volume equivalence of subgroups of free groups abstract let f n be a free group of finite rank n 2 due to kapovich levitt schupp and shpilrain 2007 1 two finitely generated subgroups h and k of f n are called volume equivalent if for every free and discrete isometric action of f n on an r tree t we have vol t h h vol t k k we give a more algebraic and combinatorial characterization of volume equivalence and discuss a counterexample of volume equivalence in order to justify our characterization we also provide a specific example to answer a question of kapovich levitt schupp and shpilrain in the negative volume equivalence does not imply equality in rank

the griffiths group of a general calabi yau threefold is not finitely generated j 2k 1 x h 2k 1 x c f k h 2k 1 x h 2k 1 x z where f k h 2k 1 x is the set ofclasses representable by a closed form in f k a 2k 1 x that is which is locally ofthe form i j i j dzi dzj with i j 2k 1 and i k griffiths 9 has defined the abel jacobi map k k x j 2k 1 x where k x is the group ofcodimension k algebraic cycles homologous to zero on x using the identification j 2k 1 x f n k 1 h 2n 2k 1 x h2n 2k 1 x z n dim x

some spectral sequences for filtered complexes and applications abstract we construct a hierarchy of spectral sequences for a filtered complex under a left exact functor as applications we prove 1 the existence of a leray spectral sequence for de rham cohomology 2 the equivalence of this sequence with the usual leray spectral spectral sequence under the comparison isomorphism and 3 the isomorphism of the bloch ogus spectral sequence with the leray spectral sequence for the morphism from the fine site to the zariski site

unirational threefolds with no universal codimension 2 cycle we prove that the general quartic double solid with k le 7 nodes does not admit a chow theoretic decomposition of the diagonal or equivalently has a nontrivial universal mathrm ch 0 group and the same holds if we replace in this statement chow theoretic by cohomological in particular it is not stably rational we also deduce that the general quartic double solid with seven nodes does not admit a universal codimension 2 cycle parameterized by its intermediate jacobian and even does not admit a parametrization with rationally connected fibers of its jacobian by a family of 1 cycles this finally implies that its third unramified cohomology group is not universally trivial

unramified cohomology of degree 3 and noether s problem let g be a finite group and w be a faithful representation of g over c the group g acts on the field of rational functions c w the question whether the field of invariant functions c w g is purely transcendental over c goes back to emmy noether using the unramified cohomology group of degree 2 of this field as an invariant saltman gave the first examples for which c w g is not rational over c around 1986 bogomolov gave a formula which expresses this cohomology group in terms of the cohomology of the group g

torsion cohomology classes and algebraic cycles on complex projective manifolds abstract atiyah and hirzebruch gave examples of even degree torsion classes in the singular cohomology of a smooth complex projective manifold which are not poincare dual to an algebraic cycle we notice that the order of these classes must be small compared to the dimension of the manifold however building upon a construction of kollar one can provide such examples with arbitrary high prime order the dimension being fixed this method also provides examples of torsion algebraic cycles which are non trivial in the griffiths group and lie in a arbitrary high level of the h saito filtration on chow groups

2 cycles sur les hypersurfaces cubiques de dimension 5 we study 2 cycles of a smooth cubic hypersurface of dimension 5 we show that the griffiths group of 2 cycles is trivial and the abel jacobi map induces an isomorphism between the chow group of algebraically trivial 2 cycles and the intermediate jacobian

on motivic cohomology with z l coefficients in this paper we prove the conjecture of bloch and kato which relates milnor s k theory of a field with its galois cohomology as well as the related comparisons results for motivic cohomology with finite coefficients in the nisnevich and etale topologies

varietes unirationnelles non rationnelles au dela de l exemple d artin et mumford keywords non rational but unirational variety grothendieck invariants rationality brauer invariants reference cma article 1989 001doi 10 1007 bf01850658 record created on 2008 12 16 modified on 2016 08 08

stably irrational hypersurfaces of small slopes we show that over any uncountable field of characteristic different from two a very general hypersurface of dimension n 2 and degree at least log 2 n 2 is not stably rational this significantly improves earlier results of kollar and totaro as a byproduct of our proof we obtain new counterexamples to the integral hodge conjecture answering a question of voisin and colliot thelene voisin

a pencil of enriques surfaces with non algebraic integral hodge classes we prove that there exists a pencil of enriques surfaces defined over mathbb q with non algebraic integral hodge classes of non torsion type this gives the first example of a threefold with the trivial chow group of zero cycles on which the integral hodge conjecture fails as an application we construct a fourfold which gives the negative answer to a classical question of murre on the universality of the abel jacobi maps in codimension three

on the rationality problem for quadric bundles we classify all positive integers n and r such that stably nonrational complex r fold quadric bundles over rational n folds exist we show in particular that for any n and r a wide class of smooth r fold quadric bundles over pcn are not stably rational if r 2n 1 1 2n 2 in our proofs we introduce a generalization of the specialization method of voisin and of colliot thelene and pirutka which avoids universally ch0 trivial resolutions of singularities

cohomologie non ramifiee et conjecture de hodge entiere en nous appuyant sur la conjecture de bloch kato en k theorie de milnor nous etablissons un lien general entre le defaut de la conjecture de hodge entiere pour la cohomologie de degre 4 et le troisieme groupe de cohomologie non ramifiee a coefficients q z ceci permet de montrer que sur un solide en anglais threefold uniregle le troisieme groupe de cohomologie non ramifiee a coefficients q z s annule ce que la k theorie algebrique ne permet d obtenir que dans certains cas ceci permet a l inverse de deduire d exemples ayant leur source en k theorie que la conjecture de hodge entiere pour la cohomologie de degre 4 peut etre en defaut pour les varietes rationnellement connexes pour certaines familles a un parametre de surfaces on etablit un lien entre la conjecture de hodge entiere et l indice de la fibre generique

unramified cohomology of quadrics i given a quadric x over a field f of characteristic 2 we compute the kernel and cokernel of the natural map in degree 4 from the mod 2 galois cohomology of f to the unramified mod 2 cohomology of f x when dim x 10 and in several smaller dimensional cases applications of these results to real quadrics and to the unramified witt ring are given

families of rationally connected varieties recall that a proper variety x is said to be rationally connected if two general points p q x are contained in the image of a map g p x this is clearly a birationally invariant property when x is smooth this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves and also to the a priori stronger condition that for any finite subset x there is a map g p x whose image contains and such that gtx is an ample bundle

zero cycles sur les fibrations au dessus d une courbe de genre quelconque resume soit x une variete propre et lisse sur un corps de nombres k des conjectures sur l image du groupe de chow des zero cycles de x dans le produit des memes groupes sur tous les completes de k ont ete proposees par colliot thelene kato et saito nous demontrons ces conjectures pour l espace total de fibrations en varietes rationnellement connexes verifiant l approximation faible au dessus de courbes dont le groupe de tate shafarevich est fini sous une hypothese d abelianite sur les fibres singulieres

on the fibration method for zero cycles and rational points conjectures on the existence of zero cycles on arbitrary smooth projective varieties over number elds were proposed by colliot th el ene sansuc kato and saito in the 1980 s we prove that these conjectures are compatible with brations for brations into rationally connected varieties over a curve in particular they hold for the total space of families of homogeneous spaces of linear groups with connected geometric stabilisers we prove the analogous result for rational points conditionally on a conjecture on locally split values of polynomials which a recent work of matthiesen establishes in the case of linear polynomials over the rationals

on the integral hodge conjecture for real varieties i we formulate the real integral hodge conjecture a version of the integral hodge conjecture for real varieties and raise the question of its validity for cycles of dimension 1 on uniruled and calabi yau threefolds and on rationally connected varieties we relate it to the problem of determining the image of the borel haefliger cycle class map for 1 cycles with the problem of deciding whether a real variety with no real point contains a curve of even geometric genus and with the problem of computing the torsion of the chow group of 1 cycles of real threefolds new results about these problems are obtained along the way

on the unramified cohomology of certain quotient varieties in this note we consider unramified cohomology with mathbb z 2 coefficients for some degree two quotient varieties and describe a method that allows one to prove the non vanishing of these groups under certain conditions we apply this method to prove a non vanishing statement in the case of kummer varieties combining this with work of colliot thelene and voisin we obtain a new type of three dimensional counterexample to the integral hodge conjecture

the gersten conjecture for milnor k theory we prove that the n th milnor k group of an essentially smooth local ring over an infinite field coincides with the n n motivic cohomology of the ring this implies levine s generalized bloch kato conjecture

efficient parallel algorithms for a class of graph theoretic problems in this paper we present efficient parallel algorithms for the following graph problems finding the lowest common ancestors for vertex pairs of a directed tree finding all fundamental cycles a directed spanning forest all bridges all bridge connected components all separation vertices all biconnected components and testing the biconnectivity of an undirected graph all these algorithms achieve the o lg 2 n time bound with the first two algorithms using n lceil n lg n rceil processors and the remaining algorithms using n lceil n lg 2 n rceil processors in all cases our algorithms are better than the previously known algorithms and in most cases reduce the number of processors used by a factor of n lg n moreover our algorithms are optimal with respect to the time processor product for dense graphs with the exception of the first two algorithms the machine model we use is the pram which is a simd model allowing simultaneous reads but not simultaneous writes to the same memory

torsion algebraic cycles and complex cobordism atiyah and hirzebruch gave the first counterexamples to the hodge conjecture with integral coefficients 3 that conjecture predicted that every integral cohomology class of hodge type p p on a smooth projective variety should be the class of an algebraic cycle but atiyah and hirzebruch found additional topological properties which must be satisfied by the integral cohomology class of an algebraic cycle here we provide a more systematic explanation for their results by showing that the classical cycle map from algebraic cycles modulo algebraic equivalence to integral cohomology factors naturally through a topologically defined ring which is richer than integral cohomology the new ring is based on complex cobordism a well developed topological theory which has been used only rarely in algebraic geometry since hirzebruch used it to prove the riemann roch theorem 17 this factorization of the classical cycle map implies the topological restrictions on algebraic cycles found by atiyah and hirzebruch it goes beyond their work by giving a topological method to show that the classical cycle map can be noninjective as well as nonsurjective the kernel of the classical cycle map is called the griffiths group and the topological proof here that the griffiths group can be nonzero is the first proof of this fact which does not use hodge theory the proof here gives nonzero torsion elements in the griffiths group whereas griffiths s hodge theoretic proof gives nontorsion elements 13 this topological argument also gives examples of algebraic cycles in the kernel of various related cycle maps where few or no examples were known before thus answering some questions posed by colliot thelene and schoen 8 p 14 37 p 13 colliot thelene asked in particular whether the map ch2 x n h4 x z n is injective for all smooth complex projective varieties x here ch x is the group of codimension i algebraic cycles modulo rational equivalence the first examples where colliot thelene s map is not injective were found by kollar and van geemen see 4 p 135 very recently bloch and esnault found examples defined over number fields 7 over nonalgebraically closed fields k there are other examples of smooth projective varieties xk with ch2 xk n hg4t xk z n not injective due to colliot thelene and sansuc as reinterpreted by salberger see 9 and 8 remark 7 6 1 and parimala and suresh 31 these elements of ch2 xk n are not shown to remain nonzero in ch2 xc n however here our topological method

on the locus of hodge classes let s be a complex algebraic variety and xs s s a family of non singular projective varieties parametrized by s the xs are the fibers of f x s with x projective and smooth over s fix s s an integer p and a class h h xs z of hodge type p p let u be an open simply connected neighborhood of s the h xt z t s form a local system on s necessarily trivial on u so that for t u they can all be identified with h xs z the hodge filtration ft of h xt c t u can be viewed as a variable filtration on the fixed complex vector space h xs c it varies holomorphically with t it follows that the locus t u where h remains of type p p i e in f is a complex analytic subspace of u