analytic formulas for the regulator of a number field letr rk andw wk be the regulator and the number of roots of unity in the number fieldk we determine allk for whichr w 1 8 there are nine such fields sincew 2 we have allk withr 1 4 in particular we find the first three minima ofr over allk there are three main ingredients to the proof i a new analytic formula theorem a below ii a refinement of remak s geometric approach iii a refinement of zimmert s analytic method

duality in dynamic fuzzy systems this paper shows the resolvent equation and the co balayage theorem for a dynamic fuzzy system next we define a dual system for the dynamic fuzzy system and give a duality for snell s optimal stopping problem by the dual system

fake projective planes a fake projective plane is a complex surface different from but has the same betti numbers as the complex projective plane it is a complex hyperbolic space form and has the smallest euler poincare characteristic among smooth surfaces of general type the first example was constructed by mumford later on two more examples were found by ishida and kato a fourth possible one was recently proposed by keum in a recent joint work with gopal prasad we showed that there are seventeen non empty classes of fake projective planes and there can at most be four more specific classes higher dimensional analogues and examples were also obtained the main purpose of the talk is to explain the joint work with prasad and other related results such as arithmeticity of the lattices involved obtained earlier by klingler and yeung independently

symmetry relations in exactly soluble models symmetry relations such as the star triangle or the inverse relation are very useful in determining the partition function of two dimensional exactly soluble models a common construction of the three dimensional equivalents of these symmetry relations is presented they are used to derive in a geometric way applying simultaneously to different kinds of spin models the consequent global properties i e the commutativity of the transfer matrices and the inverse functional equations on the transfer matrix and the partition function the usefulness of the inverse relation is illustrated by an application to the three dimensional ising model

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

construction of surfaces of general type from elliptic surfaces via q gorenstein smoothing we present methods to construct interesting surfaces of general type via mathbb q gorenstein smoothing of a singular surface obtained from an elliptic surface by applying our methods to special enriques surfaces we construct new examples of a minimal surface of general type with p g 0 pi 1 mathbb z 2 mathbb z and k2 4

computing cox rings we consider modifications for example blow ups of mori dream spaces and provide algorithms for investigating the effect on the cox ring e g testing finite generation or computing an explicit presentation in terms of generators and relations as a first application we compute the cox rings of all gorenstein log del pezzo surfaces of picard number one moreover we show computationally that all smooth rational surfaces of picard number at most six are mori dream surfaces and we provide explicit presentations of the cox ring for those not admitting a torus action finally we provide the cox rings of projective spaces blown up at a certain special point configurations

a vanishing result for a casson type instanton invariant casson type invariants emerging from donaldson theory over certain negative definite 4 manifolds were recently suggested by andrei teleman these are defined by a count of a zero dimensional moduli space of flat instantons motivated by the cobordism program of proving witten s conjecture we use a moduli space of pu 2 seiberg witten monopoles to exhibit an oriented one dimensional cobordism of the instanton moduli space to the empty space the casson type invariant must therefore vanish

a new family of surfaces of general type with k 2 7 and p_g 0 we construct a new family of smooth minimal surfaces of general type with k 2 7 and p g 0 we show that a surface in this family has ample canonical divisor and birational bicanonical morphism we also prove that these surfaces satisfy bloch s conjecture

hurwitz ball quotients we consider the analogue of hurwitz curves smooth projective curves c of genus g ge 2 that realize equality in the hurwitz bound mathrm aut c le 84 g 1 to smooth compact quotients s of the unit ball in mathbb c 2 when s is arithmetic we show that mathrm aut s le 288 e s where e s is the topological euler characteristic and in the case of equality show that s is a regular cover of a particular deligne mostow orbifold we conjecture that this inequality holds independent of arithmeticity and note that work of xiao makes progress on this conjecture and implies the best known lower bound for the volume of a complex hyperbolic 2 orbifold

spherical blow ups of grassmannians and mori dream spaces abstract in this paper we classify weak fano varieties that can be obtained by blowing up general points in prime fano varieties we also classify spherical blow ups of grassmannians in general points and we compute their effective cone these blow ups are in particular mori dream spaces furthermore we compute the stable base locus decomposition of the blow up of a grassmannian in one point and we show how it is determined by linear systems of hyperplanes containing the osculating spaces of the grassmannian at the blown up point and by the rational normal curves in the grassmannian passing through the blown up point

covolumes of nonuniform lattices in pu n 1 this paper studies the covolumes of nonuniform arithmetic lattices in pu n 1 we determine the smallest covolume nonuniform arithmetic lattices for each n the number of minimal covolume lattices for each n and study the growth of the minimal covolume as n varies in particular there is a unique lattice up to isomorphism in pu 9 1 of smallest euler poincare characteristic amongst all nonuniform arithmetic lattices in pu n 1 we also show that for each even n there are arbitrarily large families of nonisomorphic maximal nonuniform lattices in pu n 1 of equal covolume

study of a two dimensional fully frustrated model the authors consider a fully frustrated ising model on a square lattice depending on four parameters the partition function is shown to be equivalent to the two parameter ferromagnetic onsager partition function this result generalises a relation established by southern et al 1980 and can be checked in the particular case of the one parameter villain model in particular the residual entropy of the villain model is linked to the free energy of the onsager model at criticality the reduction from four to two parameters occurring in this mapping is studied in the light of the inverse relation satisfied by the partition function of the frustrated model

f purity and rational singularity we investigate singularities which are f pure respectively f pure type a ring r of characteristic p is f pure if for every r module m 0 m r m r is exact where 1r denotes the r algebra structure induced on r via the frobenius map if r e r and s e 1r then r s rps in 1r f pure type is defined in characteristic 0 by reducing to characteristicp it is proven that when r s i is the quotient of a regular local ring s r is f pure at the prime ideal q if and only if i pi i q p here j p denotes the ideal ap i a e j several theorems result from this criterion if f is a quasihomogeneous hypersurface having weights r1 r and an isolated singularity at the origin 1 17 1ri i implies k x1 x f has f pure type at m x1 x 2 1n i ri i implies k x1 xn f does not have f pure type at m 3 1 1ri i remains unsolved but does connect with a problem that number theorists have studied for many years this theorem parallels known results about rational singularities it is also proven that classifying f pure singularities for complete intersection ideals can be reduced to classifying such singularities for hypersurfaces and that the f pure locus in the maximal spectrum of k x1 xn i where k is a perfect field of characteristic p is zariski open an important conjecture is that r fr is f pure type should imply r is f pure type whenever r is a cohen macauley normal local ring it is proven that ext1 1r r 0 is a sufficient though not necessary condition a local ring r m of characteristic p is f injective if the frobenius map induces an injection on the local cohomology modules h r h 1r an example is constructed which is f injective but not f pure from this a counterexample to the conjecture that r fr is f pure implies r is f pure is constructed however it is not a domain much less normal moreover it does not lead to a counterexample to the characteristic 0 version of the conjecture 0 introduction let r be a ring of characteristic p and let r denote the ring r viewed as an r module via the frobenius map f r rp r is f pure if for every r module m 0 r m 4 r m is exact a notion of f pure type is then defined in characteristic 0 by reduction to characteristicp f pure rings are connected with invariant theory and appear in the proof that the ring of invariants of a linearly reductive affine linear group acting on a regular ring is cohen macaulay 3 it has also been demonstrated that f purity measures good singularities in the sense that it implies a great deal of simplification in the computation of local cohomology 1 received by the editors march 30 1982 and in revised form august 3 1982 1980 mathematics subject classification primary 13h99 1983 american mathematical society 0002 9947 82 0000 0829 05 25

vector bundles of rank 2 and linear systems on algebraic surfaces let s be a smooth complex algebraic surface and let l be a divisor on s it is well known that the linear system il ksi where ks is the canonical divisor plays an important role in understanding the geometry of s several authors see for example 8 11 studied various properties of this system our paper treats the same subject but the point of view is different from the works mentioned above the philosophy underlying our approach is almost classical points on s more generally effective 0 cycles in special position with respect to il ksi see definition below contain information about the geometry of s this point of view was recently revived in 5 also 10 in fact this paper contains the technique of going from the effective 0 cycle z in special position with respect to il ksi to the geometry of s for the sake of completeness we give a definition and state a theorem which can be found in 5 10

quotients of products of curves new surfaces with pg 0 and their fundamental groups we construct many new surfaces of general type with q pg 0 whose canonical model is the quotient of the product of two curves by the action of a finite group g constructing in this way many new interesting fundamental groups which distinguish connected components of the moduli space of surfaces of general type we indeed classify all such surfaces whose canonical model is singular the smooth case was classified in an earlier work as an important tool we prove a structure theorem giving a precise description of the fundamental group of quotients of products of curves by the action of a finite group g

global okounkov bodies for bott samelson varieties abstract we use the theory of mori dream spaces to prove that the global okounkov body of a bott samelson variety with respect to a natural flag of subvarieties is rational polyhedral as a corollary okounkov bodies of effective line bundles over schubert varieties are shown to be rational polyhedral in particular it follows that the global okounkov body of a flag variety g b is rational polyhedral as an application we show that the asymptotic behaviour of dimensions of weight spaces in section spaces of line bundles is given by the volume of polytopes

cox rings and combinatorics given a variety x with a finitely generated total coordinate ring we describe basic geometric properties of x in terms of certain combinatorial structures living in the divisor class group of x for example we describe the singularities we calculate the ample cone and we give simple fano criteria as we show by means of several examples the results allow explicit computations as immediate applications we obtain an effective version of the kleiman chevalley quasiprojectivity criterion and the following observation on surfaces a normal complete surface with finitely generated total coordinate ring is projective if and only if any two of its non factorial singularities admit a common affine neighbourhood

the inversion relation method for some two dimensional exactly solved models in lattice statistics the partition functions per site of several two dimensional models notably the eight vertex self dual potts and hard hexagon models can be easily obtained by using an inversion relation for local transfer matrices together with symmetry and analyticity properties this technique is discussed the analyticity properties compared and some equivalences and nonequivalences pointed out in particular the critical hard hexagon model is found to have the same as the self dualq state potts model withq 3 5 2 2 618 the temperley lieb equivalence between the potts and six vertex models is found to fail in certain nonphysical antiferromagnetic cases

the densest lattices in pgl3 q2 abstract we find the smallest possible covolume for lattices in pgl 3 q 2 show that there are exactly two lattices with this covolume and describe them explicitly they are commensurable and one of them appeared in mumford s construction of his fake projective plane we also discuss a new 2 adic uniformization of another fake projective plane

a construction of numerical campedelli surfaces with torsion z 6 we produce a family of numerical campedelli surfaces with z 6 torsion by constructing the canonical ring of the etale 6 to 1 cover using serial unprojection in section 2 we develop the necessary algebraic machinery section 3 contains the numerical campedelli surface construction while section 4 contains remarks and open questions

fibred surfaces varieties isogenous to a product and related moduli spaces a fibration of an algebraic surface s over a curve b with fibres of genus at least 2 has constant moduli iff it is birational to the quotient of a product of curves by the action of a finite group g a variety isogenous to a higher product is the quotient of a product of curves of genus at least 2 by the free action of a finite group theorem b gives a characterization of surfaces isogenous to a higher product in terms of the fundamental group and of the euler number theorem c classifies the groups thus occurring and shows that after fixing the group and the euler number one obtains an irreducible moduli space the result of theorem b is extended to higher dimension in theorem g thus generalizing cf also theorem h results of jost yau and mok concerning varieties whose universal cover is a polydisk theorem a shows that fibrations where the fibre genus and the genus of the base b are at least 2 are invariants of the oriented differentiable structure the main theorems d and e characterize surfaces carrying constant moduli fibrations as surfaces having a zariski open set satisfying certain topological conditions e g having the right euler number the right fundamental group and the right fundamental group at infinity 0 introduction the study of fibrations of a smooth algebraic surface s over a smooth algebraic curve b lies at the heart of the classification theory and of the geometry of algebraic surfaces our main results in this paper theo rems d e concern a topological characterization of surfaces admitting a constant moduli fibration these results are intimately related to a series of variations on the same theme also in higher dimension and to a circle of allied techniques which we present in the course of the paper the first new result we have in this direction and which is presented in section 2 after we recall several known results some classical some quite recent is the following theorem on the dif ferentiable invariance of the genus of fibres cf theorem 2 9 for a more precise statement theorem a let f s b be a fibration of the surface s over a curve b of genus b 2 and with fibre f of genus g 1 assume that s is another smooth compact kahler surface which is orientedly diffeomorphic to s then the subspace of h 1 s corresponding to f h 1 b determines a fibration f s b with

on the most algebraic k3 surfaces and the most extremal log enriques surfaces we shall characterize the unique k3 surface of discriminant 3 or 4 called the most algebraic k3 surfaces by vinberg in terms of the fixed locus of an automorphism on it based on this result we show that there is up to isomorphisms only one rational log eriques surface of type d 19 and one of type a 19

surfaces of general type with pg 0 k2 6 and non birational bicanonical map let s be a minimal surface of general type with p g 0 and k2 6 such that its bicanonical map open image in new window is not birational the map is a morphism of degree 4 onto a surface the case of deg 4 is completely classified in topology 40 5 2001 977 991 and the present paper completes the characterization of these surfaces it is proven that the degree of cannot be equal to 3 and the geometry of surfaces with deg 2 is analysed in detail the last section contains three examples of such surfaces two of which appear to be new

on images of mori dream spaces the purpose of this paper is to study the geometry of images of morphisms from mori dream spaces first we prove that a variety which admits a surjective morphism from a mori dream space is again a mori dream space secondly we introduce a natural fan structure on the effective cone of mori dream spaces we show that it encodes the information of zariski decompositions which in turn is equivalent to the information of the variation of git quotients of their cox rings finally we show that under a surjective morphism between mori dream spaces the fan of the target space coincides with the restriction of the fan of the source

on finite group actions on reductive groups and buildings let h be a connected reductive group over a non archimedean local field k and let f autk h be a finite group of order not divisible by p the residual characteristic of k let g h f be the identity component of the subgroup of h consisting of points fixed by f the main theorem of this paper asserts that the bruhat tits building b g of g can be identified with the set of f fixed points of b h several special cases of this theorem have been known previously when e k is a finite totally ramified galois extension h rese k g and f gal e k the condition p f is simply that e k is tamely ramified in this case our main theorem is a well known but unpublished theorem of g rousseau recently one of us g p found a simple proof of this theorem p when g is a classical group realized in the standard way as the identity component of the group of fixed points of an involution of a general linear group h bruhat and tits bt4 have given a description of b g as a subset of b h in particular it follows from their description that b g b h when p 2 recently this fact has been rediscovered by j kim and a moy km and independently a simple proof has been given by gy let h be the split form of spin 8 then one can choose an outer automorphism aut h of order 3 such that g h is of type g2 from a joint work gy by w t gan and one of us j y it is known that b g b h is true for arbitrary p

a limit theorem in some dynamic fuzzy systems abstract in this paper using a fuzzy relation we define a dynamic fuzzy system with a compact state space by introducing a contraction property we study the limit of a sequence of fuzzy states and obtain a theorem for the existence and uniqueness of the solution of a fuzzy relational equation as an application we deal with a dynamic fuzzy system with a terminal fuzzy gain and give some characterizations for the fuzzy expected gain a numerical example is given to comprehend our idea in this paper

markov chains with a transition possibility measure and fuzzy dynamic programming abstract this paper constructs a markoov fuzzy process which represents transitions of grades of fuzzy sets with a transition possibility measure and a general state space we analyse snell s optimal stopping problem for the process and we apply the results to solve fuzzy dynamic programming with optimal stopping times and with general state spaces and action spaces under fuzzy transitions

on the algebraic fundamental group of surfaces with k 2 3x let s be a minimal complex surface of general type with q s 0 we prove the following statements concerning the algebraic fundamental groupalg 1 s assume that k 2 s 3 s then s has an irregularcover if and only if s has a free pencil of hyperelliptic curves of genus 3 with at least 4 double fibres if k 2 s 3 and s 1 then s has no irregularcover if k 2 3 s and s does not have any irregularcover then alg s 9 if alg s 9 then k 2 s 2 s 1

arithmetic fake projective spaces and arithmetic fake grassmannians in a recent paper we have classified fake projective planes natural higher dimensional generalization of these surfaces are arithmetic fake bf p n 1 c and arithmetic fake bf gr m n in this paper we show that arithmetic fake bf p n 1 c can exist only if n 3 5 and an arithmetic fake bf gr m n can exist with n3 odd only if n 5 here we construct four distinct arithmetic fake bf p 4 c and four distinct fake arithmetic bf gr 2 5 furthermore we use certain results and computations of py to exhibit five irreducible arithmetic fake bf p 2 c times bf p 2 c all these are connected smooth complex projective shimura varieties

cox ring of the generic fiber abstract given a surjective morphism x y of normal varieties satisfying some regularity hypotheses we prove how to recover a cox ring of the generic fiber of from the cox ring of x as a corollary we show that in some cases it is also possible to recover the cox ring of a very general fiber and finally we give an application in the case of the blowing up of a toric fiber space

inverse functional relations for lattice models inverse functional equations are studied for the partition functions and for the correlation functions of various models the validity of the inverse relation for the partition function is justified with the help of approximations by quasi one dimensional models defined on strips of increasing size the possibility of determining the partition function through the use of the inverse and other symmetry relations coupled to analyticity hypotheses is briefly investigated on the two dimensional ising model with a field the group generated by the inverse relation and the spatial symmetries of the model is studied in a three dimensional case inverse relations are also exhibited firstly for two point and then for n point correlation functions they are first put into evidence by a geometric approach and then verified on a particular high temperature expansion

cox rings of surfaces and the anticanonical iitaka dimension abstract in this paper we investigate the relation between the finite generation of the cox ring r x of a smooth projective surface x and its anticanonical iitaka dimension k x

derived categories of keum s fake projective planes abstract we conjecture that derived categories of coherent sheaves on fake projective n spaces have a semi orthogonal decomposition into a collection of n 1 exceptional objects and a category with vanishing hochschild homology we prove this for fake projective planes with non abelian automorphism group such as keum s surface then by passing to equivariant categories we construct new examples of phantom categories with both hochschild homology and grothendieck group vanishing

globally f regular and log fano varieties abstract we prove that every globally f regular variety is log fano in other words if a prime characteristic variety x is globally f regular then it admits an effective q divisor such that k x is ample and x has controlled kawamata log terminal in fact globally f regular singularities a weak form of this result can be viewed as a prime characteristic analog of de fernex and hacon s new point of view on kawamata log terminal singularities in the non q gorenstein case we also prove a converse statement in characteristic zero every log fano variety has globally f regular type our techniques apply also to f split varieties which we show to satisfy a log calabi yau condition we also prove a kawamata viehweg vanishing theorem for globally f regular pairs

negative curves on algebraic surfaces we study curves of negative self intersection on algebraic surfaces our main result shows there exist smooth complex projective surfaces x related to hilbert modular surfaces such that x contains reduced irreducible curves c of arbitrarily negative self intersection c 2 previously the only known examples of surfaces for which c 2 was not bounded below were in positive characteristic and the general expectation was that no examples could arise over the complex numbers indeed we show that the idea underlying the examples in positive characteristic cannot produce examples over the complex number field and thus our complex examples require a different approach

on the birational geometry of spaces of complete forms ii skew forms abstract moduli spaces of complete skew forms are compactifications of spaces of skew symmetric linear maps of maximal rank on a fixed vector space where the added boundary divisor is simple normal crossing in this paper we compute their effective nef and movable cones the generators of their cox rings and for those spaces having picard rank two we give an explicit presentation of the cox ring furthermore we give a complete description of both the mori chamber and stable base locus decompositions of the effective cone of some spaces of complete skew forms having picard rank at most four

some new surfaces with pg 0 we give families of examples of surfaces of general type x with pg 0 k2 1 double covered by surfaces t with pg 0 k2 2 in chapter 2 we classify all such constructions with t 8 giving 4 parameter families of surfaces x for which x z2 and z4 there is a complete description of surfaces with pg 0 k2 l z4 in rl there was one example s with h s z z2 in 0 p the most interesting construction is the one in chapter 3 for which x l this answers negatively the following question are all simply connected surfaces with pg 0 k2 0 rational coming from severi s conjecture these constructions were motivated by reid s conjecture that if a given fundamental group h occurs there should be examples x t z2 with x h in the appendix we give an alternative proof of a formula for the arithmetic genus of a quotient surface based on a remark of hirzebruch

the recurrence of dynamic fuzzy systems abstract this paper analyses a recurrent behavior of dynamic fuzzy systems defined by fuzzy relations on a euclidean space by introducing a recurrence for crisp sets we prove probability theoretical properties for the fuzzy systems in the contractive case in kurano et al fuzzy sets and systems 51 1992 83 88 the existence of the maximum recurrent set is proved as another case we introduce a monotonicity for fuzzy relations which is extended from the linear structure in yoshida et al fuzzy sets and systems 66 1994 83 95 in the monotone case we prove the existence of the arcwise connected maximal recurrent sets

a four dimensional deformation of a numerical godeaux surface a numerical godeaux surface is a surface of general type with invariants p9 q 0 and k2 1 in this paper the moduli space of a numerical godeaux surface with order two torsion is computed to be eightdimensional whether or not the moduli space of such a surface is irreducible is still unknown the surface in this paper is constructed as one member of a four parameter family of double planes there is a natural involution on the surface inherited from the double plane construction which acts on the moduli space we show that the invariant subspace is four dimensional and coincides with the family of double planes

examples of mori dream surfaces of general type with pg 0 abstract in this paper we study effective nef and semiample cones of minimal surfaces of general type with p g 0 we provide examples of minimal surfaces of general type with p g 0 and 2 k 2 9 which are mori dream spaces on these examples we also give explicit description of their effective cones with all negative curves we also present non minimal surfaces of general type with p g 0 that are not mori dream surfaces

on the biregular geometry of the fulton macpherson compactification abstract let x n be the fulton macpherson compactification of the configuration space of n ordered points on a smooth projective variety x we prove that if either n 2 or dim x 2 then the connected component of the identity of aut x n is isomorphic to the connected component of the identity of aut x when x c is a curve of genus g c 1 we classify the dominant morphisms c n c r and thanks to this we manage to compute the whole automorphism group of c n namely aut c n s n aut c for any n 2 while aut c 2 s 2 aut c aut c furthermore we extend these results on the automorphisms to the case where x c 1 c r is a product of curves of genus g c i 2 finally using the techniques developed to deal with fulton macpherson spaces we study the automorphism groups of some kontsevich moduli spaces m 0 n p n d

bloch s conjecture for catanese and barlow surfaces catanese surfaces are regular surfaces of general type with pg 0 they specialize to double covers of barlow surfaces we prove that the ch0 group of a catanese surface is equal to z which implies the same result for the barlow surfaces

derived categories of burniat surfaces and exceptional collections we construct an exceptional collection varupsilon of maximal possible length 6 on any of the burniat surfaces with k x 2 6 a 4 dimensional family of surfaces of general type with p g q 0 we also calculate the dg algebra of endomorphisms of this collection and show that the subcategory generated by this collection is the same for all burniat surfaces the semiorthogonal complement mathcal a of varupsilon is an almost phantom category it has trivial hochschild homology and k 0 mathcal a mathbb z 2 6

new non arithmetic complex hyperbolic lattices we produce a family of new non arithmetic lattices in mathrm pu 2 1 all previously known examples were commensurable with lattices constructed by picard mostow and deligne mostow and fell into nine commensurability classes our groups produce five new distinct commensurability classes most of the techniques are completely general and provide efficient geometric and computational tools for constructing fundamental domains for discrete groups acting on the complex hyperbolic plane

discrete subgroups acting transitively on vertices of a bruhat tits building we describe all the discrete subgroups of ad g0 f o aut f which act transitively on the set of vertices of b b f g0 the bruhat tits building of a pair f g0 of a characteristic zero non archimedean local eld and a simply connected absolutely almost simple f group if b is of dimension at least 4 in fact we classify all the maximal such subgroups we show that there are exactly eleven families of such subgroups and explicitly construct them moreover we show that four of these families act simply transitively on the vertices in particular we show that there is no such actions if either the dimension of the building is larger than 7 f is not isomorphic to qp for some prime p or the building is associated to sln d0 where d0 is a non commutative division algebra along the way we also give a new proof of siegel klingen theorem on the rationality of certain dedekind zeta functions and l functions 1

sur la rigidite de certains groupes fondamentaux l arithmeticite des reseaux hyperboliques complexes et les faux plans projectifs the motivation of this work comes from the study of lattices in real simple lie groups the famous margulis s superrigidity theorem claims that finite dimensional reductive representations of any lattice of a real simple lie group of real rank 2 are superrigid as a corollary such a lattice is arithmetic these results extend to the real rank one case for lattices in sp n 1 and f4 20 by the work of corlette and gromov schoen on the other hand mostow and deligne mostow exhibited arithmetic lattices with non superrigid representations as well as non arithmetic lattices in the unitary group pu 2 1 a natural question is then to find simple sufficient conditions for superrigidity or arithmeticity of lattices in pu 2 1 rogawski conjectured the following let be a torsion free cocompact lattice in pu 2 1 such that the hyperbolic quotient m b2 verifies the cohomogical conditions b1 m 0 and h1 1 m h2 m then is arithmetic in this paper we consider a smooth complex projective surface m verifying the above cohomological assumptions and study zariski dense representations of the fundamental group 1 m in a simple k group h of k rank 2 where k denotes a local field our main result states that there are strong restrictions on such representations especially when k is non archimedean theorem 5 we then consider some cocompact lattices in pu 2 1 of special geometric interest recall that a fake p2 is a smooth complex surface distinct from p2 having the same betti numbers as p2 fake p2 exist by a result of mumford and are complex hyperbolic quotients h2 by yau s proof of the calabi conjecture they obviously verify the hypotheses of rogawski s conjecture in this case we prove that every zariski dense representation of in pgl 3 is superrigid in the sense of margulis theorem 3 as a corollary every fake p2 is an arithmetic quotient of the ball b2

fake quadrics from irreducible lattices acting on the product of upper half planes in the present article we provide examples of fake quadrics that is minimal complex surfaces of general type with the same numerical invariants as the smooth quadric in mathbb p 3 which are quotients of the bidisc by an irreducible lattice of automorphisms moreover we list classes of arithmetic lattices over a real quadratic number field which define a fake quadric and give general results towards a classification of all such fake quadrics

a simply connected surface of general type with p g 0 and k 2 2 in this paper we construct a simply connected minimal complex surface of general type with pg 0 and k2 2 using a rational blow down surgery and a gorenstein smoothing theory

inverse functional relation on the potts model the two dimensional anisotropic potts model at all temperatures satisfies a local inverse relation leading to an inverse functional equation on the transfer matrix the pertinence of the related functional equation for the partition function is discussed and verified perturbatively by introducing a diagrammatic expansion corresponding to a partial resummation on one parameter of the model possible exploitations of the inverse relation in order to determine the partition functions are sketched this indicates that the inverse relation can be used without referring to a star triangle relation

explicit equations of a fake projective plane fake projective planes are smooth complex surfaces of general type with betti numbers equal to those of the usual projective plane they come in complex conjugate pairs and have been classified as quotients of the 2 dimensional ball by explicitly written arithmetic subgroups in the following we find equations of a projective model of a conjugate pair of fake projective planes by studying the geometry of the quotient of such surface by an order 7 automorphism

chow groups are finite dimensional in some sense when s is a surface with pg s 0 mumford proved that its chow group a s is not finite dimensional in some sense in this paper we propose another definition of finite dimensionality for the chow groups using this new definition at least the chow group of some surface s with pg s 0 for example the product of two curves becomes finite dimensional the finite dimensionality of the chow groups follows from the finite dimensionality of the chow motives it turns out that the finite dimensionality of the chow motives is a very strong property for example we can prove bloch s conjecture representability of the chow groups of surfaces with pg s 0 under the assumption that the chow motive of s is finite dimensional

asymptotic base loci via okounkov bodies abstract an okounkov body is a convex subset of euclidean space associated to a divisor on a smooth projective variety with respect to an admissible flag in this paper we recover the asymptotic base loci from the okounkov bodies by studying various asymptotic invariants such as the asymptotic valuations and the moving seshadri constants consequently we obtain the nefness and ampleness criteria of divisors in terms of the okounkov bodies furthermore we compute the divisorial zariski decomposition by the okounkov bodies and find upper and lower bounds for moving seshadri constants given by the size of simplexes contained in the okounkov bodies

a simply connected numerical campedelli surface with an involution we construct a simply connected minimal complex surface of general type with p g 0 and k 2 2 which has an involution such that the minimal resolution of the quotient by the involution is a simply connected minimal complex surface of general type with p g 0 and k 2 1 in order to construct the example we combine a double covering and mathbb q gorenstein deformation especially we develop a method for proving unobstructedness for deformations of a singular surface by generalizing a result of burns and wahl which characterizes the space of first order deformations of a singular surface with only rational double points we describe the stable model in the sense of kollar and shepherd barron of the singular surfaces used for constructing the example we count the dimension of the invariant part of the deformation space of the example under the induced mathbb z 2 mathbb z action

superharmonic fuzzy sets on recurrent sets in dynamic fuzzy systems abstract we analyse p superharmonic fuzzy sets on recurrent sets in dynamic fuzzy systems and we derive a simple estimation for the fuzzy sets this paper presents a method to calculate an optimal value for snell s optimal stopping problem

cohomological goodness and the profinite completion of bianchi groups the concept of cohomological goodness was introduced by j p serre in his book on galois cohomology 31 this property relates the cohomology groups of a group to those of its profinite completion we develop properties of goodness and establish goodness for certain important groups we prove for example that the bianchi groups i e the groups psl 2 o where o is the ring of integers in an imaginary quadratic number field are good as an application of our improved understanding of goodness we are able to show that certain natural central extensions of fuchsian groups are residually finite this result contrasts with examples of p deligne 5 who shows that the analogous central extensions of sp 4 z do not have this property