subdivisions and local h vectors in part i a general theory of f vectors of simplicial subdivisions or triangulations of simplicial complexes is developed based on the concept of lo cal h vector as an application we prove that the h vector of a cohen macaulay complex increases under quasi geometric subdivision thus establishing a spe cial case of a conjecture of kalai and this author techniques include com mutative algebra homological algebra and the intersection homology of toric varieties in part ii we extend the work of part i to more general situations first a formal generalization of subdivision is given based on incidence alge bras special cases are then developed in particular one based on subdivisions of eulerian posets and involving generalized h vectors other cases deal with kazhdan lusztig polynomials ehrhart polynomials and a q analogue of eule rian posets many applications and examples are given throughout department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address rstan math mit edu this content downloaded from 157 55 39 210 on sat 30 jul 2016 05 59 16 utc all use subject to http about jstor org terms

spin holonomy of einstein manifolds berger s theorem classifies the linear holonomy groups of irreducible simply connected riemannian manifolds for physical applications however it is at least as important to have a classification of the possible spin holonomy groups defined by the parallel transport of spinors of non simply connected manifolds we establish a complete classification of the spin holonomy groups of all compact locally irreducible einstein riemannian spin manifolds of non negative scalar curvature

the map between conformal hypercomplex hyper kahler and quaternionic kahler geometry we review the general properties of target spaces of hypermultiplets which are quaternionic like manifolds and discuss the relations between these manifolds and their symmetry generators we explicitly construct a one to one map between conformal hypercomplex manifolds i e those that have a closed homothetic killing vector and quaternionic manifolds of one quaternionic dimension less an important role is played by transformations relating complex structures on conformal hypercomplex manifolds and connections on quaternionic manifolds in this map the subclass of conformal hyper kahler manifolds is mapped to quaternionic kahler manifolds we relate the curvatures of the corresponding manifolds and furthermore map the symmetries of these manifolds to each other

kahlerian twistor spaces the work of r penrose has shown the deep relationship between the conformal structure of minkowski space and the complex geometry of lines in projective three space this transformation applies to a more general class of conformal structures and in particular there is a riemannian version described in 5 which is itself of interest to mathematical physicists basically to each oriented riemannian 4manifold x one associates canonically a 6 manifold z the twistor space of x together with an almost complex structure if this structure is integrable then z becomes a complex 3 manifold with a distinguished family of projective lines on it whose geometry describes the conformal geometry of the manifold x it is natural to ask what sort of complex manifold one obtains by this process in this paper we ask the more specific question which compact 4c manifolds have ka hlerian twistor spaces 1 the answer rather surprisingly is that there are just two examples the 4 sphere s and the complex projective plane p2 c each with their standard homogeneous conformal structures the twistor space of s 4 is the 3 dimensional projective space p3 c and that of p 2 q is the manifold f2 c of flags in c 3 the essential point of the proof of this result is the remark that if z is a twistor space with a kahler metric then the canonical bundle k is negative using vanishing theorems and the riemann roch theorem we then narrow down the possibilities for z to four cases projective space the flag manifold the intersection of two quadrics in p5 and the double covering of p3 branched over a non singular quartic surface the last two candidates cannot be realized however because the value of their euler characteristic is incompatible with their being twistor spaces throughout the proof we make essential use of a real structure on z which reduces considerably the occurrence of singularities at various stages aside from this our methods are the standard ones of algebraic geometry the four examples to which the proof leads us are all fano threefolds if we drop the requirement of compactness on x then it is quite possible for an open set of a fano threefold to be a twistor space in particular we show that the family of conies lying on the intersection of two

eigenvalue estimates for the dirac operator on quaternionic kahler manifolds we consider the dirac operator on compact quaternionic kahler manifolds and prove a lower bound for the spectrum this estimate is sharp since it is the first eigenvalue of the dirac operator on the quaternionic projective space

equivariant constructible koszul duality for dual toric varieties abstract for affine toric varieties x and x defined by dual cones we define an equivalence of categories between mixed versions of the equivariant derived category d t b x and the derived category of sheaves on x which are locally constant with unipotent monodromy on each orbit this equivalence satisfies the koszul duality formalism of beilinson ginzburg and soergel

modularity quaternion kahler spaces and mirror symmetry we provide an explicit twistorial construction of quaternion kahler manifolds ob tained by deformation of c map spaces and carrying an isometric action of the modular group sl 2 z the deformation is not assumed to preserve any continuous isometry and therefore this construction presents a general framework for describing ns5 brane instanton effects in string compactifications with n 2 supersymmetry in this context the modular invariant parametrization of twistor lines found in this work yields the complete non perturbative mirror map between type iia and type iib physical fields

compact 8 manifolds with holonomy spin 7 in berger s classification 4 of the possible holonomy groups of a nonsymmetric irreducible riemannian manifold there are two special cases the exceptional holonomy groups g2 in 7 dimensions and spin 7 in 8 dimensions bryant 6 proved that such metrics exist locally using the theory of exterior differential systems and gave some explicit examples later bryant and salamon 7 constructed explicit complete metrics with holonomy g2 and spin 7 in two previous papers 12 13 the author constructed many examples of compact riemannian 7 manifolds with holonomy g2 this paper will construct examples of compact riemannian 8 manifolds with holonomy spin 7 using similar methods we believe that these are the first examples known since metrics with holonomy spin 7 are ricci flat these are also new examples of compact ricci flat riemannian 8 manifolds let m be an 8 manifold a spin 7 structure on m can be encoded in a 4 form on m a special 4 form satisfying the condition that the stabilizer of at each point should be isomorphic to spin 7 by an abuse of notation we usually identify the spin 7 structure with its associated 4 form since spin 7 so 8 the spin 7 structure also induces a riemannian metric g and an orientation on m it turns out that the holonomy group hol g of g is a subgroup of spin 7 with spin 7 structure if and only if d 0 the quantity d is called the torsion of the spin 7 structure and is called torsion free if d 0 very briefly the plan of the paper splits into four steps as follows firstly a compact 8 manifold m is given secondly a spin 7 structure on m is found with small torsion i e d is small thirdly is deformed to a nearby spin 7 structure with d 0 thus is torsion free and if g is the associated metric then hol g spin 7 fourthly it is shown that hol g is spin 7 and not some proper subgroup the structure of the paper is designed around this division of the construction into four steps there are six chapters this first chapter is of introductory material the second chapter has the full statements of the main results in

koszul duality for toric varieties we show that certain categories of perverse sheaves on affine toric varieties x and x v defined by dual cones are koszul dual in the sense of beilinson ginzburg and soergel 1996 the functor expressing this duality is constructed explicitly by using a combinatorial model for mixed sheaves on toric varieties

pairwise compatibility graphs a survey a graph g v e is a pairwise compatibility graph pcg if there exists an edge weighted tree t and two nonnegative real numbers d min and d max such that each leaf u of t is a node of v and there is an edge u v in e if and only if d min leq d t u v leq d max where d t u v is the sum of weights of the edges on the unique path from u to v in t in this article we survey the state of the art concerning this class of graphs and some of its subclasses

exact leaf powers we define and study the new notion of exact k leaf powers where a graph g v g e g is an exact k leaf power if and only if there exists a tree t v t e t an exact k leaf root of g whose set of leaves equals v g such that email protected e g holds for u email protected v g if and only if the distance of u and v in t is exactly k this new notion is closely related to but different from leaf powers and neighbourhood subtree tolerance graphs we prove characterizations of exact 3 and 4 leaf powers which imply that such graphs can be recognized in linear time and that also the corresponding exact leaf roots can be found in linear time furthermore we characterize all exact 5 leaf roots of chordless cycles and derive several properties of exact 5 leaf powers

on graphs that are not pcgs let t be an edge weighted tree and let d min d max be two nonnegative real numbers the pairwise compatibility graph pcg of t is a graph g such that each vertex of g corresponds to a distinct leaf of t and two vertices are adjacent in g if and only if the weighted distance between their corresponding leaves in t is in the interval d min d max similarly a given graph g is a pcg if there exist suitable t d min d max such that g is a pcg of t yanhaona bayzid and rahman proved that there exists a graph with 15 vertices that is not a pcg on the other hand calamoneri frascaria and sinaimeri proved that every graph with at most seven vertices is a pcg in this paper we construct a graph of eight vertices that is not a pcg which strengthens the result of yanhaona bayzid and rahman and implies optimality of the result of calamoneri frascaria and sinaimeri we then construct a planar graph with sixteen vertices that is not a pcg finally we prove a variant of the pcg recognition problem to be np complete

hypergeometric functions and moduli spaces of null correlation bundles the moduli space nn of null correlation bundles over cp2n 1 has a natural l2 kahler metric we explicitly describe that metric in terms of appell lauricella hypergeometric functions the proof uses a penrose twistor correspondence

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

perverse sheaves and actions where t x denotes the action of t on x the set xw is known to be a locally closed c stable algebraic subvariety of x isomorphic to an affine space the pieces xw form a cell decomposition x uwewxw w e w the socalled bialynicki birula decomposition we assume this decomposition to be an algebraic stratification of x the closure of a cell may not be a union of cells in general let zw be the intersection cohomology complex on the closure of a cell xw extended by 0 to the whole of x the hypercohomology h 2w has a natural structure of a graded h x c module further for any x y e w there is a natural morphism of graded spaces

mirror duality and string theoretic hodge numbers we prove in full generality the mirror duality conjecture for string theoretic hodge numbers of calabi yau complete intersections in gorenstein toric fano varieties the proof is based on properties of intersection cohomology

on certain kahler quotients of quaternionic kahler manifolds we prove that given a certain isometric action of a two dimensional abelian group a on a quaternionic kahler manifold m which preserves a submanifold n m the quotient m n a has a natural kahler structure we verify that the assumptions on the group action and on the submanifold n m are satisfied for a large class of examples obtained from the supergravity c map in particular we find that all quaternionic kahler manifolds m in the image of the c map admit an integrable complex structure compatible with the quaternionic structure such that n m is a complex submanifold finally we discuss how the existence of the kahler structure on m is required by the consistency of spontaneous mathcal n 2 to mathcal n 1 supersymmetry breaking

almost complex structures on quaternion kahler manifolds and inner symmetric spaces we prove that compact quaternionic kahler manifolds of positive scalar curvature admit no almost complex structure even in the weak sense except for the complex grassmannians gr2 n 2 we also prove that irreducible inner symmetric spaces m 4n of compact type are not weakly complex except for spheres and hermitian symmetric spaces

some classes of graphs that are not pcgs abstract a graph g v e is a pairwise compatibility graph pcg if there exists an edge weighted tree t and two non negative real numbers d m i n and d m a x d m i n d m a x such that each node u v is uniquely associated to a leaf of t and there is an edge u v e if and only if d m i n d t u v d m a x where d t u v is the sum of the weights of the edges on the unique path p t u v from u to v in t understanding which graph classes lie inside and which ones outside the pcg class is an important issue in this paper we show that some interesting classes of graphs have empty intersection with pcg they are wheels strong product of a cycle and p 2 and the square of an n node cycle with n sufficiently large as a side effect we show that the smallest planar graph not to be pcg has not 20 nodes as previously known but only 8 it is c 8 2

linear perturbations of quaternionic metrics we extend the twistor methods developed in our earlier work on linear deformations of hyperkahler manifolds 1 to the case of quaternionic kahler manifolds via swann s construction deformations of a 4d dimensional quaternionic kahler manifold mathcal m are in one to one correspondence with deformations of its 4d 4 dimensional hyperkahler cone mathcal s the latter can be encoded in variations of the complex symplectomorphisms which relate different locally flat patches of the twistor space mathcal z mathcal s with a suitable homogeneity condition that ensures that the hyperkahler cone property is preserved equivalently we show that the deformations of mathcal m can be encoded in variations of the complex contact transformations which relate different locally flat patches of the twistor space mathcal z mathcal m of mathcal m by passing the swann bundle and its twistor space we specialize these general results to the case of quaternionic kahler metrics with d 1 commuting isometries obtainable by the legendre transform method and linear deformations thereof we illustrate our methods for the hypermultiplet moduli space in string theory compactifications at tree and one loop level

special metrics and triality abstract we investigate a new 8 dimensional riemannian geometry defined by a generic closed and coclosed 3 form with stabiliser psu 3 and which arises as a critical point of hitchin s variational principle we give a riemannian characterisation of this structure in terms of invariant spinor valued 1 forms which are harmonic with respect to the twisted dirac operator on 1 we establish various obstructions to the existence of topological reductions to psu 3 for compact manifolds we also give sufficient conditions for topological psu 3 structures that can be lifted to topological su 3 structures we also construct the first known compact example of an integrable non symmetric psu 3 structure in the same vein we give a new riemannian characterisation for topological quaternionic kahler structures which are defined by an sp 1 sp 2 invariant self dual 4 form again we show that this form is closed if and only if the corresponding spinor valued 1 form is harmonic for and that these equivalent conditions produce constraints on the ricci tensor

a uniform estimate for general quaternionic calabi problem with appendix by daniel barlet abstract we prove a c 0 a priori estimate on a solution of the quaternionic calabi problem on an arbitrary compact connected hkt manifold this generalizes earlier results 9 and 7 where this result was proven under certain extra assumptions on the manifold

on pairwise compatibility graphs having dilworth number two a graph g v e is called a pairwise compatibility graph pcg if there exists a tree t a positive edge weight function w on t and two non negative real numbers d m i n and d m a x d m i n d m a x such that v coincides with the set of leaves of t and there is an edge u v e if and only if d m i n d t w u v d m a x where d t w u v is the sum of the weights of the edges on the unique path from u to v in t when the constraints on the distance between the pairs of leaves concern only d m a x or only d m i n the two subclasses lpgs leaf power graphs and mlpgs minimum leaf power graphs are defined the dilworth number of a graph is the size of the largest subset of its nodes in which the close neighborhood of no node contains the neighborhood of another it is known that lpg mlpg is not empty and that threshold graphs i e dilworth one graphs are contained in it in this paper we prove that dilworth two graphs belong to the set lpg mlpg too our proof is constructive since we show how to compute all the parameters t w d m a x and d m i n exploiting the usual representation of dilworth two graphs in terms of node weight function and thresholds for graphs with dilworth number two that are also split graphs i e split permutation graphs we provide another way to compute t w d m i n and d m a x when these graphs are given in terms of their intersection model

hard lefschetz theorem for nonrational polytopes the hard lefschetz theorem is known to hold for the intersection cohomology of the toric variety associated to a rational convex polytope one can construct the intersection cohomology combinatorially from the polytope hence it is well defined even for nonrational polytopes when there is no variety associated to it we prove the hard lefschetz theorem for the intersection cohomology of a general polytope

clifford structures on riemannian manifolds we introduce the notion of even clifford structures on riemannian manifolds which for rank r 2 and r 3 reduce to almost hermitian and quaternion hermitian structures respectively we give the complete classification of manifolds carrying parallel rank r even clifford structures kahler quaternion kahler and riemannian products of quaternion kahler manifolds for r 2 3 and 4 respectively several classes of 8 dimensional manifolds for 5 r 8 families of real complex and quaternionic grassmannians for r 8 6 and 5 respectively and rosenfeld s elliptic projective planes op 2 c o p 2 h o p 2 and o o p 2 which are symmetric spaces associated to the exceptional simple lie groups f4 e6 e7 and e8 for r 9 10 12 and 16 respectively as an application we classify all riemannian manifolds whose metric is bundle like along the curvature constancy distribution generalizing well known results in sasakian and 3 sasakian geometry 2000 mathematics subject classification primary 53c26 53c35 53c10 53c15