicp forms

martin boundary and integral representation for harmonic functions of symmetric stable processes abstract martin boundaries and integral representations of positive functions which are harmonic in a bounded domain d with respect to brownian motion are well understood unlike the brownian case there are two different kinds of harmonicity with respect to a discontinuous symmetric stable process one kind are functions harmonic in d with respect to the whole process x and the other are functions harmonic in d with respect to the process x d killed upon leaving d in this paper we show that for bounded lipschitz domains the martin boundary with respect to the killed stable process x d can be identified with the euclidean boundary we further give integral representations for both kinds of positive harmonic functions also given is the conditional gauge theorem conditioned according to martin kernels and the limiting behaviors of the h conditional stable process where h is a positive harmonic function of x d in the case when d is a bounded c 1 1 domain sharp estimate on the martin kernel of d is obtained

exact asymptotic behavior near the boundary to the solution for singular nonlinear dirichlet problems abstract we take up the existence and the exact asymptotic behavior near the boundary of the unique classical solution to a singular dirichlet problem u a x g u x u 0 in u 0 where is a c 1 1 bounded domain in r n n 2 and the function a is singular near the boundary and belongs to the kato class k

endomorphism valued cohomology and gauge neutral matter for several physically interesting calabi yau manifolds we count and parametrize gauge neutral matter particles occurring in corresponding superstring compactifications to this end we use the technique of exact and spectral sequences and then describe and discuss our results in the more familiar tensor notation

moore penrose inverse of the incidence matrix of a distance regular graph abstract let be a graph with n vertices where each edge is given an orientation and let q be the vertex edge incidence matrix we obtain a formula for the moore penrose inverse of q when is a distance regular graph the formula is illustrated by examples

calabi yau manifolds as complete intersections in products of complex projective spaces we consider constructions of manifolds withsu 3 holonomy as embedded in products of complex projective spaces by imposing certain homogeneous holomorphic constraints we prove that every such construction leads to one deformation class of manifolds withsu 3 holonomy for a subset of these manifolds we prove simple connectedness address the problem of calculating the second betti number and explicitly calculate it for a class of constructions this establishes a very wide class of manifolds withsu 3 holonomy that can give rise to yet many more constructions via dividing out the action of suitably chosen discrete groups some of the examples studied may yield phenomenologically acceptable models

betti numbers of a class of barely g2 manifolds we calculate explicitly the betti numbers of a class of barely g2 manifolds that is g2 manifolds that are realised as a product of a calabi yau manifold and a circle modulo an involution the particular class which we consider are those spaces where the calabi yau manifolds are complete intersections of hypersurfaces in products of complex projective spaces from which they inherit all their 1 1 cohomology and the involutions are free acting

stability of martin boundary under non local feynman kac perturbations recently the authors showed that the martin boundary and the minimal martin boundary for a censored or resurrected stable process y in a bounded c 1 1 open set d with 1 2 can all be identified with the euclidean boundary d of d under the gaugeability assumption we show that the martin boundary and the minimal martin boundary for the schrodinger operator obtained from y through a non local feynman kac transform can all be identified with d in other words the martin boundary and the minimal martin boundary are stable under non local feynman kac perturbations moreover an integral representation of nonnegative excessive functions for the schrodinger operator is explicitly given these results in fact hold for a large class of strong markov processes as are illustrated in the last section of this paper as an application the martin boundary for censored relativistic stable processes in bounded c 1 1 smooth open sets is studied in detail

a generalized inverse for graphs with absorption abstract we consider weighted directed graphs with a notion of absorption on the vertices related to absorbing random walks on graphs we define a generalized inverse of the graph laplacian called the absorption inverse that reflects both the graph structure as well as the absorption rates on the vertices properties of this generalized inverse are presented including a basic relationship between the absorption inverse and the group inverse of a related graph a forest theorem for interpreting the entries of the absorption inverse as well as relationships between the absorption inverse and the fundamental matrix of the absorbing random walk applications of the absorption inverse for describing the structure of graphs with absorption are given including a directed distance metric spectral partitioning algorithm and centrality measure