Showing papers in "Mathematische Zeitschrift in 2010"

First Steps in Tropical Intersection Theory

Abstract: We establish first parts of a tropical intersection theory. Namely, we define cycles, Cartier divisors and intersection products between these two (without passing to rational equivalence) and discuss push-forward and pull-back. We do this first for fans in $${\mathbb{R}^{n}}$$ and then for “abstract” cycles that are fans locally. With regard to applications in enumerative geometry, we finally have a look at rational equivalence and intersection products of cycles and cycle classes in $${\mathbb{R}^{n}}$$ .

New Besov-type spaces and Triebel–Lizorkin-type spaces including Q spaces

Abstract: Let $${s,\,\tau\in\mathbb{R}}$$ and $${q\in(0,\infty]}$$ . We introduce Besov-type spaces $${{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}$$ for $${p\in(0,\,\infty]}$$ and Triebel–Lizorkin-type spaces $${{{{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}\,{\rm for}\, p\in(0,\,\infty)}$$ , which unify and generalize the Besov spaces, Triebel–Lizorkin spaces and Q spaces. We then establish the $${\varphi}$$ -transform characterization of these new spaces in the sense of Frazier and Jawerth. Using the $${\varphi}$$ -transform characterization of $${{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\, {\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}$$ , we obtain their embedding and lifting properties; moreover, for appropriate τ, we also establish the smooth atomic and molecular decomposition characterizations of $${{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\,{\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}$$ . For $${s\in\mathbb{R}}$$ , $${p\in(1,\,\infty), q\in[1,\,\infty)}$$ and $${\tau\in[0,\,\frac{1}{(\max\{p,\,q\})'}]}$$ , via the Hausdorff capacity, we introduce certain Hardy–Hausdorff spaces $${{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}}$$ and prove that the dual space of $${{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}}$$ is just $${\dot{B}^{-s,\,\tau}_{p',\,q'}(\mathbb{R}^{n})}$$ , where t′ denotes the conjugate index of $${t\in (1,\infty)}$$ .

On Bogovskiĭ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains

Abstract: We study integral operators related to a regularized version of the classical Poincare path integral and the adjoint class generalizing Bogovskiĭ’s integral operator, acting on differential forms in \({\mathbb{R}^n}\) . We prove that these operators are pseudodifferential operators of order −1. The Poincare-type operators map polynomials to polynomials and can have applications in finite element analysis. For a domain starlike with respect to a ball, the special support properties of the operators imply regularity for the de Rham complex without boundary conditions (using Poincare-type operators) and with full Dirichlet boundary conditions (using Bogovskiĭ-type operators). For bounded Lipschitz domains, the same regularity results hold, and in addition we show that the cohomology spaces can always be represented by \({\fancyscript{C}^{\infty}}\) functions.

Centers of F-purity

Abstract: In this paper, we study a positive characteristic analogue of the centers of log canonicity of a pair (R, Δ). We call these analogues centers of F-purity. We prove positive characteristic analogues of subadjunction-like results, prove new stronger subadjunction-like results, and in some cases, lift these new results to characteristic zero. Using a generalization of centers of F-purity which we call uniformly F-compatible ideals, we give a characterization of the test ideal (which unifies several previous characterizations). Finally, in the case that Δ = 0, we show that uniformly F-compatible ideals coincide with the annihilators of the $${\mathcal{F}(E_R(k))}$$ -submodules of E R (k) as defined by Lyubeznik and Smith.

Biharmonic submanifolds of \({\mathbb{C}P^n}\)

Abstract: We give some general results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space. These results are mainly concerned with submanifolds with constant mean curvature or parallel mean curvature vector field. We find the relation between the bitension field of the inclusion of a submanifold $${\bar{M}}$$ in $${\mathbb{C}P^n}$$ and the bitension field of the inclusion of the corresponding Hopf-tube in $${\mathbb{S}^{2n+1}}$$ . Using this relation we produce new families of proper-biharmonic submanifolds of $${\mathbb{C}P^n}$$ . We study the geometry of biharmonic curves of $${\mathbb{C}P^n}$$ and we characterize the proper-biharmonic curves in terms of their curvatures and complex torsions.

Linear Systems on Tropical Curves

Abstract: A tropical curve $\Gamma$ is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system $|D|$ of a divisor $D$ on a tropical curve $\Gamma$ analogously to the classical counterpart. We investigate the structure of $|D|$ as a cell complex and show that linear systems are quotients of tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, $|D|$ defines a map from $\Gamma$ to a tropical projective space, and the image can be modified to a tropical curve of degree equal to $\mathrm{deg}(D)$. The tropical convex hull of the image realizes the linear system $|D|$ as a polyhedral complex.

The Fujita–Kato approach to the Navier–Stokes equations in the rotational framework

Abstract: Consider the Navier-Stokes equations in the rotational framework. It is proved that these equations possess a unique global mild solution for arbitrary speed of rotation provided the initial data u 0 is small enough in the $${H^{\frac12}_{\sigma}(\mathbb{R}^3)}$$ -norm.

Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation

Abstract: Given a knot complement X and its p-fold cyclic cover X p → X, we identify twisted polynomials associated to $${GL_1\left({\bf F}[t^{\pm 1}]\right)}$$ representations of π 1(X p ) with twisted polynomials associated to related $${GL_p\left({\bf F}[t^{\pm 1}]\right)}$$ representations of π 1(X) which factor through metabelian representations. This provides a simpler and faster algorithm to compute these polynomials, allowing us to prove that 16 (of 18 previously unknown) algebraically slice knots of 12 or fewer crossings are not slice. We also use this improved algorithm to prove that the 24 mutants of the pretzel knot P(3, 7, 9, 11, 15), corresponding to permutations of (7, 9, 11, 15), represent distinct concordance classes.

Global L p estimates for degenerate Ornstein–Uhlenbeck operators

Abstract: We consider a class of degenerate Ornstein–Uhlenbeck operators in $${\mathbb{R}^{N}}$$ , of the kind $$\mathcal{A}\equiv\sum_{i, j=1}^{p_{0}}a_{ij}\partial_{x_{i}x_{j}}^{2} + \sum_{i, j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}$$ where (a ij ), (b ij ) are constant matrices, (a ij ) is symmetric positive definite on $${\mathbb{R} ^{p_{0}}}$$ (p 0 ≤ N), and (b ij ) is such that $${\mathcal{A}}$$ is hypoelliptic. For this class of operators we prove global L p estimates (1 < p < ∞) of the kind: $$\left\Vert \partial_{x_{i}x_{j}}^{2}u\right\Vert _{L^{p}\left(\mathbb{R}^{N}\right)} \leq c \left\{\left\Vert \mathcal{A}u\right\Vert _{L^{p}\left(\mathbb{R}^{N}\right)} + \left\Vert u\right\Vert_{L^{p}\left(\mathbb{R}^{N}\right)}\right\} {\rm for} \, i, j = 1, 2, \ldots, p_{0}$$ and corresponding weak type (1,1) estimates. This result seems to be the first case of global estimates, in Lebesgue L p spaces, for complete Hormander’s operators $$\sum X_{i}^{2}+X_{0},$$ proved in absence of a structure of homogeneous group. We obtain the previous estimates as a byproduct of the following one, which is of interest in its own: $$\left\Vert \partial_{x_{i}x_{j}}^{2}u\right\Vert _{L^{p}\left( S\right)}\leq c\left\Vert Lu\right\Vert _{L^{p}\left( S\right)}$$ for any $${u \in C_{0}^{\infty} \left(S\right)}$$ , where S is the strip $${\mathbb{R}^{N} \times \left[-1, 1\right]}$$ and L is the Kolmogorov-Fokker-Planck operator $${\mathcal{A} - \partial_{t}}$$ . To get this estimate we use in a crucial way the left invariance of L with respect to a Lie group structure in $${\mathbb{R}^{N+1}}$$ and some results on singular integrals on nonhomogeneous spaces recently proved in Bramanti (Revista Matematica Iberoamericana, 2009, in press).

The cyclic sliding operation in Garside groups

Abstract: We present a new operation to be performed on elements in a Garside group, called cyclic sliding, which is introduced to replace the well known cycling and decycling operations. Cyclic sliding appears to be a more natural choice, simplifying the algorithms concerning conjugacy in Garside groups and having nicer theoretical properties. We show, in particular, that if a super summit element has conjugates which are rigid (that is, which have a certain particularly simple structure), then the optimal way of obtaining such a rigid conjugate through conjugation by positive elements is given by iterated cyclic sliding.

On the size of the set A ( A + 1)

Abstract: Let F p be the field of a prime order p. For a subset $${A \subset F_p}$$ we consider the product set A(A + 1). This set is an image of A × A under the polynomial mapping f(x, y) = xy + x : F p × F p → F p . In the present note we show that if |A| < p 1/2, then $$|A(A + 1)| \ge |A|^{106/105+o(1)}.$$ If |A| > p 2/3, then we prove that $$|A(A + 1)| \gg \sqrt{p\, |A|}$$ and show that this is optimal in general settings bound up to the implied constant. We also estimate the cardinality of A(A + 1) when A is a subset of real numbers. We show that in this case one has the Elekes type bound $$|A(A + 1)| \gg |A|^{5/4}.$$

Harmonic maps from degenerating Riemann surfaces

Abstract: We study harmonic maps from degenerating Riemann surfaces with uniformly bounded energy and show the so-called generalized energy identity We find conditions that are both necessary and sufficient for the compactness in W 1,2 and C 0 modulo bubbles of sequences of such maps

Algebras that satisfy Auslander's condition on vanishing of cohomology

Abstract: Auslander conjectured that every Artin algebra satisfies a certain condition on vanishing of cohomology of finitely generated modules. The failure of this conjecture—by a 2003 counterexample due to Jorgensen and Sega—motivates the consideration of the class of rings that do satisfy Auslander’s condition. We call them AC rings and show that an AC Artin algebra that is left-Gorenstein is also right-Gorenstein. Furthermore, the Auslander–Reiten Conjecture is proved for AC rings, and Auslander’s G-dimension is shown to be functorial for AC rings that are commutative or have a dualizing complex.

The Grothendieck ring of varieties and piecewise isomorphisms

Abstract: Let K0(Var k) be the Grothendieck ring of algebraic varieties over a field k .L et X, Y be two algebraic varieties over k which are piecewise isomorphic (i.e. X and Y admit finite partitions X1 ,..., Xn, Y1 ,..., Yn into locally closed subvarieties such that Xi is iso- morphic to Yi for all i ≤ n), then (X )=( Y ) in K0(Var k). Larsen and Lunts ask whether the converse is true. For characteristic zero and algebraically closedfield k, we answer positively this question when dim X ≤ 1o rX is a smooth connected projective surface or if X contains only finitely many rational curves.

Comparison of relative cohomology theories with respect to semidualizing modules

Abstract: We compare and contrast various relative cohomology theories that arise from resolutions involving semidualizing modules. We prove a general balance result for relative cohomology over a Cohen-Macaulay ring with a dualizing module, and we demonstrate the failure of the naive version of balance one might expect for these functors. We prove that the natural comparison morphisms between relative cohomology modules are isomorphisms in several cases, and we provide a Yoneda-type description of the first relative Ext functor. Finally, we show by example that each distinct relative cohomology construction does in fact result in a different functor.

On transfer inequalities in Diophantine approximation, II

Abstract: Let Θ be a point in Rn. We are concerned with the approximation to Θ by rational linear subvarieties of dimension d for 0 ≤ d ≤ n−1. To that purpose, we introduce various convex bodies in the Grassmann algebra Λ(Rn+1). It turns out that our convex bodies in degree d are the dth compound, in the sense of Mahler, of convex bodies in degree one. A dual formulation is also given. This approach enables us both to split and to refine the classical Khintchine transference principle.

On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series

Abstract: For every positive integer n, consider the linear operator U n on polynomials of degree at most d with integer coefficients defined as follows: if we write $${\frac{h(t)}{(1 - t)^{d + 1}}=\sum_{m \geq 0} g(m) \, t^{m}}$$ , for some polynomial g(m) with rational coefficients, then $${\frac{{\rm{U}}_{n}h(t)}{(1- t)^{d+1}} = \sum_{m \geq 0}g(nm) \, t^{m}}$$ . We show that there exists a positive integer n d , depending only on d, such that if h(t) is a polynomial of degree at most d with nonnegative integer coefficients and h(0) = 1, then for n ≥ n d , U n h(t) has simple, real, negative roots and positive, strictly log concave and strictly unimodal coefficients. Applications are given to Ehrhart δ-polynomials and unimodular triangulations of dilations of lattice polytopes, as well as Hilbert series of Veronese subrings of Cohen–Macauley graded rings.

Combinatorics of binomial primary decomposition

Abstract: An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero This decomposition is derived from a characteristic-free combinatorial description of certain primary components of binomial ideals in affine semigroup rings, namely those that are associated to faces of the semigroup These results are intimately connected to hypergeometric differential equations in several variables

Cluster structures from 2-Calabi–Yau categories with loops

Abstract: We generalise the notion of cluster structures from the work of Buan–Iyama–Reiten–Scott to include situations where the endomorphism rings of the clusters may have loops. We show that in a Hom-finite 2-Calabi–Yau category, the set of maximal rigid objects satisfies these axioms whenever there are no 2-cycles in the quivers of their endomorphism rings. We apply this result to the cluster category of a tube, and show that this category forms a good model for the combinatorics of a type B cluster algebra.

On some arithmetic properties of Siegel functions

Abstract: We deal with several arithmetic properties of the Siegel functions which are modular units. By modifying the ideas in Kubert and Lang (Modular Units. Grundlehren der mathematischen Wissenschaften, vol 244. Spinger, Heidelberg, 1981), we establish certain criterion for determining a product of Siegel functions to be integral over \({\mathbb{Z}[j]}\) . We also find generators of the function fields \({\mathcal{K}(X_1(N))}\) by examining the orders of Siegel functions at the cusps and apply them to evaluate the Ramanujan’s cubic continued fraction systematically. Furthermore we construct ray class invariants over imaginary quadratic fields in terms of singular values of j and Siegel functions.

Some results about the existence of critical points for the Willmore functional

Abstract: Using a perturbative approach, it is shown existence and multiplicity of critical points for the Willmore functional in ambient manifold \({(\mathbb{R}^3, g_\epsilon)}\) —where \({g_\epsilon}\) is a metric close and asymptotic to the Euclidean one. With the same technique it is proved a non-existence result in general Riemannian manifold (M, g) of dimension three.

Dimensions of triangulated categories via Koszul objects

Abstract: Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for the representation dimensions of certain Artin algebras.

Schubert polynomials for the affine Grassmannian of the symplectic group

Abstract: We study the Schubert calculus of the affine Grassmannian Gr of the symplectic group. The integral homology and cohomology rings of Gr are identified with dual Hopf algebras of symmetric functions, defined in terms of Schur’s P and Q functions. An explicit combinatorial description is obtained for the Schubert basis of the cohomology of Gr, and this is extended to a definition of the affine type C Stanley symmetric functions. A homology Pieri rule is also given for the product of a special Schubert class with an arbitrary one.

The shifted plactic monoid

Abstract: We introduce a shifted analog of the plactic monoid of Lascoux and Schutzenberger, the shifted plactic monoid. It can be defined in two different ways: via the shifted Knuth relations, or using Haiman’s mixed insertion. Applications include: a new combinatorial derivation (and a new version of) the shifted Littlewood–Richardson Rule; similar results for the coefficients in the Schur expansion of a Schur P-function; a shifted counterpart of the Lascoux–Schutzenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more.

The classification of special cohen-macaulay modules

Abstract: In this paper we completely classify all the special Cohen–Macaulay (=CM) modules corresponding to the exceptional curves in the dual graph of the minimal resolutions of all two dimensional quotient singularities. In every case we exhibit the specials explicitly in a combinatorial way. Our result relies on realizing the specials as those CM modules whose first Ext group vanishes against the ring R, thus reducing the problem to combinatorics on the AR quiver; such possible AR quivers were classified by Auslander and Reiten. We also give some general homological properties of the special CM modules and their corresponding reconstruction algebras.

Packing dimension and Ahlfors regularity of porous sets in metric spaces

Abstract: Let X be a metric measure space with an s-regular measure μ. We prove that if \({A\subset X}\) is \({\varrho}\) -porous, then \({{\rm {dim}_p}(A)\le s-c\varrho^s}\) where dimp is the packing dimension and c is a positive constant which depends on s and the structure constants of μ. This is an analogue of a well known asymptotically sharp result in Euclidean spaces. We illustrate by an example that the corresponding result is not valid if μ is a doubling measure. However, in the doubling case we find a fixed \({N\subset X}\) with μ(N) = 0 such that \({{\rm {dim}_p}(A)\le{\rm {dim}_p}(X)-c(\log \tfrac1\varrho)^{-1}\varrho^t}\) for all \({\varrho}\) -porous sets \({A \subset X{\setminus} N}\) . Here c and t are constants which depend on the structure constant of μ. Finally, we characterize uniformly porous sets in complete s-regular metric spaces in terms of regular sets by verifying that A is uniformly porous if and only if there is t < s and a t-regular set F such that \({A\subset F}\) .

The cone of pseudo-effective divisors of log varieties after Batyrev

Abstract: In these notes, we investigate the cone of nef curves of projective varieties, which is the dual cone to the cone of pseudo-effective divisors. We prove a structure theorem for the cone of nef curves of projective $${\mathbb Q}$$ -factorial klt pairs of arbitrary dimension from the point of view of the Minimal Model Program. This is a generalization of Batyrev’s structure theorem for the cone of nef curves of projective terminal threefolds.

On Beauville structures on the groups Sn and An

Abstract: It is known that the symmetric group Sn, for n ≥ 5, and the alternating group An, for large n, admit a Beauville structure. In this paper we prove that An admits a Beauville (resp. strongly real Beauville) structure if and only if n ≥ 6 (resp n ≥ 7). We also show that Sn admits a strongly real Beauville structure for n ≥ 5.

End-point maximal regularity and its application to two-dimensional Keller–Segel system

Abstract: We prove the maximal L ρ regularity of the Cauchy problem of the heat equation in the Besov space $${\dot{B}_{1,\rho}^0(\mathbb{R}^n)}$$ , 1 < ρ ≤ ∞, which is not UMD space And as its application, we establish the time local well-posedness of the solution of two dimensional nonlinear parabolic system with the Poisson equation in $${\dot{B}_{1,2}^0(\mathbb{R}^2)}$$ , where the equation is considered in the space invariant by a scaling and particularly the natural free energy is well defined from the initial time The small data global existence is also obtained in the same class

Réalisation ℓ -adique des motifs triangulés géométriques II

Abstract: For geometrical triangulated motives with rational coefficients over a ground field of characteristic zero which is embeddable into \({\mathbb{C}}\) , Huber (J. Algebraic Geom. 9(4):755–799, 2000; J. Algebraic Geom. 13(1):195–207, 2004) has constructed a realization functor with values in the category of mixed realization of Huber (Mixed motives and their realization in derived categories. Lecture Notes in Mathematics, vol. 1604. Springer, Berlin, 1995). In this sequel to Ivorra (Doc Math 12:607–671, 2007), we prove that the l-adic realization functor obtained in Theorem 4.3 of Ivorra (Doc Math 12:607–671, 2007) is the same up to a canonical isomorphism as the l-adic component of A. Huber’s construction. In this way (Ivorra in Doc Math 12:607–671, 2007) might be viewed as an integral generalization to all noetherian separed schemes of the work (Huber in J. Algebraic Geom. 9(4):755–799, 2000; J. Algebraic Geom. 13(1):195–207, 2004) as far as the l-adic setting is concerned. We also prove a comparison theorem with the classical l-adic cycle class map over a perfect field using a naive motivic cycle class map.