Showing papers in "Crelle's Journal in 2007"

Sheaves on Artin stacks

Abstract: We develop a theory of quasi–coherent and constructible sheaves on algebraic stacks correcting a mistake in the recent book of Laumon and Moret-Bailly. We study basic cohomological properties of such sheaves, and prove stack–theoretic versions of Grothendieck’s Fundamental Theorem for proper morphisms, Grothendieck’s Existence Theorem, Zariski’s Connectedness Theorem, as well as finiteness Theorems for proper pushforwards of coherent and constructible sheaves. We also explain how to define a derived pullback functor which enables one to carry through the construction of a cotangent complex for a morphism of algebraic stacks due to Laumon and Moret–Bailly. 1.1. In the book ([LM-B]) the lisse-etale topos of an algebraic stack was introduced, and a theory of quasi–coherent and constructible sheaves in this topology was developed. Unfortunately, it was since observed by Gabber and Behrend (independently) that the lisse-etale topos is not functorial as asserted in (loc. cit.), and hence the development of the theory of sheaves in this book is not satisfactory “as is”. In addition, since the publication of the book ([LM-B]), several new results have been obtained such as finiteness of coherent and etale cohomology ([Fa], [Ol]) and various other consequences of Chow’s Lemma ([Ol]). The purpose of this paper is to explain how one can modify the arguments of ([LM-B]) to obtain good theories of quasi–coherent and constructible sheaves on algebraic stacks, and in addition we provide an account of the theory of sheaves which also includes the more recent results mentioned above. 1.2. The paper is organized as follows. In section 2 we recall some aspects of the theory of cohomological descent ([SGA4], V) which will be used in what follows. In section 3 we review the basic definitions of the lisse-etale site, cartesian sheaves over a sheaf of algebras, and verify some basic properties of such sheaves. In section 4 we relate the derived category of cartesian sheaves over some sheaf of rings to various derived categories of sheaves on the simplicial space obtained from a covering of the algebraic stack by an algebraic space. Loosely speaking the main result states that the cohomology of a complex with cartesian cohomology sheaves can be computed by restricting to the simplicial space obtained from a covering and computing cohomology on this simplicial space using the etale topology. In section 5 we generalize these results to comparisons between Ext–groups computed in the lisse-etale topos and Ext–groups computed using the etale topology on a hypercovering. In section 6 we specialize the discussion of sections 3-5 to quasi–coherent sheaves. We show that if X is an algebraic stack and OXlis-et denotes the structure sheaf of the lisse-etale topos, then the triangulated category D qcoh(X ) of bounded below complexes of OXlis-et–modules with quasi–coherent cohomology sheaves satisfies all the basic properties that one would expect from the theory for schemes. For example we show in this section that if f : X → Y is a quasi–compact morphism of algebraic stacks and M is a quasi–coherent sheaf on X Date: November 2, 2005. 1

Pinching estimates and motion of hypersurfaces by curvature functions

Abstract: Second derivative pinching estimates are proved for a class of elliptic and parabolic equations, including motion of hypersurfaces by curvature functions such as quotients of elementary symmetric functions of curvature. The estimates imply convergence of convex hypersurfaces to spheres under these flows, improving earlier results of B. Chow and the author. The result is obtained via a detailed analysis of gradient terms in the equations satisfied by second derivatives.

The maximum size of L-functions

Abstract: We conjecture the true rate of growth of the maximum size of the Riemann zeta-function and other L-functions. We support our conjecture using arguments from random matrix theory, conjectures for moments of L-functions, and also by assuming a random model for the primes.

On the first and second variations of a nonlocal isoperimetric problem

Abstract: We consider a nonlocal perturbation of an isoperimetric variational prob- lem. The problem may be viewed as a mathematical paradigm for the ubiquitous phenom- enon of energy-driven pattern formation associated with competing short and long-range interactions. In particular, it arises as a G-limit of a model for microphase separation of diblock copolymers. In this article, we establish precise conditions for criticality and stability (i.e. we explicitly compute the first and second variations). We also present some applications.

Bifurcation currents in holomorphic dynamics on P^k

Abstract: Potential theory has been introduced in one dimensional rational dynamics by Brolin and Tortrat ([4], [29]) but does not play a central role there. In higher dimension however, as the classical tools are not any longer efficient, pluri-potential theory has revealed itself to be essential. The fundamental works of Hubbard-Papadopol, Fornaess-Sibony, BedfordSmillie, Briend-Duval (see [26] for precise references) enlight the remarkable efficency of pluri-potential theory in holomorphic dynamics on Pk or Ck. It is therefore tempting to study the parameter spaces in a similar way. More precisely, one would like to relate the bifurcations of an holomorphic family {fλ}λ∈X of endomorphisms of Pk to the powers of a certain current on the parameter space X. Let us recall that in dimension k = 1, a bifurcation is said to occur at some point λ0 ∈ X if the Julia set of fλ does not move continuously around λ0. The famous work of Mane-SadSullivan [16], which is based on the λ-lemma and the Fatou-Cremer-Sullivan classification, relates the bifurcations with the instability of the critical orbits. It also establishes that the bifurcations concentrate on the complement of an open dense subset of X (for the quadratic family {z2 + λ}λ∈X=C the bifurcation locus is precisely the boundary of the Mandelbrot set).

First steps towards p-adic Langlands functoriality

Abstract: — By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the Weil-Deligne group equipped with an admissible filtration on the underlying vector space. Using a modification of the classical local Langlands correspondence, we associate with any pair consisting of a Weil-Deligne group representation and a type of a filtration (admissible or not) a specific locally algebraic representation of a general linear group. We advertise the conjecture that this pair comes from a de Rham representation if and only if the corresponding locally algebraic representation carries an invariant norm. In the crystalline case, the Weil-Deligne group representation is unramified and the associated locally algebraic representation can be studied using the classical Satake isomorphism. By extending the latter to a specific norm completion of the Hecke algebra, we show that the existence of an invariant norm implies that our pair, indeed, comes from a crystalline representation. We also show, by using the formalism of Tannakian categories, that this latter fact is compatible with classical unramified Langlands functoriality and therefore generalizes to arbitrary split reductive groups. Resume. — Par la theorie de Colmez et Fontaine, une representation de de Rham du groupe de Galois d’un corps local correspond essentiellement a une representation du groupe de Weil-Deligne dont l’espace sous-jacent est muni d’une filtration admissible. En modifiant la correspondance locale de Langlands, on associe a chaque couple forme d’une representation du groupe de Weil-Deligne et des poids d’une filtration (admissible ou pas) une representation localement algebrique particuliere d’un groupe lineaire general. On conjecture qu’un couple provient d’une representation de de Rham si et seulement si la representation localement algebrique correspondante possede une norme invariante. Dans le cas cristallin, la representation du groupe de Weil-Deligne est non-ramifiee et la representation localement algebrique associee peut s’etudier grâce a l’isomorphisme de Satake classique. En prolongeant ce dernier a une completion de l’algebre de Hecke, on montre que l’existence d’une norme invariante comme ci-dessus implique que le couple provient effectivement d’une representation cristalline. On montre aussi, en utilisant le formalisme des categories tannakiennes, que ce dernier fait est compatible avec la fonctorialite de Langlands non-ramifiee classique, et donc qu’il se generalise a tout groupe reductif deploye. 2 C. BREUIL & P. SCHNEIDER

The numbers of tropical plane curves through points in general position

Abstract: We show that the number of tropical curves of given genus and degree through some given general points in the plane does not depend on the position of the points. In the case when the degree of the curves contains only primitive integral vectors this statement has been known for a while now, but the only known proof was indirect with the help of Mikhalkin’s Correspondence Theorem that translates this question into the well-known fact that the numbers of complex curves in a toric surface through some given points do not depend on the position of the points. This paper presents a direct proof entirely within tropical geometry that is in addition applicable to arbitrary degree of the curves.

Pseudodifferential Operators on Locally Compact Abelian Groups and Sjoestrand's Symbol Class

Abstract: We investigate pseudodifferential operators on arbitrary locally compact abelian groups. As symbol classes for the Kohn-Nirenberg calculus we introduce a version of Sjoestrand's class. Pseudodifferential operators with such symbols form a Banach algebra that is closed under inversion. Since "hard analysis" techniques are not available on locally compact abelian groups, a new time-frequency approach is used with the emphasis on modulation spaces, Gabor frames, and Banach algebras of matrices. Sjoestrand's original results are thus understood as a phenomenon of abstract harmonic analysis rather than "hard analysis" and are proved in their natural context and generality.

A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates

Abstract: In this paper we obtain multifractal generalizations of classical results by L\'evy and Khintchin in metrical Diophantine approximations and measure theory of continued fractions. We give a complete multifractal analysis for Stern--Brocot intervals, for continued fractions and for certain Diophantine growth rates. In particular, we give detailed discussions of two multifractal spectra closely related to the Farey map and the Gauss map.

Nilpotent orbits of a generalization of Hodge structures

Abstract: We study a generalization of Hodge structures which first appeared in the work of Cecotti and Vafa. It consists of twistors, that is, holomorphic vector bundles on P 1 , with additional structure, a flat connection on C � , a real subbundle and a pairing. We call these objects TERP-structures. We generalize to TERPstructures a correspondence of Cattani, Kaplan and Schmid between nilpotent orbits of Hodge structures and polarized mixed Hodge structures. The proofs use work of Simpson and Mochizuki on variations of twistor structures and a control of the Stokes structures of the poles at zero and infinity. The results are applied to TERP-structures which arise via oscillating integrals from holomorphic functions with isolated singularities.

Multiple recurrence and convergence for sequences related to the prime numbers

Abstract: For any measure preserving system $(X,\mathcal{X},\mu,T)$ and $A\in\mathcal{X}$ with $\mu(A)>0$, we show that there exist infinitely many primes $p$ such that $\mu\bigl(A\cap T^{-(p-1)}A\cap T^{-2(p-1)}A\bigr) > 0$ Furthermore, we show the existence of the limit in $L^2(\mu)$ of the associated double average over the primes A key ingredient is a recent result of Green and Tao on the von Mangoldt function A combinatorial consequence is that every subset of the integers with positive upper density contains an arithmetic progression of length three and common difference of the form $p-1$ for some prime $p$

Convergence of quantum cohomology by quantum Lefschetz

Abstract: Quantum Lefschetz theorem by Coates and Givental gives a relationship between the genus 0 Gromov-Witten theory of X and the twisted theory by a line bundle L on X. We prove the convergence of the twisted theory under the assumption that the genus 0 theory for original X converges. As a byproduct, we prove the semi-simplicity and the Virasoro conjecture for the Gromov-Witten theories of (not necessarily Fano) projective toric manifolds.

Three-dimensional Ricci solitons which project to surfaces

Abstract: We study $3$-dimensional Ricci solitons which project via a semi-conformal mapping to a surface. We reformulate the equations in terms of parameters of the map; this enables us to give an ansatz for constructing solitons in terms of data on the surface. A complete description of the soliton structures on all the $3$-dimensional geometries is given, in particular, non-gradient solitons are found on Nil and Sol.

Uniform embeddability of relatively hyperbolic groups

Abstract: Let Γ be a finitely generated group which is hyperbolic relative to a finite family {H1, . . . ,Hn} of subgroups. We prove that Γ is uniformly embeddable in a Hilbert space if and only if each subgroup Hi is uniformly embeddable in a Hilbert space.

Heat semigroup and functions of bounded variation on Riemannian manifolds

Abstract: Abstract Let M be a connected Riemannian manifold without boundary with Ricci curvature bounded from below and such that the volume of the geodesic balls of centre x and fixed radius r > 0 have a volume bounded away from 0 uniformly with respect to x, and let (T(t)) t≧0 be the heat semigroup on M. We show that the total variation of the gradient of a function u ∈ L 1(M) equals the limit of the L 1-norm of ∇T(t)u as t → 0. In particular, this limit is finite if and only if u is a function of bounded variation.

Purely infinite C*-algebras of real rank zero

Abstract: We show that a separable purely infinite C*-algebra is of real rank zero if and only if its primitive ideal space has a basis consisting of compact-open sets and the natural map K_0(I) -> K_0(I/J) is surjective for all closed two-sided ideals J contained in I in the C*-algebra. It follows in particular that if A is any separable C*-algebra, then A tensor O_2 is of real rank zero if and only if the primitive ideal space of A has a basis of compact-open sets, which again happens if and only if A tensor O_2 has the ideal property, also known as property (IP).

Massey products and ideal class groups

Abstract: We consider certain Massey products in the cohomology of a Galois extension of fields with coefficients in p-power roots of unity. We prove formulas for these products both in general and in the special case that the Galois extension in question is the maximal extension of a number field unramified outside a set of primes S including those above p and any archimedean places. We then consider those Zp-Kummer extensions L∞ of the maximal p-cyclotomic extension K∞ of a number field K that are unramified outside S. We show that Massey products describe the structure of a certain “decomposition-free” quotient of a graded piece of the maximal unramified abelian pro-p extension of L∞ in which all primes above those in S split completely, with the grading arising from the augmentation filtration on the group ring of the Galois group of L∞/K∞. We explicitly describe examples of the maximal unramified abelian pro-p extensions of unramified outside p Kummer extensions of the cyclotomic field of all p-power roots of unity, for irregular primes p.

Twisted burnside-frobenius theory for discrete groups

Abstract: For a wide class of groups including polycyclic and finitely generated poly- nomial growth groups it is proved that the Reidemeister number of an automorphism � is equal to the number of finite-dimensional fixed points of the induced map b � on the unitary dual, if one of these numbers is finite. This theorem is a natural generalization of the classical Burnside-Frobenius theorem to infinite groups. This theorem also has important consequences in topological dynamics and in some sense is a reply to a remark of J.-P. Serre. The main technical results proved in the paper yield a tool for a further progress.

The existence of regular boundary points for non-linear elliptic systems: To the memory of Sergio Campanato

Abstract: Abstract We consider non-linear elliptic systems of the type with Hölder continuous dependence on (x, u), and give conditions guaranteeing that almost every boundary point is a regular point for the gradient of solutions to related Dirichlet problems. We also introduce a new comparison technique, in order to deal with difference quotients.

On classical Saito-Kurokawa liftings

Abstract: There exist two different generalizations of the classical Saito–Kurokawa lifting to modular forms with (square-free) level; one lifting produces modular forms with respect to Γ0(m), the other one with respect to the paramodular group Γ(m). We shall give an alternative and unified construction of both liftings using group theoretic methods. The construction shows that a single elliptic modular form may in fact have many Saito–Kurokawa liftings. We also obtain precise information about the spin L–function of the resulting Siegel modular forms.

Bivariant algebraic k-theory

Abstract: We show how methods from K-theory of operator algebras can be applied in a completely algebraic setting to define a bivariant, M1-stable, homotopy-invariant, excisive K- theory of algebras over a fixed unital ground ring H, (A, B) 7→ kk�(A, B), which is universal in the sense that it maps uniquely to any other such theory. It turns out kk is related to C. Weibel's homotopy algebraic K-theory, KH. We prove that, if H is commutative and A is central as an H-bimodule, then kk�(H, A) = KH�(A). We show further that some calculations from operator algebra KK-theory, such as the exact sequence of Pimsner-Voiculescu, carry over to algebraic kk.

On the genericity of cuspidal automorphic forms of SO2n+1

Abstract: This is a sequel of our work [JS06]. We extend Moeglin’s results ([M97a] and [M97b]) from the even orthogonal groups to old orthogonal groups and complete our proof of the CAP conjecture for irreducible cuspidal automoprhic representations of SO2n+1(A) with special Bessel models. We also give characterization of the vanishing of the central value of the standard L-function of SO2n+1(A) in terms of theta correspondence. As result, we obtain the weak Langlands functorial transfer from SO2n+1(A) to GL2n(A) for irreducible cuspidal automoprhic representations of SO2n+1(A) with special Bessel models.

CR-manifolds of dimension 5: A Lie algebra approach

Abstract: We study real-analytic Levi degenerate hypersurfaces M in complex manifolds of dimension 3, for which the CR-automorphism group Aut(M) is a real Lie group acting transitively on M. We provide large classes of examples for such M, compute the corresponding groups Aut(M) and determine the maximal subsets of M that cannot be separated by global continuous CR-functions. It turns out that all our examples, although partly arising in different contexts, are locally CR-equivalent to the tube over the future light cone in 3-dimensional space-time.

The Anick automorphism of free associative algebras

Abstract: We prove that the well-known Anick automorphism (see [3, p. 343]) of the free associative algebra F over an arbitrary field F of characteristic 0 is wild.

Shelling totally nonnegative flag varieties

Abstract: In this paper we study the partially ordered set Q^J of cells in Rietsch's cell decomposition of the totally nonnegative part of an arbitrary flag variety P^J_{\geq 0}. Our goal is to understand the geometry of P^J_{\geq 0}: Lusztig has proved that this space is contractible, but it is unknown whether the closure of each cell is contractible, and whether P^J_{\geq 0} is homeomorphic to a ball. The order complex |Q^J| is a simplicial complex which can be thought of as a combinatorial approximation of P^J_{\geq 0}. Using combinatorial tools such as Bjorner's EL-labellings and Dyer's reflection orders, we prove that Q^J is graded, thin and EL-shellable. As a corollary, we deduce that Q^J is Eulerian and that the Euler characteristic of the closure of each cell is 1. Additionally, our results imply that |Q^J| is homeomorphic to a ball, and moreover, that Q^J is the face poset of some regular CW complex homeomorphic to a ball.

On the density of rational and integral points on algebraic varieties

Abstract: Let X ⊂ ℙn be a projective geometrically integral variety over of dimension r and degree d ≧ 4. Suppose that there are only finitely many (r − 1)-planes over on X. The main result of this paper is a proof of the fact that the number N(X;B) of rational points on Xwhich have height at most B satisfies for any ɛ > 0. The implied constant depends at most on d, n and ɛ.

Preperiodic points of polynomials over global fields

Abstract: Given a global field K and a polynomial φ defined over K of degree at least two, Morton and Silverman conjectured in 1994 that the number of K-rational preperiodic points of φ is bounded in terms of only the degree of K and the degree of φ. In 1997, for quadratic polynomials over K = Q, Call and Goldstine proved a bound which was exponential in s, the number of primes of bad reduction of φ. By careful analysis of the filled Julia sets at each prime, we present an improved bound on the order of s log s. Our bound applies to polynomials of any degree (at least two) over any global field K. Let K be a field, and let φ ∈ K(z) be a rational function. Let φ denote the n iterate of φ under composition; that is, φ is the identity function, and for n ≥ 1, φ = φ ◦ φn−1. We will study the dynamics φ on the projective line P(K). In particular, we say a point x is preperiodic under φ if there are integers n > m ≥ 0 such that φ(x) = φ(x). The point y = φ(x) satisfies φn−m(y) = y and is said to be periodic (of period n −m). Note that x ∈ P(K) is preperiodic if and only if its orbit {φ(x) : n ≥ 0} is finite. For example, let K = Q and φ(z) = z−29/16. Then {5/4,−1/4,−7/4} forms a periodic cycle (of period 3), and −5/4, 1/4, 7/4, and ±3/4 each land on this cycle after one or two iterations. In addition, the point ∞ is of course fixed. These nine Q-rational points are all preperiodic under φ. Meanwhile, it is not difficult to see that no other point in P(Q) is preperiodic by showing that the denominator of a rational preperiodic point must be 4, and that the absolute value must be less than 2. In general, for any global field K, any dimension N ≥ 1, and any morphism φ : P → P over K of degree at least two, Northcott proved in 1950 that the number of K-rational preperiodic points of φ is finite [25]. More precisely, he showed that the preperiodic points form a set of bounded arithmetic height. Years later, by analogy with the Theorems of Mazur [19] and Merel [20] on K-rational torsion of elliptic curves, Morton and Silverman proposed the following Conjecture [23]. Uniform Boundedness Conjecture. (Morton and Silverman, 1994) Given integers D,N ≥ 1 and d ≥ 2, there is a constant κ = κ(D,N, d) with the following property. Let K be a number field with [K :Q] = D, and let φ : P → P be a morphism of degree d defined over K. Then φ has at most κ preperiodic points in P(K). The analogy between preperiodic points and torsion comes from the fact that the torsion points of an elliptic curve E are precisely the preperiodic points of the multiplication-bytwo map [2] : E → E. In fact, taking x-coordinates, the map [2] induces a rational function (known to dynamicists as a Lattes map) φ : P → P of degree 4 whose preperiodic points Date: June 21, 2005; revised December 21, 2005. 2000 Mathematics Subject Classification. Primary: 11G99 Secondary: 11D45, 37F10.

The colored Jones polynomials of the figure-eight knot and its Dehn surgery spaces

Abstract: We calculate limits of the colored Jones polynomials of the figure-eight knot and conclude that in most cases they determine the volumes and the Chern--Simons invariants of the three-manifolds obtained by Dehn surgeries along it.

On the logarithmic Kobayashi conjecture

Abstract: We study the hyperbolicity of the log variety (Pn, X), where X is a very general hypersurface of degree d ≥ 2n + 1 (which is the bound predicted by the Kobayashi conjecture). Using a positivity result for the sheaf of (twisted) logarithmic vector fields, which may be of independent interest, we show that any log-subvariety of (Pn, X) is of log-general type, give a new proof of the algebraic hyperbolicity of (Pn, X), and exclude the existence of maximal rank families of entire curves in the complement of the universal degree d hypersurface. Moreover, we prove that, as in the compact case, the algebraic hyperbolicity of a log-variety is a necessary condition for the metric one.