scalar curvature and warped products of riemann manifolds we establish the relationship between the scalar curvature of a warped product m xf n of riemann manifolds and those ones of m and n then we search for weights f to obtain constant scalar curvature on m xf n when m is compact

the topological trace formula the topological trace formula is a computation of the lefschetz number of a hecke correspondence c acting on the weighted cohomology groups defined in ghm of a locally symmetric space x it expresses this lefschetz number as a sum of contributions from fixed point components of c on the reductive borel serre compactifi cation of x the proof uses the lefschetz fixed point formula of gm2

hodge cohomology of gravitational instantons we study the space of l2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type these metrics generalize the geometric structures at infinity of several different well known classes of metrics including asymptotically locally euclidean manifolds the known types of gravitational instantons and also poincare metrics on q rank 1 ends of locally symmetric spaces and on the complements of smooth divisors in kahler manifolds the answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold the l2 signature formula implied by our result is closely related to the one proved by dai 25 and more generally by vaillant 67 and identifies dai s invariant directly in terms of intersection cohomology of differing perversities this work is also closely related to a recent paper of carron 12 and the forthcoming paper of cheeger and dai 17 we apply our results to a number of examples gravitational instantons among them arising in predictions about l2 harmonic forms in duality theories in string theory

simplicial differential forms with poles shadow forms consider a simplicial complex k whitney associated to each simplicial cochain 4 a differential form v 4 on k whitney s map v from simplicial cochains to differential forms is a map of cochain complexes which induces the derham isomorphism we call the forms v 4 whitney forms they have been useful in several contexts most notably integral geometry and rational homotopy theory in this paper we introduce a generalization of whitney forms called shadow forms we associate to certain simplicial chains g with respect to the first derived subdivision k of k a shadow form w on k let us illustrate the construction of shadow forms when k is a single n simplex which we now denote by a in this case w g is defined for k simplices g of ax which do not lie in the boundary of a or for linear combinations of these for example if a is the 2 simplex then either of the 1 simplices or 2 pictured below has an associated shadow form

infinitesimal rigidity for the action of sl n ℤ on n let r sl n z or any subgroup of finite index then the action of r on tn by automorphisms is infinitesimally rigid for n 7 i e the cohomology h i r vec tn 0 where vec t n denotes the module of c vector fields on t

the l class of non witt spaces characteristic classes for oriented pseudomanifolds can be defined using appropriate self dual complexes of sheaves on non witt spaces self dual complexes compatible to intersection homology are determined by choices of lagrangian structures at the strata of odd codimension we prove that the associated signature and l classes are independent of the choice of lagrangian structures so that singular spaces with odd codimensional strata such as e g certain compactifications of locally symmetric spaces have well defined l classes provided lagrangian structures exist we illustrate the general results with the example of the reductive borel serre compactification of a hilbert modular surface

on the reductive borel serre compactification l p cohomology of arithmetic groups for large p the l 2 cohomology of an arithmetic quotient of an hermitian symmetric space i e of a locally symmetric variety is known to have the topological interpretation as the intersection homology of its baily borel satake compactification in this article we observe that even without the hermitian hypothesis the l p cohomology of an arithmetic quotient for p finite and sufficiently large is isomorphic to the ordinary cohomology of its reductive borel serre compactification we use this to generalize a theorem of mumford concerning homogeneous vector bundles their invariant chern forms and the canonical extensions of the bundles here though we are referring to canonical extensions to the reductive borel serre compactification of any arithmetic quotient to achieve that we give a systematic discussion of vector bundles and chern classes on stratified spaces

relative artin motives and the reductive borel serre compactification of a locally symmetric variety we introduce the notion of artin motives and cohomological motives over a scheme x given a cohomological motive m over x we consider the universal artin motive mapping to m and denote it omega 0 x m we use this to define a motive mathbb e x over x which is an invariant of the singularities of x the first half of the paper is devoted to the study of the functors omega 0 x and the computation of the motives mathbb e x

toroidal and reductive borel serre compactifications of locally symmetric spaces by hermitian locally symmetric space we mean an arithmetic quotient of a bounded symmetric domain both the toroidal and the reductive borel serre compactifications of such a space come equipped with canonical mappings to the baily borel satake compactification in this article we show that there is a mapping from the toroidal compactification to the reductive borel serre compactification whose composition with the projection to the baily borel compactification agrees with the canonical projection up to an arbitrarily small homotopy we also consider arithmetic quotients of a self adjoint homogeneous cone there is a canonical mapping from the reductive borel serre compactification to the standard compactification of such a locally symmetric cone we show that this projection when restricted to the closure of a polyhedral cone has contractible fibers

with respect to whom are you critical abstract for any compact riemannian surface s and any point y in s q y 1 denotes the set of all points in s for which y is a critical point and q y 1 its cardinality we proved 2 together with imre barany that q y 1 1 and that equality for all y s characterizes the surfaces homeomorphic to the sphere here we show for any orientable surface s and any point y s the following two main results there exists an open and dense set of riemannian metrics g on s for which y is critical with respect to an odd number of points in s and this is sharp if s is the torus then q y 1 5 and if s has genus g 2 then q y 1 8 g 5 properties involving points at globally maximal distance on s are eventually presented

lefschetz properties for noncompact arithmetic ball quotients we prove a lefschetz property for the restriction of the cohomology of noncompact arithmetic ball quotients to ball quotients of smaller dimension

on the reductive borel serre compactification ii excentric quotients and least common modifications let x be a locally symmetric variety i e the quotient of a bounded symmetric domain by a say neat arithmetically defined group of isometries let overline x rm exc and x rm tor exc denote its excentric borel serre and toroidal compactifications respectively we determine their least common modification and use it to prove a conjecture of goresky and tai concerning canonical extensions of homogeneous vector bundles in the process we see that overline x rm exc and x rm tor exc are homotopy equivalent

l2 index and the selberg trace formula abstract a method is developed for computing the l 2 index of a locally symmetric elliptic differential operator d acting on a locally symmetric manifold m g k with g semisimple of real rank one and of finite co volume based on applying the selberg trace formula to the difference of the two heat kernels associated to d the applications include an extension of the osborne warner multiplicity formula to certain non integrable discrete series derived from the l 2 spinor formula and showing the existence in some cases of non invariant l 2 cohomology classes in the middle dimension via the l 2 signature formula

geometry of proteins hydrogen bonding sterics and marginally compact tubes the functionality of proteins is governed by their structure in the native state protein structures are made up of emergent building blocks of helices and almost planar sheets a simple coarse grained geometrical model of a flexible tube barely subject to compaction provides a unified framework for understanding the common character of globular proteins we argue that a recent critique of the tube idea is not well founded

sur la l 2 cohomologie des varietes a courbure negative we give a topological interpretation of the space of l2 harmonic forms on finite volume manifolds with sufficiently pinched negative curvature we give examples showing that this interpretation fails if the curvature is not sufficiently pinched and that our result is sharp with respect to the pinching constants the method consists first in comparing l2 cohomology with weighted l2 cohomology thanks to previuos works done by t ohsawa and then in identifying these weighted spaces

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

chern classes of automorphic vector bundles 1 1 suppose x is a compact n dimensional complex manifold each partition i i1 i2 ir of n corresponds to a chern number c x ci1 x ci2 x cir x x z where c x h x z are the chern classes of the tangent bundle x h2n x z is the fundamental class and h0 x z z is the augmentation many invariants of x such as its complex cobordism class may be expressed in terms of its chern numbers mi st during the last 25 years characteristic classes of singular spaces have been defined in a variety of contexts whitney classes of euler spaces su h t ak todd classes of singular varieties bfm chern classes of singular algebraic varieties mac l classes of stratified spaces with even codimension strata gm1 wu classes of singular spaces go2 gp to name a few however these characteristic classes are invariably homology classes and as such they cannot be multiplied with each other in some cases it has been found possible to lift these classes from homology to intersection homology where some characteristic numbers may be formed bbf bw go2 gp t the case of locally symmetric spaces is particularly interesting suppose is a torsionfree arithmetic group acting on a complex n dimensional hermitian symmetric domain d g k where g is the group of real points of a semisimple algebraic group g defined over q with g q and where k g is a maximal compact subgroup then x d is a hermitian locally symmetric space when x is compact hirzebruch s proportionality theorem hr1 says that there is a number v q so that for every partition i i1 i2 ir of n the chern number satisfies c x v c where gu k is the compact dual symmetric space and gu is a compact real form of g c containing k if x d is noncompact it has a canonical baily borel satake compactification x this is a usually highly singular complex projective algebraic variety to formulate a proportionality theorem in the noncompact case one might hope that the tangent bundle tx extends as a complex vector bundle over x but this is false in mu1 d mumford showed that tx has a particular extension e x over any toroidal resolution x x of the baily borel compactification and that for any partition i of n the resulting chern numbers c e c e c e c e x 1 1 1

the cauchy riemann equations on product domains we establish the l 2 theory for the cauchy riemann equations on product domains provided that the cauchy riemann operator has closed range on each factor we deduce regularity of the canonical solution on p 1 forms in special sobolev spaces represented as tensor products of sobolev spaces on the factors of the product this leads to regularity results for smooth data

foliated stratified spaces and a de rham complex describing intersection space cohomology the method of intersection spaces associates cell complexes depending on a perversity to certain types of stratified pseudomanifolds in such a way that poincare duality holds between the ordinary rational cohomology groups of the cell complexes associated to complementary perversities the cohomology of these intersection spaces defines a cohomology theory hi for singular spaces which is not isomorphic to intersection cohomology ih mirror symmetry tends to interchange ih and hi the theory ih can be tied to type iia string theory while hi can be tied to iib theory for pseudomanifolds with stratification depth 1 and flat link bundles the present paper provides a de rham theoretic description of the theory hi by a complex of global smooth differential forms on the top stratum we prove that the wedge product of forms introduces a perversity internal cup product on hi for every perversity flat link bundles arise for example in foliated stratified spaces and in reductive borel serre compactifications of locally symmetric spaces a precise topological definition of the notion of a stratified foliation is given

hodge decompositions and dolbeault complexes on normal surfaces give the smooth subset of a normal singular complex projective surface the metric induced from the ambient projective space the l2 cohomology of this incomplete manifold is isomorphic to the surface s intersection cohomology which has a natural hodge decomposition this paper identifies dolbeault complexes whose 0 closed and 0 coclosed forms represent the classes of pure type in the corresponding hodge decomposition of l2 cohomology

l2 cohomology of arithmetic varieties abstract the l2 cohomology of arithmetic quotients of bounded symmetric domains is studied we establish the conjecture of zucker equating the l2 cohomology of these spaces to the intersection cohomology of their baily borel compactifications

on the cohomology groups of local systems over hilbert modular varieties via higgs bundles let x be a hilbert modular variety and v a non trivial local system over x in this paper we study deligne saito s mixed hodge structure mhs on the cohomology groups h k x v using the method of higgs bundles among other results we give a dimension formula for the hodge numbers and show that the mixed hodge structure is split over r these results are analogous to 18 in the cocompact case and complement the results in 8 for constant coefficients

fredholm hodge and liouville theorems on noncompact manifolds fredholm liouville hodge and l2 cohomology theorems are proved for laplacians associated with a class of metrics defined on manifolds that have finitely many ends the metrics are conformal to ones that are asymptotically translation invariant they are not necessarily complete the fredholm results are of necessity with respect to weighted sobolev spaces embedding and compact embedding theorems are also proved for these spaces two of the most useful facts in analysis on a compact riemannian manifold are that the laplacian is fredholm and its kernel consists of closed and coclosed forms that provide unique representatives for all the de rham cohomology classes naturally one would like to extend these results to noncompact manifolds the first such result in this direction is due to kodaira 9 12 p 165 it is that l2 a m g is the orthogonal direct sum of dco aq 1m dg co aq lm and k2 awm g ae l2 am g ida dga 0 unfortunately if no restriction is made on the manifold or the metric one cannot improve on this the laplacian need not be fredholrn l2 harmonic forms need not be closed or coclosed even if they are closed and coclosed the space of harmonic l2 forms need not be finite dimensional see 7 and even if the laplacian is fredholm and l2 harmonic forms are closed and coclosed those forms need not provide unique or total representation of de rham cohomology thus one of the main questions in analysis on noncompact manifolds is what conditions on m and g allow one to carry over the fredholm and hodge theorems for compact manifolds and if they cannot be carried over completely to what extent can they be in the case of hodge s theorem i e properties of the space of l2 harmonic forms this question has been quite actively investigated recently see 2 8 11 and 13 for instance one of the results of atiyah patodi and singer in 2 is that if a manifold has cylindrical ends then k 2 a m g is naturally isomorphic to the image of hcomp m in hfdr m see 2 proposition 4 9 in 11 muller investigates the spectrum of the laplacian on manifolds that outside a compact set are q x r with received by the editors august 30 1985 and in revised form april 10 1986 1980 mathematics subject classification 1985 revision primary 58gi0 35j05 1987 american mathematical society 0002 9947 87 1 00 25 per page this content downloaded from 157 55 39 45 on tue 19 jul 2016 04 20 24 utc all use subject to http about jstor org terms

the noether lefschetz conjecture and generalizations we prove the noether lefschetz conjecture on the moduli space of quasi polarized k3 surfaces this is deduced as a particular case of a general theorem that states that low degree cohomology classes of arithmetic manifolds of orthogonal type are dual to the classes of special cycles i e sub arithmetic manifolds of the same type for compact manifolds this was proved in 3 here we extend the results of 3 to non compact manifolds this allows us to apply our results to the moduli spaces of quasi polarized k3 surfaces

intersection numbers of hecke cycles on hilbert modular varieties let o be the ring of integers of a totally real number field e and set g res e q gl 2 fix an ideal c 0 for each ideal m 0 let t m denote the mth hecke operator associated to the standard compact open subgroup u 0 c of g a f setting x 0 c g q g a k u 0 c where k is a certain subgroup of g r we use t m to define a hecke cycle z m ih 2 e q x 0 c x 0 c here ih denotes intersection homology we use zucker s conjecture proven by looijenga and independently by saper and stern to obtain a formula relating the intersection number z m z n to the trace of t m o t n considered as an endomorphism of the space of hilbert cusp forms on u 0 c

cohomologie l 2 des varietes qale we give a topological interpretation of the space of l 2 harmonis forms of some qale manifolds introduced by d joyce we introduce a analytical criterium which make possible the used of mayer vietoris sequence

weighted cohomology of arithmetic groups let g be a semisimple algebraic group defined over the rational numbers k a maximal compact subgroup ofg g r and g q a neat arithmetic subgroup letx g k be the locally symmetric space associated to and e the local system on x constructed out of a finite dimensional irreducible algebraic representation e of g fix a maximally q split torus s in g s is assumed to be nontrivial so that x is noncompact let a s r 0 and by a slight abuse of notation let x a denote the group of rational characters of s choose a minimal rational parabolic subgroup p0 s the choice determines a notion of positivity in x a and a set of positive roots among the roots of s in g let 0 be the half sum of the positive roots weighted cohomology is an invariant of introduced by m goresky g harder and r macpherson 12 in the study 15 16 12 14 of the trace of hecke operators in the cohomology of for each p x a q there are weight profiles p and p upper and lower p see 1 5 and groups w ph i e and w h i e the notion of profile here differs from but is equivalent to that of 12 see remark 1 2 below if p q 2 0 then the profiles p and q are dual and the corresponding groups are in poincare duality for very positive p one gets the compactly supported cohomology h i c x e and dually for very negative p the full cohomology h i x e the definition of the weighted cohomology groups is somewhat involved so i will attempt to give some idea of the motivation behind it detailed definitions are recalled in section 1 associated to p and p are complexes of sheaves wpc e and wpc e on the reductive borel serre compactification x of x let p be a rational parabolic subgroup np its unipotent radical sp the split centre of p np and xp the p boundary component in x the idea behind the construction in 12 is as follows the stalk cohomology of the direct image ri e by i x x at x xp is