Showing papers in "Commentarii Mathematici Helvetici in 2008"
TL;DR: In this paper, a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary algebras, was given. But this interpretation is restricted to finite types.
Abstract: Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary alge- bras. Using this, we obtain a representation theoretic interpretation of cluster mutation in case of acyclic cluster algebras of finite type.
171 citations
TL;DR: In this paper, the authors estimate the diameter of a closed in-dimensional manifold W immersed in R-n in terms of its mean curvature H by considering the mean curvatures of vertical bars.
Abstract: Given a closed in-dimensional manifold W immersed in R-n, we estimate its diameter d in terms of its mean curvature H by
d <= C(m)integral(M)vertical bar H vertical bar(m-1)(d mu).
90 citations
TL;DR: In this article, the authors show that automorphisms of the free group strongly resemble straight-line programs, which are widely studied in the theory of compressed data structures and give a detailed exposition of the necessary results from computer science.
Abstract: We find polynomial-time solutions to the word problem for free-by-cyclic groups, the word problem for automorphism groups of free groups, and the membership problem for the handlebody subgroup of the mapping class group. All of these results follow from observing that automorphisms of the free group strongly resemble straight-line programs, which are widely studied in the theory of compressed data structures. In an effort to be self-contained we give a detailed exposition of the necessary results from computer science.
50 citations
TL;DR: In this article, a proof of the integrality of Perrin-Riou's exponential was given using the theory of π, π-Gamma modules and Iwasawa-theoretic descent techniques.
Abstract: Let $K$ be a finite unramified extension of $\Qp$ and let $V$ be a crystalline representation of $\mathrm{Gal}(\Qpbar/K)$. In this article, we give a proof of the $C_{\mathrm{EP}}(L,V)$ conjecture for $L \subset \Qp^{\mathrm{ab}}$ as well as a proof of its equivariant version $C_{\mathrm{EP}}(L/K,V)$ for $L \subset \cup_{n=1}^\infty K(\zeta_{p^n})$. The main ingredients are the $\delta_{\Zp}(V)$ conjecture about the integrality of Perrin-Riou's exponential, which we prove using the theory of $(\phi,\Gamma)$-modules, and Iwasawa-theoretic descent techniques used to show that $\delta_{\Zp}(V)$ implies $C_{\mathrm{EP}}(L/K,V)$.
45 citations
TL;DR: In this article, the authors combine tools and ideology from two different fields, symplectic geometry and asymptotic geometric analysis, to arrive at some new results, including a dimension-independent bound for the symplectic capacity of a convex body.
Abstract: In this work we bring together tools and ideology from two different fields, symplectic geometry and asymptotic geometric analysis, to arrive at some new results. Our main result is a dimension-independent bound for the symplectic capacity of a convex body ,
44 citations
TL;DR: In this paper, a sharp vanishing theorem for the L p cohomology torsion of Riemannian manifolds with pinched negative curvature is given, and it follows that certain negatively curved homogeneous spaces cannot be quasiisometric to better pinched manifolds.
Abstract: A sharp vanishing theorem for the L p cohomology torsion of Riemannian manifolds with pinched negative curvature is given. It follows that certain negatively curved homogeneous spaces cannot be quasiisometric to better pinched manifolds.
42 citations
TL;DR: In this article, it was shown that for any line bundle L over X, the rank-p vector bundle FL is stable and the rational map V has base points, i.e., there exist stable bundles E over X1 such that FE is not semistable.
Abstract: Let X be a smooth projective curve of genus g ≥ 2 over an algebraically closed field k of characteristic p > 0. Let MX be the moduli space of semistable rank-2 vector bundles over X with trivial determinant. The relative Frobenius map F : X → X1 induces by pull-back a rational map V : MX1 99K MX. In this paper we show the following results. (1) For any line bundle L over X, the rank-p vector bundle FL is stable. (2) The rational map V has base points, i.e., there exist stable bundles E over X1 such that FE is not semistable. (3) Let B ⊂ MX1 denote the scheme-theoretical base locus of V. If g = 2, p > 2 and X ordinary, then B is a 0-dimensional local complete intersection of length 2 p(p 2 − 1) and the degree of V equals 1 p(p 2 + 2).
36 citations
TL;DR: In this paper, the authors constructed a minimal algebraic surface S of general type over the complex numbers with K 2 = 45 and pg = 4, and with birational canonical map.
Abstract: Answering a question posed by Enriques, we construct a minimal smooth algebraic surface S of general type over the complex numbers with K 2 = 45 and pg = 4, and with birational canonical map. The canonical system |KS| has a fixed part and the degree of the canonical image is 19. The surface we construct is rigid, S is indeed a ball quotient. It is obtained as an Abelian covering of the plane branched over an arrangement of lines already considered by Hirzebruch, and it is the first such example which is regular (q = 0).
34 citations
TL;DR: In this article, a lower bound on the first Betti number was shown to be sufficient for the local regularity theorem, but not a full Sobolev constant bound for the volume growth assumption.
Abstract: We make some improvements to our previous results in [TV05a] and [TV05b]. First, we prove a version of our volume growth theorem which does not require any assumption on the first Betti number. Second, we show that our local regularity theorem only requires a lower volume growth assumption, not a full Sobolev constant bound. As an application of these results, we can weaken the assumptions of several of our theorems in [TV05a] and [TV05b].
32 citations
TL;DR: In this article, it was shown that the separating systole of a closed Riemannian surface M of genus and area g satisfies an upper bound similar to M.C. Gromov's asymptotic estimate on the (homotopy) systoles.
Abstract: The separating systole on a closed Riemannian surface M, denoted by sys0(M) ,i s defined as the length of the shortest noncontractible loops which are homologically trivial. We answer positively a question of M. Gromov (Gr96, 2.C.2.(d)) about the asymptotic estimate on the separating systole. Specifically, we show that the separating systole of a closed Riemannian surface M of genus and area g satisfies an upper bound similar to M. Gromov's asymptotic estimate on the (homotopy) systole. That is, sys0(M) log g.
32 citations
Abstract: In this paper we fully describe the rational homotopy Lie algebra of any component of a given (free or pointed) function space. Also, we characterize higher order Whitehead products on these spaces. From this, we deduce the existence of H-structures on a given component of a pointed mapping space F*(X,Y;f) between rational spaces, assuming the cone length of X is smaller than the order of any non trivial generalized Whitehead product in p*(Y)
TL;DR: In this article, the authors consider magnetic geodesic flows of the normal metrics on a class of homogeneous spaces, in particular (co)adjoint orbits of compact Lie groups.
Abstract: We consider magnetic geodesic flows of the normal metrics on a class of homogeneous spaces, in particular (co)adjoint orbits of compact Lie groups. We give the proof of the non-commutative integrability of flows and show, in addition, for the case of (co)adjoint orbits, the usual Liouville integrability by means of analytic integrals. We also consider the potential systems on adjoint orbits, which are generalizations of the magnetic spherical pendulum. The complete integrability of such system is proved for an arbitrary adjoint orbit of a compact semisimple Lie group. to appear in Commentarii Mathematici Helvetici
TL;DR: In this paper, the existence of a complete, properly embedded, genus-one minimal surface that is asymptotic to a helicoid at infinity was proved using variational methods.
Abstract: In this paper, we use variational methods to prove existence of a complete, properly embedded, genus-one minimal surface that is asymptotic to a helicoid at infinity. We also prove some new properties of such helicoid-like surfaces.
TL;DR: In this paper, the limiting behavior of parabolic and hyperbolic Eisenstein series on a degenerating family of finite volume Riemann surfaces was studied. And the following result was proved.
Abstract: Let G be a Fuchsian group of the first kind acting on the hyperbolic upper half plane H, and let M = G \ H be the associated finite volume hyperbolic Riemann surface. If ? is parabolic, there is an associated (parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If ? is hyperbolic, then, following ideas due to Kudla�Millson, there is a corresponding hyperbolic Eisenstein series. In this article, we study the limiting behavior of parabolic and hyperbolic Eisenstein series on a degenerating family of finite volume hyperbolic Riemann surfaces. In particular, we prove the following result. If ? ? G corresponds to a degenerating hyperbolic element, then a multiple of the associated hyperbolic Eisenstein series converges to parabolic Eisenstein series on the limit surface.
TL;DR: In this paper, the constructible Witt groups of triangulated categories of sheaves (of modules over a ring R in which 2 is invertible) equipped with Poincare-Verdier duality were investigated.
Abstract: This paper investigates the Witt groups of triangulated categories of sheaves (of modules over a ring R in which 2 is invertible) equipped with Poincare-Verdier duality. We consider two main cases, that of perfect complexes of sheaves on locally compact Hausdorff spaces and that of cohomologically constructible complexes of sheaves on polyhedra. We show that the Witt groups of the latter form a generalised homology theory for polyhedra and continuous maps. Under certain restrictions on the ring R, we identify these constructible Witt groups of a finite simplicial complex with Ranicki's free symmetric L-groups. Witt spaces are the natural class of spaces for which the rational intersection homology groups have Poincare duality. When the ring R is the rationals we identify the constructible Witt groups with the 4-periodic colimit of the bordism groups of PL Witt spaces. This allows us to interpret L-classes of singular spaces as stable homology operations from the constructible Witt groups to rational homology.
TL;DR: In this article, the Galois groupoid of the first Painleve equation is computed using E. Cartan's classification of structural equations of pseudogroups acting on $C^2$ and the degeneration of the Painlesve equation on an elliptic equation, and a definition of reducibility for singular holomorphic foliations is proposed.
Abstract: In this article, the Galois groupoid of the first Painleve equation is computed. This computation use E. Cartan's classification of structural equations of pseudogroups acting on $C^2$ and the degeneration of the first Painleve equation on an elliptic equation ($y'' = 6y^2$). A definition of reducibility for singular holomorphic foliations is proposed. A characterisation of reducible foliations on their Galois groupoid is given and applied to prove the foliation irreducibility of the first Painleve equation.
TL;DR: The main result of as discussed by the authors is a classification of rational or Fano manifolds of degree d = n. The proof of their theorem makes essential use of the adjunction mapping and, in particular, plays a crucial role in the argument.
Abstract: Let X ? Pn be a complex connected projective, non-degenerate, linearly normal manifold of degree d = n. The main result of this paper is a classification of such manifolds. As a by-product of the classification it follows that these manifolds are either rational or Fano. In particular, they are simply connected (hence regular) and of negative Kodaira dimension. Moreover, easy examples show that d = n is the best possible bound for such properties to hold true. The proof of our theorem makes essential use of the adjunction mapping and, in particular, the main result of [15] plays a crucial role in the argument.
TL;DR: In this article, the space of Riemannian metrics for which the Dirac operator is invertible was studied on a compact spin manifold and the first main result is a surgery theorem stating that such a metric can be extended.
Abstract: On a compact spin manifold we study the space of Riemannian metrics for which the Dirac operator is invertible. The first main result is a surgery theorem stating that such a metric can be extended ...
TL;DR: In this article, certaines familles d'espaces de longueur compacts have been examined and shown to be complete for the distance of Gromov�Hausdorff equivariante.
Abstract: Nous etudions certaines familles d'espaces de longueur compacts dont l'entropie volumique est majoree. Nous montrons que ces familles sont completes pour la distance de Gromov�Hausdorff et nous prouvons l'existence d'une constante explicite e0 > 0 telle que, sur les boules de rayon e0 pour la distance de Gromov�Hausdorff, le groupe fondamental est constant, les revetements universels sont proches pour la distance de Gromov�Hausdorff equivariante, le spectre des longueurs est continu, l'entropie est Lipschitzienne. Si l'on se restreint a certains sous-ensembles des varietes riemanniennes compactes, nous montrons de plus que, sur ces boules de rayon e0, le volume est semi-continu inferieurement et que l'integrale de la courbure de Ricci est minoree uniformement.
TL;DR: In this paper, the authors give the first counterexamples to the Nielsen realization problem about lifting homotopy actions of finite groups to honest group actions, showing that one cannot guarantee that a given action of a finitely generated group π on Euclidean space extends to an action of �, a group containing π as a subgroup of finite index.
Abstract: The main goal of this paper is to give the first examples of equivariant aspherical Poincare complexes, that are not realized by group actions on closed aspherical manifolds M. These will also provide new counterexamples to the Nielsen realization problem about lifting homotopy actions of finite groups to honest group actions. Our examples show that one cannot guarantee that a given action of a finitely generated group π on Euclidean space extends to an action of � , a group containing π as a subgroup of finite index, even when all the torsion of � lives in π.
TL;DR: In this paper, it was shown that there exists a unique Jenkins-Strebel differential on the Riemann surface with prescribed heights, and that the differential has second order poles at distinguished punctures with prescribed leading coefficients.
Abstract: Given any compact Riemann surface with finitely many punctures, we show that there exists a unique Jenkins-Strebel differential on the Riemann surface with prescribed heights. In addition, the differential has second order poles at the distinguished punctures with prescribed leading coefficients. As a corollary, we obtain the solution of the moduli problem.
TL;DR: In this paper, the spectre, the volume, and the classe conformé de Hodge-de-Rham en restriction aux formes de degre $p\in[2,n-2], en excluant $p=n/2$ si $n$ est pair, and imposant a metrique d'appartenir a une classe conforme donnee.
Abstract: Sur toute variete compacte de dimension $n\geq5$, on prescrit le volume et toute partie finie du spectre du laplacien de Hodge-de~Rham en restriction aux formes de degre $p\in[2,n-2]$, en excluant $p=n/2$ si $n$ est pair, et en imposant a la metrique d'appartenir a une classe conforme donnee. On sait que pour $n\leq4$, ainsi que pour $p=0,1,n-1,n$, et $p=n/2$ si $n$ est pair, on ne peut pas prescrire simultanement le spectre, le volume et la classe conforme.
TL;DR: In this paper, it was shown that for a smooth subcanonical subvariety X ⊂ P, n ≥ 5 (the degree d, the integer e such that ωX OX(e), the least degree, s, of a hypersurface containing X, s ≥ n+ 1 if X is not a complete intersection).
Abstract: Let X ⊂ P be a smooth codimension 2 subvariety. We first prove a “positivity lemma” (Lemma 1.1) which is a direct application of the positivity of NX(−1). Then we first derive two consequences: 1) Roughly speaking the family of “biliaison classes” of smooth subvarieties of P5 lying on a hypersurface of degree s is limited. 2) The family of smooth codimension 2 subvarieties of P6 lying on a hypersurface of degree s is limited. The result in 1) is not effective, but 2) is. Then we obtain precise inequalities connecting the usual numerical invariants of a smooth subcanonical subvariety X ⊂ P, n ≥ 5 (the degree d, the integer e such that ωX OX(e), the least degree, s, of a hypersurface containing X). In particular we prove: s ≥ n+ 1 if X is not a complete intersection. Mathematics Subject Classification (2000). 14MO7, 14MlO, 14C20.
TL;DR: For countable groups, a criterion for irreducible representability is given in this paper, which generalises a result obtained for finite groups by W. Gaschutz in 1954.
Abstract: A group is irreducibly represented if it has a faithful irreducible unitary representation. For countable groups, a criterion for irreducible representability is given, which generalises a result obtained for finite groups by W. Gaschutz in 1954. In particular, torsionfree groups and infinite conjugacy class groups are irreducibly represented. We indicate some consequences of this for operator algebras. In particular, we characterise up to isomorphism the countable subgroups of the unitary group of a separable infinite dimensional Hilbert space H of which the bicommutants '' (in the sense of the theory of von Neumann algebras) coincide with the algebra of all bounded linear operators on H.
TL;DR: In this article, it was shown that a biLipschitz map with a constant close to one can be pointwise approximated, quantitatively on any xed ball, by an isometry.
Abstract: In this paper we prove some approximation results for biLipschitz maps in the Heisenberg group. Namely, we show that a biLipschitz map with biLipschitz constant close to one can be pointwise approximated, quantitatively on any xed ball, by an isometry. This leads to an approximation in BMO norm for the map’s Pansu derivative. We also prove that a global quasigeodesic can be approximated by a geodesic on any xed segment.
TL;DR: In this paper, it was shown that a positive semidefinite analytic function can be expressed as a sum of squares of global meromorphic functions on a neighborhood of a curve.
Abstract: Among the invariant factors g of a positive semidefinite analytic function f on R-3, those g whose zero set Y is a curve are called special. We show that if each special g is a sum of squares of global meromorphic functions on a neighbourhood of Y, then f is a sum of squares of global meromorphic functions. Here sums can be (convergent) infinite, but we also find some sufficient conditions to get finite sums of squares. In addition, we construct several examples of positive semidefinite analytic functions which are infinite sums of squares but maybe could not be finite sums of squares.
TL;DR: In this paper, a concrete description of the whitehead group for the simply connected triality forms G of k-rank 1 is given, and it is shown that if k is a global field, then the Kneser�Tits problem for these forms has an affirmative solution.
Abstract: The purpose of this paper is to provide a concrete description of the �Whitehead group�� W(k,G) := G(k)/G(k)+ for the simply connected triality forms G of k-rank 1, and to use this description to prove that if k is a global field, then the Kneser�Tits problem for these forms has an affirmative solution.
TL;DR: In this article, the Hopf boundary point lemma for sections of a vector bundle over a manifold with boundary was established, which may be viewed as a counterpart to the tensor maximum principle obtained by R. Hamilton in 1986.
Abstract: The paper establishes a version of the Hopf boundary point lemma for sections of a vector bundle over a manifold with boundary. This result may be viewed as a counterpart to the tensor maximum principle obtained by R. Hamilton in 1986. Potential applications include the study of various geometric flows and the construction of invariant sets for geometric boundary value problems.
TL;DR: In this paper, the authors introduce the space of cycles of type fini (les cycles analytiques fermes n'ayant qu'un nombre fini de composantes irreductible) which is a complex space analytique complexe donne de dimension finie, muni d'une topologie naturelle.
Abstract: Cet article presente un nouveau point de vue a propos des principaux resultats de David Mathieu [M00] sur les relations d'equivalence meromorphes. Nous introduisons l'espace des cycles de type fini (les cycles analytiques fermes n'ayant qu'un nombre fini de composantes irreductibles) d'un espace analytique complexe donne de dimension finie, muni d'une topologie naturelle, ce qui permet d'eviter la condition de («?regularite?» des familles analytiques de cycles qui est utilisee dans loc. cit. et egalement les deux notions de «?fuite a l'infini?» qui sont ici encodees de facon naturelle dans notre contexte. Les resultats obtenus sont meilleurs et surtout d'enonces et d'utilisation beaucoup plus simples. Ils contiennent, avec un langage un peu different, une version plus claire et plus generale des travaux de H. Grauert [G83] et [G86] et de B. Siebert [S93] et [S94] sur les relations d'equivalence meromorphes.
TL;DR: In this paper, the authors characterize connected Lie groups that have a dense, finitely generated subgroup with property (T) and show that T is the property of the subgroup.
Abstract: We characterize connected Lie groups that have a dense, finitely generated subgroup with Property (T)