stratifiable maps and topological invariants let x and y be spaces and f x y be a map between them then a large family of problems in topology and geometry involve the attempt to relate the invariants of x and y via f this attempt has been successful often for the case x and y manifolds and f the projection of a fibre bundle for many types of important invariants for example the result of chernhirzebruch serre chs asserts that for y simply connected the signature of x is the product of the signatures of y and of the fibre f 1 y in this paper we undertake the study of maps in which the fiber may vary from point to point the spaces x and y will be any whitney stratified spaces with even codimension strata e g algebraic varieties and f will be a stratified map this means essentially that for v a component of a stratum of y thus v is an open manifold f 1v is a union of strata of x and the restriction of f to f71v is a locally trivial fibre bundle for example subanalytic maps of subanalytic sets can be stratified with respect to whitney

on the homology of the hilbert scheme of points in the plane geir ellingsrud 1 and stein arild stromme 2 i matematisk institutt universitetet i oslo blindern n oslo 3 norway 2 matematisk institutt universitetet i bergen n 5014 bergen norway although several authors have been interested in the hilbert scheme hilba ip z parametrizing finite subschemes of length d in the projective plane i1 i2 f 1 f2 br among others not much is known about the topological properties of this space the picard group has been calculated i f2 and the homology groups of hilb3 ip 2 have been computed hi in this paper we give a precise description of the additive structure of the homology of hilba ip2 applying the results of birula bialynicki b1 b2 on the cellular decompositions de fined by a torus action to the natural action of a maximal torus of sl 3 on hilbe ipz a rather easy consequence of the fact that this action has finitely many fixpoints is that the cycle maps between the chow groups and the homology groups are isomorphisms in particular there is no odd homology and the homology groups are all free the main objective of this work is to compute their ranks the betti numbers of hilba ipz as a byproduct of our method we get similar results on the homology of the punctual hilbert scheme and of the hilbert scheme of points in the affine plane it seems natural to generalize our results to any toric smooth surface however we give the results only for the rational ruled surfaces if with an indication of the necessary changes in the proofs for simplicity we work over the field of complex numbers but with an appropriate interpretation of the word homology our results remain valid over any base field

an abelian category of motivic sheaves abstract a category of motivic sheaves is constructed over a variety in characteristic 0 using nori s method although the relationship with many alternative constructions remains to be clarified it does have many of the properties one expects for example it is abelian and has betti hodge and l adic realizations and it has a tannakian subcategory of motivic local systems

weak limit and blowup of approximate solutions to h systems abstract let h r 3 r be a continuous function such that h p h 0 r as p fixing a domain in r 2 we study the behaviour of a sequence u n of approximate solutions to the h system u 2 h u u x u y in assuming that sup p r 3 h p h 0 p 1 we show that the weak limit of the sequence u n solves the h system and u n u strongly in h 1 apart from a countable set s made by isolated points moreover if in addition h p h 0 o 1 p as p then in correspondence of each point of s we prove that the sequence u n blows either an h bubble or an h 0 sphere

bubbles with prescribed mean curvature the variational approach abstract let h r 3 r be a c 1 mapping such that h p h 0 as p we show that when h satisfies some global conditions then there exists an h bubble namely a sphere s in r 3 such that the mean curvature of s at any regular point p s equals h p

blow up analysis for the prescribed mean curvature equation on r2 abstract we consider the h system u 2 h u u x u y on r 2 where h c 0 r 3 r satisfies h q h o 1 q as q and sup q r 3 h q h q 1 for some h r 0 we show that a sequence of approximate solutions of the h system on r 2 admits a limit configuration made by h bubbles namely nonconstant solutions of h systems on r 2 and h can be the mapping h or the constant h

the douady space of a complex surface abstract we prove that a standard realization of the direct image complex via the so called douady barlet hilbert chow in the algebraic case morphism associated with a smooth complex analytic surface admits a natural decomposition in the form of an injective quasi isomorphism of complexes this is a more precise form of a special case of the decomposition theorems of beilinson bernstein deligne gabber and m saito of which the proof we present is independent in addition it is elementary and transparent and does not use either perverse sheaves the methods of positive characteristic nor saito s theory of mixed hodge modules the proof hinges on the special case of the bi disk in the complex affine plane where we make explicit use of a construction of nakajima s and of the corresponding representation theoretic interpretation foreseen by vafa and witten some consequences of the decomposition theorem the gottsche formula holds for complex surfaces interpretation of the rational cohomologies of douady spaces as a kind of fock space new proofs of results of briancon and ellingsrud and stromme on punctual hilbert schemes and computation of the mixed hodge structure of the douady spaces in the kahler case we also derive a natural connection with equivariant k theory for which in the case of algebraic surfaces bezrukavnikov and ginzburg have proposed a different approach

construction of central elements in the affine hecke algebra via nearby cycles 0 1 overview let g be a connected reductive group over a finite field fq and let g k be the corresponding group over the local field k fq t let g o g k be a maximal compact subgroup of g k here o fq t and let hsph denote the hecke algebra of g k with respect to g o in other words hsph as a vector space consists of compactly supported bi g o invariant functions g k q and the product is defined by

modular equations and the genus zero property of moonshine functions and let n be the normalizer of 0 n in sl 2 r we shall call a discrete subgroup of sl 2 r a congruence group if it contains n for some n necessarily the index of n in is finite and acts on the extended upper half plane h h q i by fractional linear transformations the quotient h has the structure of a compact riemann surface and will be denoted by x conway and norton cn conjectured relationships between certain congruence groups and m known as the moonshine conjectures these have been proved by borcherds

multiple solutions of h systems and rellich s conjecture soit x y r 2 x 2 y 2 1 on considere une fonction u r 3 satisfaisant le systeme h u 2hu x u y sur avec l une des conditions suivantes soit dirichlet u sur soit plateau u x 2 u y 2 u x u y 0 sur u et u non decroissante sur

mixed hodge complexes on algebraic varieties using the theory of mixed hodge modules we introduce the notion of mixed hodge complex on an algebraic variety and establish the relation between the filtered complex of du bois and the corresponding complex of mixed hodge modules some application to the du bois singularity is given

normalisers of subgroups of the modular group abstract let g be a subgroup of finite index of the modular group and let n g be the normaliser of g in psl2 r in this article we give an algorithm that determines n g

the decomposition theorem perverse sheaves and the topology of algebraic maps we give a motivated introduction to the theory of perverse sheaves culminating in the decomposition theorem of beilinson bernstein deligne and gabber a goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties while most proofs are omitted we discuss several approaches to the decomposition theorem indicate some important applications and examples

the dirichlet problem for h systems with small boundary data blowup phenomena and nonexistence results given h 3 of class c1 and bounded we consider a sequence un of solutions of the h system open image in new window in the unit open disc open image in new window satisfying the boundary condition un n on open image in new window in the first part of this paper assuming that un is bounded in h1 open image in new window 3 we study the behavior of un when the boundary data n shrink to zero we show that either un 0 strongly in h1 open image in new window 3 or un blows up at least one h bubble namely a nonconstant conformal solution of the h system on 2 under additional assumptions on h we can obtain more precise information on the blow up in the second part of this paper we investigate the multiplicity of solutions for the dirichlet problem on the disc with small boundary datum we detect a family of nonconstant functions h even close to a nonzero constant in any reasonable topology for which the dirichlet problem cannot admit a large solution at a mountain pass level when the boundary datum is small

hodge genera of algebraic varieties ii we study the behavior of hodge genera under algebraic maps we prove thatthemotivic c y genussatisfiesthe stratifiedmultiplicativeproperty whichshows how to compute the invariant of the source of a morphism from its values on varieties arising from the singularities of the map by considering morphisms to a curve we obtain a hodge theoretic version of the riemann hurwitz formula we also study the monodromy contributions to the y genus of a family of compact complex manifolds and prove an atiyah meyer type formula in the algebraic and analytic contexts this formula measures the deviation from multiplicativity of the y genus and expresses the correction terms as higher genera associated to the period map these higher gen era are hodge theoretic extensions of novikov higher signatures to analytic and alge braic settings characteristic class formulae of atiyah meyer type are also obtained by making use of saito s theory of mixed hodge modules

the structure of the tautological ring in genus one we prove getzler s claims about the cohomology of the moduli space of stable curves of genus one that is that the even cohomology ring is spanned by the strata classes and that all relations between these classes follow from the witten dijkgraaf verlinde verlinde wdvv relation and getzler s relation in particular the even cohomology ring is isomorphic to the tautological ring

h bubbles in a perturbative setting the finite dimensional reduction s method given a regular function h colon mathbb r 3 to mathbb r we look for h bubbles that is regular surfaces in mathbb r 3 parametrized on the sphere mathbb s 2 with mean curvature h at every point here we study the case of h u h 0 varepsilon h 1 u h varepsilon u where h 0 is a nonzero constant varepsilon is the smallness parameter and h 1 is any c 2 function we prove that if bar p in mathbb r 3 is a good stationary point for the melnikov type function gamma p int q p h 0 1 h 1 q dq then for varepsilon small there exists an h varepsilon bubble omega varepsilon that converges to a sphere of radius h 0 1 centered at bar p as varepsilon to 0

level and index in the modular group it is shown that the index of a congruence subgroup of the modular group cannot be less than the level of the subgroup this allows a number of existence theorems about non congruence subgroups the level of a subgroup of the modular group can be defined in terms of the action on q we define a similar action to get information on congruence subgroups in fact we get a more powerful result but this appears to be the most useful version

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

groups commensurable with the modular group abstract let a and b be maximal subgroups of psl 2 r that commensurable with psl 2 z s and t are commensurable with each other if s t is of finite index in both s and t in this article we determine the index a a b and the level of a b appendix a

motivic random variables and representation stability ii hypersurface sections abstract we prove geometric and cohomological stabilization results for the universal smooth degree d hypersurface section of a fixed smooth projective variety as d goes to infinity we show that relative configuration spaces of the universal smooth hypersurface section stabilize in the completed grothendieck ring of varieties and deduce from this the stabilization of the hodge euler characteristic of natural families of local systems constructed from the vanishing cohomology we prove explicit formulas for the stable values using a probabilistic interpretation along with the natural analogs in point counting over finite fields we explain how these results provide new geometric examples of a weak version of representation stability for symmetric symplectic and orthogonal groups this interpretation of representation stability was studied in the prequel 20 for configuration spaces

frobenius semisimplicity for convolution morphisms this article concerns properties of mixed ell adic complexes on varieties over finite fields related to the action of the frobenius automorphism we establish a fiberwise criterion for the semisimplicity and frobenius semisimplicity of the direct image complex under a proper morphism of varieties over a finite field we conjecture that the direct image of the intersection complex on the domain is always semisimple and frobenius semisimple this conjecture would imply that a strong form of the decomposition theorem of beilinson bernstein deligne gabber is valid over finite fields we prove our conjecture for generalized convolution morphisms associated with partial affine flag varieties for split connected reductive groups over finite fields as a crucial tool we develop a new schematic theory of big cells for loop groups with suitable reformulations the main results are valid over any algebraically closed ground field

multiple solutions to the plateau problem for nonconstant mean curvature we prove the existence of at least two solutions to the plateau problem for any curvature function h close enough in l norm to a nonzero constant h 0 such that h 0 1 where denotes the l norm of the boundary condition

perverse sheaves on real loop grassmannians the aim of this paper is to identify a certain tensor category of perverse sheaves on the loop grassmannian gr of a real form g of a connected reductive complex algebraic group g with the category of finite dimensional representations of a connected reductive complex algebraic subgroup check h of the dual group check g the root system of check h is closely related to the restricted root system of g the fact that check h is reductive implies that an interesting family of real algebraic maps satisfies the conclusion of the decomposition theorem of beilinson bernstein deligne

hodge genera of algebraic varieties i the aim of this paper is to study the behavior of hodge theoretic intersection homology genera and their associated characteristic classes under proper morphisms of complex algebraic varieties we obtain formulae that relate parametrized families of global invariants of a complex algebraic variety x to such invariants of singularities of proper algebraic maps defined on x such formulae severely constrain both topologically and analytically the singularities of complex maps even between smooth varieties similar results were announced by the first and third author in 13 32 2007 wiley periodicals inc

euler characteristics of algebraic varieties this note studies the behavior of euler characteristics and of intersection homology euler characteristics under proper morphisms of algebraic respectively analytic varieties the methods also yield for algebraic respectively analytic varieties formulae comparing these two kinds of euler characteristics the main results are direct consequences of the calculus of constructible functions and grothendieck groups of constructible sheaves similar formulae for hodgetheoretic invariants of algebraic varieties under morphisms were announced by the first and third authors in 5 14 c 2007 wiley periodicals inc

topology of hitchin systems and hodge theory of character varieties the case a1 for g gl2 pgl2 sl2 we prove that the perverse ltration associated with the hitchin map on the rational cohomology of the moduli space of twisted g higgs bundles on a compact riemann surface c agrees with the weight ltration on the rational cohomology of the twisted g character variety of c when the cohomologies are identied via non abelian hodge theory the proof is accomplished by means of a study of the topology of the hitchin map over the locus of integral spectral curves

on rademacher s conjecture congruence subgroups of genus zero of the modular group we list all genus zero congruence subgroups of psl2 z there are altogether 132 of them up to conjugation in psl2 z geometrical invariants genus v2 v3 number of cusps index in psl2 z fundamental polygons farey symbol and independent generators of all such groups are determined generators of the function fields associated to such groups are also determined http www math nus edu sg matlml

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