Showing papers in "Manuscripta Mathematica in 2009"

Everywhere regularity of functionals with φ-growth

Abstract: We prove C 1,α -regularity for local minimizers of functionals with φ-growth, giving also the decay estimate. In particular, we present a unified approach in the case of power-type functions.

Constant mean curvature spheres in Riemannian manifolds

Abstract: We prove the existence of embedded spheres with large constant mean curvature in any compact Riemannian manifold (M, g). This result partially generalizes a result of R. Ye which handles the case where the scalar curvature function of the ambient manifold (M, g) has non-degenerate critical points.

Tropical descendant Gromov–Witten invariants

Abstract: We define tropical Psi-classes on\({\mathcal{M}_{0,n}(\mathbb{R}^2, d)}\) and consider intersection products of Psi-classes and pull-backs of evaluations on this space. We show a certain WDVV equation which is sufficient to prove that tropical numbers of curves satisfying certain Psi- and evaluation conditions are equal to the corresponding classical numbers. We present an algorithm that generalizes Mikhalkin’s lattice path algorithm and counts rational plane tropical curves satisfying certain Psi- and evaluation conditions.

Small resolutions and non-liftable Calabi-Yau threefolds

Abstract: We use properties of small resolutions of the ordinary double point in dimension three to construct smooth non-liftable Calabi-Yau threefolds. In particular, we construct a smooth projective Calabi-Yau threefold over $${\mathbb{F}_3}$$ that does not lift to characteristic zero and a smooth projective Calabi-Yau threefold over $${\mathbb{F}_5}$$ having an obstructed deformation. We also construct many examples of smooth Calabi-Yau algebraic spaces over $${\mathbb{F}_p}$$ that do not lift to algebraic spaces in characteristic zero.

Multipeak solutions for some singularly perturbed nonlinear elliptic problems on Riemannian manifolds

Abstract: Given (M, g) a smooth compact Riemannian N-manifold, we prove that for any fixed positive integer K the problem $$-\varepsilon^2\Delta_g u +u=u^{p-1}\,{{\rm in}\, M}, \quad u > 0\,{{\rm in}\,M}$$ has a K-peaks solution, whose peaks collapse, as e goes to zero, to an isolated local minimum point of the scalar curvature. Here p > 2 if N = 2 and \({2 < p < 2^*={2N \over N-2}\,if\,N\ge3}\).

A remark on dynamical degrees of automorphisms of hyperkähler manifolds

Abstract: We describe all the dynamical degrees of automorphisms of hyperkahler manifolds in terms of the first dynamical degree. We also present two explicit examples of different geometric flavours.

Géométrie des nombres adélique et lemmes de Siegel généralisés

Abstract: A Siegel’s lemma provides an explicit upper bound for a non-zero vector of minimal height in a finite dimensional vector spaces over a number field. This article explains how to obtain Siegel’s lemmas for which the minimal vectors do not belong to a finite union of vector subspaces (Siegel’s lemmas with conditions). The proofs mix classical results of adelic geometry of numbers and an adelic variant of a theorem of Henk about the number of lattice points of a centrally symmetric convex body in terms of the successive minima of the body.

Non vanishing loci of Hodge numbers of local systems

Abstract: We show that closures of families of unitary local systems on quasiprojective varieties for which the dimension of a graded component of Hodge filtration has a constant value can be identified with a finite union of polytopes. We also present a local version of this theorem. This yields the “Hodge decomposition” of the set of unitary local systems with a non-vanishing cohomology extending Hodge decomposition of characteristic varieties of links of plane curves studied by the author earlier. We consider a twisted version of the characteristic varieties generalizing the twisted Alexander polynomials. Several explicit calculations for complements to arrangements are made.

Soliton dynamics for the nonlinear Schrödinger equation with magnetic field

Abstract: The semiclassical regime of a nonlinear focusing Schrodinger equation in presence of non-constant electric and magnetic potentials V, A is studied by taking as initial datum the ground state solution of an associated autonomous stationary equation. The concentration curve of the solutions is a parameterization of the solutions of the second order ordinary equation $${\ddot x=- abla V(x)-\dot x\times B(x)}$$ , where $${B= abla\times A}$$ is the magnetic field of a given magnetic potential A.

The spectral curve of a quaternionic holomorphic line bundle over a 2-torus

Abstract: A conformal immersion of a 2-torus into the 4-sphere is characterized by an auxiliary Riemann surface, its spectral curve. This complex curve encodes the monodromies of a certain Dirac type operator on a quaternionic line bundle associated to the immersion. The paper provides a detailed description of the geometry and asymptotic behavior of the spectral curve. If this curve has finite genus the Dirichlet energy of a map from a 2-torus to the 2-sphere or the Willmore energy of an immersion from a 2-torus into the 4-sphere is given by the residue of a specific meromorphic differential on the curve. Also, the kernel bundle of the Dirac type operator evaluated over points on the 2-torus linearizes in the Jacobian of the spectral curve. Those results are presented in a geometric and self contained manner.

Prime filtrations and Stanley decompositions of squarefree modules and Alexander duality

Abstract: In this paper we study how prime filtrations and squarefree Stanley decompositions of squarefree modules over the polynomial ring and over the exterior algebra behave with respect to Alexander duality. The results which we obtained suggest a lower bound for the regularity of a \({\mathbb {Z}^n}\)-graded module in terms of its Stanley decompositions. For squarefree modules this conjectured bound is a direct consequence of Stanley’s conjecture on Stanley decompositions. We show that for pretty clean rings of the form R/I, where I is a monomial ideal, and for monomial ideals with linear quotient our conjecture holds.

Stability of quadratic modules

Abstract: A finitely generated quadratic module or preordering in the real polynomial ring is called stable, if it admits a certain degree bound on the sums of squares in the representation of polynomials. Stability, first defined explicitly in Powers and Scheiderer (Adv Geom 1, 71–88, 2001), is a very useful property. It often implies that the quadratic module is closed; furthermore, it helps settling the Moment Problem, solves the Membership Problem for quadratic modules and allows applications of methods from optimization to represent nonnegative polynomials. We provide sufficient conditions for finitely generated quadratic modules in real polynomial rings of several variables to be stable. These conditions can be checked easily. For a certain class of semi-algebraic sets, we obtain that the nonexistence of bounded polynomials implies stability of every corresponding quadratic module. As stability often implies the non-solvability of the Moment Problem, this complements the result from Schmudgen (J Reine Angew Math 558, 225–234, 2003), which uses bounded polynomials to check the solvability of the Moment Problem by dimensional induction. We also use stability to generalize a result on the Invariant Moment Problem from Cimpric et al. (Trans Am Math Soc, to appear).

Log canonical thresholds of binomial ideals

Abstract: We prove that the log canonical thresholds of a large class of binomial ideals, such as complete intersection binomial ideals and the defining ideals of space monomial curves, are computable by linear programming.

Jumping numbers of a unibranch curve on a smooth surface

Abstract: A formula for the jumping numbers of a curve unibranch at a singular point is established. The jumping numbers are expressed in terms of the Enriques diagram of the log resolution of the singularity, or equivalently in terms of the canonical set of generators of the semigroup of the curve at the singular point.

The first Hochschild cohomology group of a schurian cluster-tilted algebra

Abstract: Given a cluster-tilted algebra B we study its first Hochschild coho-mology group HH 1 (B) with coefficients in the B-B-bimodule B. We findseveral consequences when B is representation-finite, and also in the casewhere B is cluster-tilted of type A˜.2000 Mathematics Subject Classification : 16E40 1 Introduction Cluster categories were introduced in [11] and also in [17] for type A, in order tounderstand better the cluster algebras of Fomin and Zelevinsky [20]. Cluster-tiltedalgebras were defined in [12] and also in [18] for type A. These algebras have beenstudied by several authors (see, for instance, [1, 18, 12, 13]). Our objective here is,for a cluster-tilted algebra B, to study its first Hochschild cohomology group HH 1 (B)with coefficients in the B-B-bimodule B, see [19]. As a first step, we consider thecase where B is schurian: this includes the case of all representation-finite cluster-tiltedalgebras. There are several reasons for this restriction. Indeed, it was shown in [1]that, if C is a tilted algebra, then the trivial extension C ⋉ Ext

Elliptic curve configurations on Fano surfaces

Abstract: The elliptic curves on a surface of general type constitute an obstruction for the cotangent sheaf to be ample. In this paper, we give the classification of the configurations of the elliptic curves on the Fano surface of a smooth cubic threefold. That means that we give the number of such curves, their intersections and a plane model. This classification is linked to the classification of the automorphism groups of theses surfaces.

On pointed Hopf algebras associated with the symmetric groups

Abstract: Let G be the symmetric group $${{\mathbb S}_m}$$ . It is an important open problem whether the dimension of the Nichols algebra $${\mathfrak{B} (\mathcal{O},\rho)}$$ is finite when $$\mathcal{O}$$ is the class of the transpositions and ρ is the sign representation, with m ≥ 6. In the present paper, we discard most of the other conjugacy classes showing that very few pairs $${(\mathcal{O},\rho)}$$ might give rise to finite-dimensional Nichols algebras.

Remarks on the nef cone on symmetric products of curves

Abstract: Let C be a very general curve of genus g and let C (2) be its second symmetric product. This paper concerns the problem of describing the convex cone Nef (C (2) )R of all numerically eective R-divisors classes in the N eron-Severi space N 1 (C (2) )R. In a recent work, Julius Ross improved the bound on Nef (C (2) )R in the case of genus ve. By using his techniques and by studying the gonality of the curves lying on C (2) , we give new bounds on the nef cone of C (2) when C is a very general curve of genus 5 g 8.

On automorphisms of split metacyclic groups

Abstract: Let D(m, n; k) be the semi-direct product of two finite cyclic groups $${\mathbb{Z}/m=\langle x\rangle}$$ and $${\mathbb{Z}/n=\langle y\rangle}$$ , where the action is given by yxy −1 = x k . In particular, this includes the dihedral groups D 2m . We calculate the automorphism group Aut (D(m, n; k)).

The excess intersection formula for Grothendieck-Witt groups

Abstract: We prove the excess intersection and self intersection formulae for Grothendieck–Witt groups.

Invariants of simple algebras

Abstract: We determine the group of invariants with values in Galois cohomology with coefficients \({\mathbb{Z}/2\mathbb{Z}}\) of central simple algebras of degree at most 8 and exponent dividing 2.

Existence for a degenerate Cauchy problem

Abstract: We prove the existence of a solution to the degenerate parabolic Cauchy problem with a possibly unbounded Radon measure as an initial data. To accomplish this, we establish a priori estimates and derive a compactness result. We also show that the result is optimal in the Euclidian setting.

Gradient bounds for the horizontal p -Laplacian on a Carnot group and some applications

Abstract: In a Carnot group we prove a new priori bound for the right-invariant horizontal gradient of smooth solutions of a class of quasilinear equations which are modeled on the so-called horizontal p-Laplacian. Exploiting such bound and a regularization procedure based on difference quotients we obtain the $${C^{1,\alpha}_{loc}}$$ regularity of weak solutions which possess some special symmetries. For instance, in the first Heisenberg group $${\mathbb{H}^{1}}$$ we obtain such regularity for all weak solutions of the horizontal p-Laplacian, with p ≥ 2, which are of the form u(z, t) = u(|z|, t).

Nonlinear subelliptic equations

Abstract: In this paper, we study the higher order regularity for weak solutions of a class of quasilinear subelliptic equations. We introduce the notion of ν-closed Hormander system of vector fields, which includes all the previously studied nilpotent systems and extends them to some classes of non-nilpotent systems of vector fields, including those generating the Lie Algebra of the rotation group SO(n) and other non-compact semisimple and solvable Lie groups.

Homological properties of Orlik–Solomon algebras

Abstract: The Orlik–Solomon algebra of a matroid can be considered as a quotient ring over the exterior algebra E. At first, we study homological properties of E-modules as e.g., complexity, depth and regularity. In particular, we consider modules with linear injective resolutions. We apply our results to Orlik–Solomon algebras of matroids and give formulas for the complexity, depth and regularity of such rings in terms of invariants of the matroid. Moreover, we characterize those matroids whose Orlik–Solomon ideal has a linear projective resolution and compute in these cases the Betti numbers of the ideal.

The semiflow of a reaction diffusion equation with a singular potential

Abstract: We study the semiflow \({\mathcal{S}(t)_{t\geq 0}}\) defined by a semilinear parabolic equation with a singular square potential \({V(x)=\frac{\mu}{|x|^2}}\). It is known that the Hardy-Poincare inequality and its improved versions, have a prominent role on the definition of the natural phase space. Our study concerns the case 0 < μ ≤ μ*, where μ* is the optimal constant for the Hardy-Poincare inequality. On a bounded domain of \({\mathbb{R}^N}\), we justify the global bifurcation of nontrivial equilibrium solutions for a reaction term f(s) = λs − |s|2γs, with λ as a bifurcation parameter. We remark some qualitative differences of the branches in the subcritical case μ < μ* and the critical case μ = μ*. The global bifurcation result is used to show that any solution \({\phi(t)}\), initiating form initial data \({\phi_0\geq 0}\) tends to the unique nonnegative equilibrium.

Generalized connected sum construction for scalar flat metrics

Abstract: In this paper we construct constant scalar curvature metrics on the generalized connected sum M = M1 � K M2 of two compact Riemannian scalar flat manifolds (M1,g1) and (M2,g2) along a common Riemannian submanifold (K ,gK ) whose codimension is ≥3. Here we present two constructions: the first one produces a family of "small" (in general nonzero) constant scalar curvature metrics on the generalized connected sum of M1 and M2. It yields an extension of Joyce's result for point-wise connected sums in the spirit of our previous issues for nonzero constant scalar curvature metrics. When the initial manifolds are not Ricci flat, and in particular they belong to the (1+) class in the Kazdan-Warner clas- sification, we refine the first construction in order to produce a family of scalar flat metrics on M. As a consequence we get new solutions to the Einstein constraint equations on the generalized connected sum of two compact time symmetric initial data sets, extending the Isenberg-Mazzeo-Pollack gluing construction.

Strong density results in trace spaces of maps between manifolds

Abstract: We deal with strong density results of smooth maps between two manifolds $${\mathcal X}$$ and $${\mathcal Y}$$ in the fractional spaces given by the traces of Sobolev maps in W 1,p .

Smoothing of quiver varieties

Abstract: We show that Gorenstein singularities that are cones over singular Fano varieties provided by so-called flag quivers are smoothable in codimension three. Moreover, we give a precise characterization about the smoothability in codimension three of the Fano variety itself.

On connectedness of sets in the real spectra of polynomial rings

Abstract: Let R be a real closed field. The Pierce–Birkhoff conjecture says that any piecewise polynomial function f on Rn can be obtained from the polynomial ring R[x1,..., xn] by iterating the operations of maximum and minimum. The purpose of this paper is threefold. First, we state a new conjecture, called the Connectedness conjecture, which asserts, for every pair of points \({{\alpha,\beta\in\,{\rm {Sper}}\ R[x_1,\ldots,x_n]}}\) , the existence of connected sets in the real spectrum of R[x1,..., xn], satisfying certain conditions. We prove that the Connectedness conjecture implies the Pierce–Birkhoff conjecture. Secondly, we construct a class of connected sets in the real spectrum which, though not in itself enough for the proof of the Pierce–Birkhoff conjecture, is the first and simplest example of the sort of connected sets we really need, and which constitutes the first step in our program for a proof of the Pierce–Birkhoff conjecture in dimension greater than 2. Thirdly, we apply these ideas to give two proofs that the Connectedness conjecture (and hence also the Pierce–Birkhoff conjecture in the abstract formulation) holds locally at any pair of points \({{\alpha,\beta\in\,{\rm {Sper}}\ R[x_1,\ldots,x_n]}}\) , one of which is monomial. One of the proofs is elementary while the other consists in deducing this result as an immediate corollary of the main connectedness theorem of this paper.