Showing papers in "Crelle's Journal in 2010"

Noncompact shrinking four solitons with nonnegative curvature

Abstract: Abstract We prove that if (M, g, X) is a noncompact four dimensional shrinking soliton with bounded nonnegative curvature operator, then (M, g) is isometric to or a finite quotient of or S 3 × ℝ. In the process we also show that a complete shrinking soliton (M, g, X) with bounded curvature is gradient and κ-noncollapsed and the dilation of a Type I singularity is a shrinking soliton. Further in dimension three we show shrinking solitons with bounded curvature can be classified under only the assumption of Rc ≧ 0. The proofs rely on the technical construction of a singular reduced length function, a function which behaves as the reduced length function but can be extended to singular times.

The Jiang-Su algebra revisited

Abstract: We give a number of new characterizations of the Jiang–Su algebra Z, both intrinsic and extrinsic, in terms of C∗-algebraic, dynamical, topological and K-theoretic conditions. Along the way we study divisibility properties of C∗-algebras, we give a precise characterization of those unital C∗-algebras of stable rank one that admit a unital embedding of the dimensiondrop C∗-algebra Zn,n+1, and we prove a cancellation theorem for the Cuntz semigroup of C∗-algebras of stable rank one.

Smooth toric Deligne-Mumford stacks

Abstract: We give a geometric definition of smooth toric Deligne-Mumford stacks using the action of a“torus”. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a stacky fan. We also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks.

Free analysis questions II: The Grassmannian completion and the series expansions at the origin

Abstract: The fully matricial generalization in part I, of the difference quotient derivation on holomorphic functions, in which ${\mathbb C}$ is replaced by a Banach algebra $B$, is extended from the affine case to a Grassmannian completion. The infinitesimal bialgebra duality, the duality transform generalizing the Stieltjes transform and the spectral theory with non-commuting scalars all extend to this completion. The series expansions of fully matricial analytic functions are characterized, providing a new way to generate fully matricial functions.

Homeomorphisms in the Sobolev space W 1,n–1

Abstract: Abstract Let be open. We show that each homeomorphism satisfies . If we moreover assume that ƒ has finite distortion, then and ƒ–1 has finite distortion. The main ingredient is a new result on change of variables in integral (area and coarea formula) for such mappings.

Hyper-Kähler fourfolds and Grassmann geometry

Abstract: We construct a new 20-dimensional family of projective hyper-Kahler fourfolds and prove that they are deformationequivalent to the second punctual Hilbert scheme of a K3 surface of genus 12.

Decomposition of residue currents

Abstract: Given a submodule J ⊂ O⊕r 0 and a free resolution of J one can de ne a certain vector-valued residue current whose annihilator is J . We make a decomposition of the current with respect to AssJ that corresponds to a primary decomposition of J . As a tool we introduce a class of currents that includes usual residue and principal value currents; in particular these currents admit a certain type of restriction to analytic varieties and more generally to constructible sets.

The dirac operator on compact quantum groups

Abstract: For the q-deformation Gq, 0 < q < 1, of any simply connected simple compact Lie group G we construct an equivariant spectral triple which is an isospectral deformation of that defined by the Dirac operator D on G. Our quantum Dirac operator Dq is a unitary twist of D considered as an element of Ug Cl(g). The commutator of Dq with a regular function on Gq consists of two parts. One is a twist of a classical commutator and so is automatically bounded. The second is expressed in terms of the commutator of the associator with an extension of D. We show that in the case of the Drinfeld associator the latter commutator is also bounded.

Crossed products by minimal homeomorphisms

Abstract: Let X be an infinite compact metric space with finite covering dimension and let h : X ! X be a minimal homeomorphism. We show that the associated crossed product C -algebra A 1⁄4 C ðZ;X ; hÞ has tracial rank zero whenever the image of K0ðAÞ in A¤ TðAÞ is dense. As a consequence, we show that these crossed product C -algebras are in fact simple AH algebras with real rank zero. When X is connected and h is further assumed to be uniquely ergodic, then the above happens if and only if the rotation number associated to h has irrational values. By applying the classification result for nuclear simple C -algebras with tracial rank zero, we show that two such dynamical systems have isomorphic crossed products if and only if they have isomorphic scaled ordered K-theory.

Subshifts and perforation

Abstract: We demonstrate that the perforative phenomena shown to occur among simple amenable C -algebras by Villadsen and Toms can be realized within a dynamical framework. More specically, we construct a minimal homeomorphism for which the K0 group of the crossed product fails to be weakly unperforated, and a minimal homeomor- phism for which the crossed product has the same Elliott invariant as an AT-algebra but has Cuntz semigroup which fails to be almost unperforated.

La filtration de Harder-Narasimhan des schémas en groupes finis et plats

Abstract: (rang et degre) Elle n’est pas abelienne mais verifie tout de meme les proprietes suivantes : – Soit η le point generique de la courbe Les germes de fibres sur des ouverts non vides de X forment une categorie abelienne Cη, la categorie des OX,η-espaces vectoriels de dimension finie – Il y a un foncteur fibre generique C −→ Cη – Si E est un fibre vectoriel sur X , il y a un foncteur d’adherence schematique des sous-fibres de Eη vers ceux de E et les deux foncteurs precedents induisent des bijections inverses

Kuga-Satake abelian varieties of K3 surfaces in mixed characteristic

Abstract: Abstract Kuga and Satake associate with every polarized complex K3 surface (X, ℒ) a complex abelian variety called the Kuga-Satake abelian variety of (X, ℒ). We use this construction to define morphisms between moduli spaces of polarized K3 surfaces with certain level structures and moduli spaces of polarized abelian varieties with level structure over ℂ. In this note we study these morphisms. We prove first that they are defined over finite extensions of ℚ. This is done by proving analogues of the main theorems of complex multiplication for abelian varieties for K3 surfaces. Then we show that they extend in positive characteristic. In this way we give an indirect construction of Kuga-Satake abelian varieties over an arbitrary base. We also give some applications of this construction to canonical lifts of ordinary K3 surfaces.

Kähler-Ricci solitons on homogeneous toric bundles

Abstract: This is the first of a sequence of two papers. Here, a simple alge- braic characterization of the Fano manifolds in the class of homogeneous toric bundles over a flag manifold G C /P is provided in terms of symplectic data. The result of this paper is used in the second paper, where it is proved that an homogeneous toric bundle over a flag manifold admits a Kahler-Ricci solitonic metric if and only if it is Fano.

Conic-connected manifolds

Abstract: We study a particular class of rationally connected manifolds, $X\subset \p^N$, such that two general points $x,x' \in X$ may be joined by a conic contained in $X$. We prove that these manifolds are Fano, with $b_2\leq 2$. Moreover, a precise classification is obtained for $b_2=2$. Complete intersections of high dimension with respect to their multi-degree provide examples for the case $b_2=1$. The proof of the classification result uses a general characterization of rationality, in terms of suitable covering families of rational curves.

Classification of the simple factors appearing in composition series of totally disconnected contraction groups

Abstract: Let G be a totally disconnected, locally compact group admitting a contractive automorphism \alpha. We prove a Jordan-Holder theorem for series of \alpha-stable closed subgroups of G, classify all possible composition factors and deduce consequences for the structure of G.

On the facial structure of the unit ball in a JB*-triple

Abstract: Abstract It is shown that every norm-closed face of the closed unit ball A 1 in a JB*-triple A is norm-semi-exposed, thereby completing the description of the facial structure of A 1.

The twisted fourth moment of the riemann zeta function

Abstract: We compute the asymptotics of the fourth moment of the Riemann zeta func- tion times an arbitrary Dirichlet polynomial of length T 1 11 " . The study of the moments of the Riemann zeta function has a long and distinguished history, starting with the work of Hardy and Littlewood in 1918 and continuing to the present day. One motivation for understanding moments is that they yield information about the maximum size of the zeta function (the Lindelof Hypothesis); another application is to zero density estimates which in turn have consequences for primes in short intervals. However they have become an interesting topic in their own right. Very few rigorous results are known, just the second and fourth power moments. Indeed, it is only recently that a believable conjecture for higher powers has been made. The twisted moments (that is, moments of the Riemann zeta function times an arbitrary Dirichlet polynomial) are important too, for example Levinson's method of detecting zeros of the zeta function lying on the critical line requires knowing the asymptotics of the mollified second moment. In a series of papers, Duke, Friedlander, and Iwaniec used estimates for amplified moments of a family of L-functions in order to deduce a subconvexity bound for an individual member of the family. Of course, there are far easier methods to give a subconvexity bound for zeta, but there are close analogies between different families and it is desirable to understand the structure of these amplified moments in general. In this paper, we prove an asymptotic formula for the twisted fourth moment of the Riemann zeta function, where we may take a Dirichlet polynomial of length up to T 1 11 −e .

Monotone volume formulas for geometric flows

Abstract: We consider a closed manifold M with a Riemannian metric gij(t) evolving by @t gij = 2Sij where Sij(t) is a symmetric two-tensor on (M;g(t)). We prove that if Sij satises the tensor inequality D(Sij;X) 0 for all vector elds X on M, where D(Sij;X) is dened in (1.6), then one can construct a forwards and a backwards reduced volume quantity, the former being non-increasing, the latter being non-decreasing along the ow @t gij = 2Sij. In the case where Sij = Rij, the Ricci curvature of M, the result corresponds to Perelman’s well-known reduced volume monotonicity for the Ricci

Moduli of parabolic Higgs bundles and Atiyah algebroids

Abstract: In this paper we study the geometry of the moduli space of (non-strongly) parabolic Higgs bundles over a Riemann surface with marked points. We show that this space possesses a Poisson structure, extending the one over the dual of an Atiyah al- gebroid over the moduli space of parabolic vector bundles. By con- sidering the case of full flags, we get a Grothendieck-Springer reso- lution for all other flag types, in particular for the moduli spaces of twisted Higgs bundles, as studied by Markman and Bottacin and used in the recent work of Laumon-Ngo. We discuss the Hitchin system, and demonstrate that all these moduli spaces are inte- grable systems in the Poisson sense.

Über Pro-p-Fundamentalgruppen markierter arithmetischer Kurven

Abstract: Let k be a global field, p an odd prime number dierent from char (k) and S, T disjoint, finite sets of primes of k. Let G T(k)(p) = G(k T(p)|k) be the Galois group of the maximal p-extension of k which is unramified outside S and completely split at T. We prove the existence of a finite set of primes S0, which can be chosen disjoint from any given setM of Dirichlet density zero, such that the cohomology of G T[S0 (k)(p) coincides with the etale cohomology of the associated marked arithmetic curve. In particular, cd G T[S0 (k)(p) = 2. Furthermore, we can choose S0 in such a way that k T S[S0 (p) realizes the maximal p-extension kp(p) of the local field kp for all p2 S[S0, the cup-product H 1 (G T[S0 (k)(p),Fp) H 1 (G T[S0 (k)(p),Fp)! H 2 (G T[S0 (k)(p),Fp) is surjective and the decomposition groups of the primes in S establish a free product inside G T[S0 (k)(p). This generalizes previous work of the author where similar results were shown in the case T = ? under the restrictive assumption p - #Cl(k) and p / 2 k.

Waring's problem in function fields

Abstract: Let Fq(t) denote the ring of polynomials over the nite eld Fq of characteristic p, and write J k (t) for the additive closure of the set of kth powers of polynomials in Fq(t). Dene Gq(k) to be the least integer s satisfying the property that every polynomial in J k (t) of suciently large degree admits a strict representation as a sum of s kth powers. We employ a version of the Hardy-Littlewood method involving the use of smooth polynomials in order to establish a bound of the shape Gq(k) Ck logk + O(k log logk). Here, the coecient C is equal to 1 when k p, but in any case satises C 4=3. There are associated conclusions for the solubility of diagonal equations over Fq(t), and for exceptional set estimates in Waring's problem. Z to Fq(t), therefore, is the derivation of results uniform in the characteristic. In this paper we investigate the analogue of Waring's problem over Fq(t), our aim being to establish conclusions that are relatively robust to changes in the characteristic of Fq. We concentrate, in particular, on methods having the potential to impact questions that concern the density of rational points on algebraic varieties in function elds, a topic to which we intend to return on a future occasion. Some preparation is required before we can announce our principal conclusions. Let k be an integer with k 2, let s2 N, and consider a polynomial m in Fq(t). We seek to

Smoothness of Lipschitz minimal intrinsic graphs in Heisenberg groups H^n

Abstract: We prove that Lipschitz intrinsic graphs in the Heisenberg groups H n , with n > 1, which are vanishing viscosity solutions of the minimal surface equation are smooth.

Graph algebras, exel-laca algebras, and ultragraph algebras coincide up to morita equivalence

Abstract: We prove that the classes of graph algebras, Exel-Laca algebras, and ultra- graph algebras coincide up to Morita equivalence. This result answers the long-standing open question of whether every Exel-Laca algebra is Morita equivalent to a graph algebra. Given an ultragraph G we construct a directed graph E such that C � (G) is isomorphic to a full corner of C � (E). As applications, we characterize real rank zero for ultragraph algebras and describe quotients of ultragraph algebras by gauge-invariant ideals.

Adjoint ideals along closed subvarieties of higher codimension

Abstract: In this paper, we introduce a notion of adjoint ideal sheaves along closed subvarieties of higher codimension and study its local properties using char- acteristic p methods. When X is a normal Gorenstein closed subvariety of a smooth complex variety A, we formulate a restriction property of the adjoint ideal sheaf adj X (A) of A along X involving the l.c.i. ideal sheaf DX of X. The proof relies on a modification of generalized test ideals of Hara and Yoshida (11).

Duality theorems for slice hyperholomorphic functions

Abstract: Abstract The aim of this paper is to provide a characterization of the dual of the ℝ n -module of slice monogenic functions on a class of compact sets in the Euclidean space . Despite the fact that the Cauchy formulas which are essential to such a characterization are based on different kernels, depending on whether one considers right or left slice monogenic functions, we are still able to establish a duality theorem which, since holomorphic functions are a very special case of slice monogenic functions, is the generalization of Köthe's theorem. The duality results are also obtained in the setting of quaternionic valued slice regular functions.

Resolutions for representations of reductive p-adic groups via their buildings

Abstract: Schneider–Stuhler and Vigneras have used cosheaves on the affine Bruhat–Tits building to construct natural projective resolutions of finite type for admissible representations of reductive p-adic groups in characteristic not equal to p. We use a system of idempotent endomorphisms of a representation with certain properties to construct a cosheaf and a sheaf on the building and to establish that these are acyclic and compute homology and cohomology with these coefficients. This implies Bernstein’s result that certain subcategories of the category of representations are Serre subcategories. Furthermore, we also get results for convex subcomplexes of the building. Following work of Korman, this leads to trace formulas for admissible representations.

A remark on the codimension of the Green-Griffiths locus of generic projective hypersurfaces of high degree.

Abstract: We show that for every smooth generic projective hypersurface X in P^{n+1} there exists a proper subvariety Y in X of codimension at least 2 such that for every non-constant holomorphic map f: C--->X one has f(C) is contained in Y, provided that the degree of X is greater than 2^{n^5}. In particular we obtain an affirmative confirmation of the Kobayashi conjecture for threefolds in P^4.

Regularity of optimal transport on compact, locally nearly spherical, manifolds

Abstract: Given a couple of smooth positive measures of same total mass on a compact connected Riemannian manifold $M$, we look for a smooth optimal transportation map $G$, pushing one measure to the other at a least total squared distance cost, directly by using the continuity method to produce a classical solution of the elliptic equation of Monge--Ampere type satisfied by the potential function $u$, such that $G =\exp(\grad u)$. This approach boils down to proving an \textit{a priori} upper bound on the Hessian of $u$, which was done on the flat torus by the first author. The recent local $C^2$ estimate of Ma--Trudinger--Wang enabled Loeper to treat the standard sphere case by overcoming two difficulties, namely: in collaboration with the first author, he kept the image $G(m)$ of a generic point $m\in M$, uniformly away from the cut-locus of $m$; he checked a fourth-order inequality satisfied by the squared distance cost function, proving the uniform positivity of the so-called $c$-curvature of $M$. In the present paper, we treat along the same lines the case of manifolds with curvature sufficiently close to 1 in $C^2$ norm -- specifying and proving a conjecture stated by Trudinger.