Showing papers in "Tohoku Mathematical Journal in 2015"

THE SPECIAL McKAY CORRESPONDENCE AND EXCEPTIONAL COLLECTIONS

Abstract: We show that the derived category of coherent sheaves on the quotient stack of the affine plane by a finite small subgroup of the general linear group is obtained from the derived category of coherent sheaves on the minimal resolution by adding a semiorthogonal summand with a full exceptional collection. The proof is based on an explicit construction in the abelian case, together with the analysis of the behavior of the derived categories of coherent sheaves under root constructions.

Biharmonic hypersurfaces with three distinct principal curvatures in Euclidean space

Abstract: The well known Chen's conjecture on biharmonic submanifolds states that a biharmonic submanifold in a Euclidean space is a minimal one ([10--13, 16, 18--21, 8]). For the case of hypersurfaces, we know that Chen's conjecture is true for biharmonic surfaces in $\mathbb E^3$ ([10], [24]), biharmonic hypersurfaces in $\mathbb E^4$ ([23]), and biharmonic hypersurfaces in $\mathbb E^m$ with at most two distinct principal curvature ([21]). The most recent work of Chen-Munteanu [18] shows that Chen's conjecture is true for $\delta(2)$-ideal hypersurfaces in $\mathbb E^m$, where a $\delta(2)$-ideal hypersurface is a hypersurface whose principal curvatures take three special values: $\lambda_1, \lambda_2$ and $\lambda_1+\lambda_2$. In this paper, we prove that Chen's conjecture is true for hypersurfaces with three distinct principal curvatures in $\mathbb E^m$ with arbitrary dimension, thus, extend all the above-mentioned results. As an application we also show that Chen's conjecture is true for $O(p)\times O(q)$-invariant hypersurfaces in Euclidean space $\mathbb E^{p+q}$.

Almost complex surfaces in the nearly Kähler $S^3\times S^3$

Abstract: In this paper we initiate the study of almost complex surfaces in the nearly Kahler $S^3\times S^3$. We show that on such a surface it is possible to define a global holomorphic differential, which is induced by an almost product structure on the nearly Kahler $S^3\times S^3$. We also find a local correspondence between almost complex surfaces in the nearly Kahler $S^3\times S^3$ and solutions of the general $H$-system equation introduced by Wente ([13]), thus obtaining a geometric interpretation of solutions of the general $H$-system equation. From this we deduce a correspondence between constant mean curvature surfaces in $\mathbb R^3$ and almost complex surfaces in the nearly Kahler $S^3\times S^3$ with vanishing holomorphic differential. This correspondence allows us to obtain a classification of the totally geodesic almost complex surfaces. Moreover, we prove that almost complex topological 2-spheres in $S^3\times S^3$ are totally geodesic. Finally, we also show that every almost complex surface with parallel second fundamental form is totally geodesic.

Contact properties of surfaces in R 3 with corank 1 singularities

Abstract: We study the geometry of surfaces in R 3 with corank 1 singularities. At a singular point we define the curvature parabola using the first and second fundamental forms of the surface, which contains all the local second order geometrical information about the surface. The curvature parabola is used to introduce the concepts of asymptotic directions and umbilic curvature, which are related to contact properties of the surface with planes and spheres.

Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces II

Abstract: In this paper we determine the Hamiltonian stability of Gauss images, i.e., the images of the Gauss maps, of homogeneous isoparametric hypersurfaces of exceptional type with $g=6$ or $4$ distinct principal curvatures in spheres. Combining it with our previous results in [12] and Part I [14], we determine the Hamiltonian stability for the Gauss images of all homogeneous isoparametric hypersurfaces. In addition, we discuss the exceptional Riemannian symmetric space $(E_6, U(1)\cdot Spin(10))$ and the corresponding Gauss image, which have their own interest from the viewpoint of symmetric space theory.

SKT and tamed symplectic structures on solvmanifolds

Abstract: We study the existence of strong Kahler with torsion (SKT) metrics and of symplectic forms taming invariant complex structures $J$ on solvmanifolds $G/\Gamma$ providing some negative results for some classes of solvmanifolds. In particular, we show that if either $J$ is invariant under the action of a nilpotent complement of the nilradical of $G$ or $J$ is abelian or $G$ is almost abelian (not of type (I)), then the solvmanifold $G/\Gamma$ cannot admit any symplectic form taming the complex structure $J$, unless $G/\Gamma$ is Kahler. As a consequence, we show that the family of non-Kahler complex manifolds constructed by Oeljeklaus and Toma cannot admit any symplectic form taming the complex structure.

On good reduction of some K3 surfaces related to abelian surfaces

Abstract: The Neron--Ogg--Safarevic criterion for abelian varieties tells that the Galois action on the $l$-adic etale cohomology of an abelian variety over a local field determines whether the variety has good reduction or not. We prove an analogue of this criterion for a certain type of K3 surfaces closely related to abelian surfaces. We also prove its $p$-adic analogue. This paper includes T. Ito's unpublished result on Kummer surfaces.

On the $r$-nuclearity of some integral operators on Lebesgue spaces

Abstract: In this paper we will exhibit a class of kernels generating $r$-nuclear operators. The class includes the Fox-Li and related operators. Estimates for the corresponding asymptotic behaviour of the eigenvalues are also derived.

Möbius invariant energies and average linking with circles

Abstract: We introduce a Mobius invariant energy associated to planar domains, as well as a generalization to space curves. This generalization is a Mobius version of Banchoff-Pohl's notion of area enclosed by a space curve. A relation with Gauss-Bonnet theorems for complete surfaces in hyperbolic space is also described.

Set-valued and fuzzy stochastic differential equations in M-type 2 Banach spaces

Abstract: In this paper we study set-valued stochastic differential equations in M-type 2 Banach spaces. Their drift terms and diffusion terms are assumed to be set-valued and single-valued respectively. These coefficients are considered to be random which makes the equations to be truely nonautonomous. Firstly we define set-valued stochastic Lebesgue integral in a Banach space. This integral is a set-valued random variable. We state its properties such as additivity with respect to the interval of integration, continuity as a function of the upper limit of integration, integrable boundedness. The existence and uniqueness of solution to set-valued differential equations in M-type 2 Banach space is obtained by a method of successive approximations. We show that the approximations are uniformly bounded and converge to the unique solution. A distance between $n$th approximation and exact solution is estimated and a continuous dependence of solution with respect to the data of the equation is proved. Finally, we construct a fuzzy stochastic Lebesgue integral in a Banach space and examine fuzzy stochastic differential equations in M-type 2 Banach spaces. We investigate properties like those in set-valued cases. All the results are achieved without assumption on separability of underlying sigma-algebra.

Strong convergence theorem of Cesàro means with respect to the Walsh system

Abstract: We prove that Cesaro means of one-dimensional Walsh-Fourier series are uniformly bounded operators in the martingale Hardy space $H_p$ for $0

On minimal Lagrangian surfaces in the product of Riemannian two manifolds

Abstract: Let $(\Sigma_1,g_1)$ and $(\Sigma_2,g_2)$ be connected, complete and orientable 2-dimensional Riemannian manifolds. Consider the two canonical Kahler structures \linebreak $(G^{\epsilon},J,\Omega^{\epsilon})$ on the product 4-manifold $\Sigma_1\times\Sigma_2$ given by $ G^{\epsilon}=g_1\oplus \epsilon g_2$, $\epsilon=\pm 1$ and $J$ is the canonical product complex structure. Thus for $\epsilon=1$ the Kahler metric $G^+$ is Riemannian while for $\epsilon=-1$, $G^-$ is of neutral signature. We show that the metric $G^{\epsilon}$ is locally conformally flat if and only if the Gauss curvatures $\kappa(g_1)$ and $\kappa(g_2)$ are both constants satisfying $\kappa(g_1)=-\epsilon\kappa(g_2)$. We also give conditions on the Gauss curvatures for which every $G^{\epsilon}$-minimal Lagrangian surface is the product $\gamma_1\times\gamma_2\subset\Sigma_1\times\Sigma_2$, where $\gamma_1$ and $\gamma_2$ are geodesics of $(\Sigma_1,g_1)$ and $(\Sigma_2,g_2)$, respectively. Finally, we explore the Hamiltonian stability of projected rank one Hamiltonian $G^{\epsilon}$-minimal surfaces.

Toric modifications of cyclic orbifolds and an extended Zagier reciprocity for Dedekind sums

Abstract: We study a toric modification of Fujiki-Oka type for cyclic quotient singularities. Especially the behavior of rational Chow rings, orbifold signatures and so on are explicitly calculated. As a result, we extend Zagier's reciprocity for higher-dimensional Dedekind sums. Namely, we define Dedekind sums with weight by using Atiyah-Singer's equivariant signature with non-isolated fixed point locus, and then prove our reciprocity among them.

Extremal Lorentzian surfaces with null $r$-planar geodesics in space forms

Abstract: We show a congruence theorem for oriented Lorentzian surfaces with horizontal reflector lifts in pseudo-Riemannian space forms of neutral signature. As a corollary, a characterization theorem is obtained for the Lorentzian Boruvka spheres, that is, a full real analytic null $r$-planar geodesic immersion with vanishing mean curvature vector field is locally congruent to the Lorentzian Boruvka sphere in a $2r$-dimensional space form of neutral signature.

Mackey's criterion for subgroup restriction of Kronecker products and harmonic analysis on Clifford groups

Abstract: We present a criterion for multiplicity-freeness of the decomposition of the restriction ${\rm Res}^G_H(\rho_1 \otimes \rho_2)$ of the Kronecker product of two generic irreducible representations $\rho_1, \rho_2$ of a finite group $G$ with respect to a subgroup $H \leq G$. This constitutes a generalization of a well-known criterion due to Mackey (which corresponds to the case $H = G$). The corresponding harmonic analysis is illustated by detailed computations on the Clifford groups $G=\mathbb{CL}(n)$, together with the subgroups $H=\mathbb{CL}(n-1)$, for $n \geq 1$, which lead to an explicit decomposition of the restriction of Kronecker products.

Structure of symplectic Lie groups and momentum map

Abstract: We describe the structure of the Lie groups endowed with a left-invariant symplectic form, called symplectic Lie groups, in terms of semi-direct products of Lie groups, symplectic reduction and principal bundles with affine fiber. This description is particularly nice if the group is Hamiltonian, that is, if the left canonical action of the group on itself is Hamiltonian. The principal tool used for our description is a canonical affine structure associated with the symplectic form. We also characterize the Hamiltonian symplectic Lie groups among the connected symplectic Lie groups. We specialize our principal results to the cases of simply connected Hamiltonian symplectic nilpotent Lie groups or Frobenius symplectic Lie groups. Finally we pursue the study of the classical affine Lie group as a symplectic Lie group. MSC Classes 53D20,70G65 Key words:Symplectic Lie groups,Hamiltonian Lie groups, Symplectic reduction,Symplectic double extension.

An SDE approach to leafwise diffusions on foliated spaces and its applications

Abstract: We construct leafwise diffusions on foliated spaces via SDE approach. The obtained diffusions are stochastically continuous and hence have the Feller property. Moreover our construction enables us to prove a central limit theorem for the leafwise diffusion on a compact foliated space in the same way as for a diffusion on a compact manifold.

Bilinear differential operators: Projectively equivariant symbol and quantization maps

Abstract: We study the space of bilinear differential operators on weighted densities as a module over $\mathrm{sl}(2,{\mathbb R})$. We introduce the corresponding space of symbols and we prove the existence and the uniqueness of canonical projective equivariant symbol and quantization maps.

Minimal singular metrics of a line bundle admitting no Zariski decomposition

Abstract: We give a concrete expression of a minimal singular metric on a big line bundle on a compact Kahler manifold which is the total space of a toric bundle over a complex torus. In this class of manifolds, Nakayama constructed examples which have line bundles admitting no Zariski decomposition even after modifications. As an application, we discuss the Zariski closedness of non-nef loci.

On smooth Gorenstein polytopes

Abstract: A Gorenstein polytope of index $r$ is a lattice polytope whose $r$th dilate is a reflexive polytope. These objects are of interest in combinatorial commutative algebra and enumerative combinatorics, and play a crucial role in Batyrev's and Borisov's computation of Hodge numbers of mirror-symmetric generic Calabi-Yau complete intersections. In this paper we report on what is known about smooth Gorenstein polytopes, i.e., Gorenstein polytopes whose normal fan is unimodular. We classify $d$-dimensional smooth Gorenstein polytopes with index larger than $(d+3)/3$. Moreover, we use a modification of Obro's algorithm to achieve classification results for smooth Gorenstein polytopes in low dimensions. The first application of these results is a database of all toric Fano $d$-folds whose anticanonical divisor is divisible by an integer $r$ satisfying $r \ge d-7$. As a second application we verify that there are only finitely many families of Calabi-Yau complete intersections of fixed dimension that are associated to a smooth Gorenstein polytope via the Batyrev-Borisov construction.

On the dead-core problem for the $p$-Laplace equation with a strong absorption

Abstract: We study an initial boundary value problem for the $p$-Laplace equation with a strong absorption. We are concerned with the dead-core behavior of the solution. First, some criteria for developing dead-core are given. Also, the temporal dead-core rate for certain initial data is determined. Then we prove uniqueness theorem for the backward self-similar solutions.

On the structure of linearization of the scalar curvature

Abstract: For a compact $n$-dimensional manifold a critical point metric of the total scalar curvature functional satisfies the critical point equation (1) below, if the functional is restricted to the space of constant scalar curvature metrics of unit volume. The right-hand side in this equation is nothing but the adjoint operator of the linearization of the total scalar curvature acting on functions. The structure of the kernel space of the adjoint operator plays an important role in the geometry of the underlying manifold. In this paper, we study some geometric structure of a given manifold when the kernel space of the adjoint operator is nontrivial. As an application, we show that if there are two distinct solutions satisfying the critical point equation mentioned above, then the metric should be Einstein. This generalizes a main result in [6] to arbitrary dimension.

Order of operators determined by operator mean

Abstract: Let $\sigma$ be an operator mean and $f$ a non-constant operator monotone function on $(0,\infty)$ associated with $\sigma$. If operators $A, B$ satisfy $0\le A \le B$, then it holds that $Y \sigma (tA+X) \le Y \sigma (tB+X)$ for any non-negative real number $t$ and any positive, invertible operators $X,Y$. We show that the condition $ Y \sigma (tA+X) \le Y \sigma (tB+X)$ for a sufficiently small $t>0$ implies $A \le B$ if and only if $X$ is a positive scalar multiple of $Y$ or the associated operator monotone function $f$ with $\sigma$ has the form $f(t) = (at+b)/(ct+d)$, where $a,b,c,d$ are real numbers satisfying $ad-bc>0$.

Ramification and nearby cycles for $\ell$-adic sheaves on relative curves

Abstract: Deligne and Kato proved a formula computing the dimension of the nearby cycles complex of an $\ell$-adic sheaf on a relative curve over an excellent strictly henselian trait. In this article, we reprove this formula using Abbes-Saito's ramification theory.

Large deviation principle for certain spatially lifted Gaussian rough path

Abstract: In rough stochastic PDE theory of Hairer type, rough path lifts with respect to the space variable of two-parameter continuous Gaussian processes play a main role. A prominent example of such processes is the solution of the stochastic heat equation under the periodic condition. The main objective of this paper is to show that the law of the spatial lift of this process satisfies a Schilder type large deviation principle on the continuous path space over a geometric rough path space.

Alexandrov's isodiametric conjecture and the cut locus of a surface

Abstract: We prove that Alexandrov's conjecture relating the area and diameter of a convex surface holds for the surface of a general ellipsoid. This is a direct consequence of a more general result which estimates the deviation from the optimal conjectured bound in terms of the length of the cut locus of a point on the surface. We also prove that the natural extension of the conjecture to general dimension holds among closed convex spherically symmetric Riemannian manifolds. Our results are based on a new symmetrization procedure which we believe to be interesting in its own right.

Group algebras and normal basis problem

Abstract: We formulate the notion of cleft extensions in the Hopf-Galois theory in the framework of algebraic geometry. The unit group scheme of the algebra of a finite flat group scheme plays a key role.

Classification of hypersurfaces with constant Möbius Ricci curvature in $\mathbb{R}^{n+1}$

Abstract: Let $f: M^n\rightarrow \mathbb{R}^{n+1}$ be an immersed umbilic-free hypersurface in an $(n+1)$-dimensional Euclidean space $\mathbb{R}^{n+1}$ with standard metric $I=df\cdot df$. Let $II$ be the second fundamental form of the hypersurface $f$. One can define the Mobius metric $g=\frac{n}{n-1}(\|II\|^2-n|{\rm tr}II|^2)I$ on $f$ which is invariant under the conformal transformations (or the Mobius transformations) of $\mathbb{R}^{n+1}$. The sectional curvature, Ricci curvature with respect to the Mobius metric $g$ is called Mobius sectional curvature, Mobius Ricci curvature, respectively. The purpose of this paper is to classify hypersurfaces with constant Mobius Ricci curvature.

A note on Ribaucour transformations in Lie sphere geometry

Abstract: Following Burstall and Hertrich-Jeromin we study the Ribaucour transformation of Legendre submanifolds in Lie sphere geometry. We give an explicit parametrization of the resulted Legendre submanifold $\hat{F}$ of a Ribaucour transformation, via a single real function $\tau$ which represents the regular Ribaucour sphere congruence $s$ enveloped by the original Legendre submanifold $F$.

Differentiable pinching theorems for submanifolds via Ricci flow

Abstract: Two differentiable pinching theorems are verified via the Ricci flow and stable currents. We first prove a differentiable sphere theorem for positively pinched submanifolds in a space form. Moreover, we obtain a differentiable sphere theorem for submanifolds in the sphere $\mathbb{S}^{n+p}$ under extrinsic restriction.