Showing papers in "Tohoku Mathematical Journal in 2018"

On the K-stability of Fano varieties and anticanonical divisors

Abstract: We apply a recent theorem of Li and the first author to give some criteria for the K-stability of Fano varieties in terms of anticanonical $\mathbb{Q}$-divisors. First, we propose a condition in terms of certain anti-canonical $\mathbb{Q}$-divisors of given Fano variety, which we conjecture to be equivalent to the K-stability. We prove that it is at least a sufficient condition and also related to the Berman-Gibbs stability. We also give another algebraic proof of the K-stability of Fano varieties which satisfy Tian's alpha invariants condition.

Worpitzky partitions for root systems and characteristic quasi-polynomials

Abstract: For a given irreducible root system, we introduce a partition of (coweight) lattice points inside the dilated fundamental parallelepiped into those of partially closed simplices. This partition can be considered as a generalization and a lattice points interpretation of the classical formula of Worpitzky. This partition, and the generalized Eulerian polynomial, recently introduced by Lam and Postnikov, can be used to describe the characteristic (quasi)polynomials of Shi and Linial arrangements. As an application, we prove that the characteristic quasi-polynomial of the Shi arrangement turns out to be a polynomial. We also present several results on the location of zeros of characteristic polynomials, related to a conjecture of Postnikov and Stanley. In particular, we verify the “functional equation” of the characteristic polynomial of the Linial arrangement for any root system, and give partial affirmative results on “Riemann hypothesis” for the root systems of type $E_6, E_7, E_8$, and $F_4$.

An elementary proof of Cohen-Gabber theorem in the equal characteristic $p>0$ case

Abstract: The aim of this article is to give a new proof of Cohen-Gabber theorem in the equal characteristic $p>0$ case.

Sharp $L^p$-bounds for the martingale maximal function

Abstract: The paper studies sharp weighted $L^p$ inequalities for the martingale maximal function. Proofs exploit properties of certain special functions of four variables and self-improving properties of $A_p$ weights.

Polar foliations on quaternionic projective spaces

Abstract: We classify irreducible polar foliations of codimension q on quaternionic projective spaces ℍPn, for all (n, q) ≠ (7,1). We prove that all irreducible polar foliations of any codimension (resp. of codimension one) on ℍPn are homogeneous if and only if n + 1 is a prime number (resp. n is even or n = 1). This shows the existence of inhomogeneous examples of codimension one and higher.

Closed three-dimensional Alexandrov spaces with isometric circle actions

Abstract: We obtain a topological and weakly equivariant classification of closed three-dimensional Alexandrov spaces with an effective, isometric circle action. This generalizes the topological and equivariant classifications of Raymond [26] and Orlik and Raymond [23] of closed three-dimensional manifolds admitting an effective circle action. As an application, we prove a version of the Borel conjecture for closed three-dimensional Alexandrov spaces with circle symmetry.

The primitive spectrum for $\mathfrak{gl}(m|n)$

Abstract: We study inclusions between primitive ideals in the universal enveloping algebra of general linear superalgebras. For classical simple Lie superalgebras, any primitive ideal is the annihilator of a simple highest weight module. It therefore suffices to study the quasi-order on highest weights determined by the relation of inclusion between primitive ideals. For the specific case of reductive Lie algebras, this quasi-order is essentially the left Kazhdan-Lusztig quasi-order. For Lie superalgebras, a description of the poset structure on the set primitive ideals is at the moment not known, apart from some low dimensional specific cases. We derive an alternative definition of the left Kazhdan-Lusztig quasi-order which extends to classical Lie superalgebras. We denote this quasi-order by $\unlhd$ and show that a relation in $\unlhd$ implies an inclusion between primitive ideals. For $\mathfrak{gl}(m|n)$ the new quasi-order $\unlhd$ is defined explicitly in terms of Brundan's Kazhdan-Lusztig theory. We prove that $\unlhd$ induces an actual partial order on the set of primitive ideals. We conjecture that this is the inclusion order. By the above paragraph one direction of this conjecture is true. We prove several consistency results concerning the conjecture and prove it for singly atypical and typical blocks of $\mathfrak{gl}(m|n)$ and in general for $\mathfrak{gl}(2|2)$. An important tool is a new translation principle for primitive ideals, based on the crystal structure underlying Brundan's categorification on category ${\mathcal O}$. Finally we focus on an interesting explicit example; the poset of primitive ideals contained in the augmentation ideal for $\mathfrak{gl}(m|1)$.

Parallel mean curvature tori in $\mathbb{C}P^{2}$ and $\mathbb{C}H^{2}$

Abstract: We explicitly determine tori that have a parallel mean curvature vector, both in the complex projective plane and the complex hyperbolic plane.

The equivalence of weak and very weak supersolutions to the porous medium equation

Abstract: We prove that various notions of supersolutions to the porous medium equation are equivalent under suitable conditions. More spesifically, we consider weak supersolutions, very weak supersolutions, and $m$-superporous functions defined via a comparison principle. The proofs are based on comparison principles and a Schwarz type alternating method, which are also interesting in their own right. Along the way, we show that Perron solutions with merely continuous boundary values are continuous up to the parabolic boundary of a sufficiently smooth space-time cylinder.

The structure of the space of polynomial solutions to the canonical central systems of differential equations on the block Heisenberg groups: A generalization of a theorem of Korányi

Abstract: A result of Koranyi that describes the structure of the space of polynomial solutions to the Heisenberg Laplacian operator is generalized to the canonical central systems on the block Heisenberg groups. These systems of differential operators generalize the Heisenberg Laplacian and, like it, admit large algebras of conformal symmetries. The main result implies that in most cases all polynomial solutions can be obtained from a single one by the repeated application of conformal symmetry operators.

$\sigma$-actions and symmetric triads

Abstract: For a given compact connected Lie group and an involution on it, we can define a hyperpolar action. We study the orbit space and the properties of each orbit of the action. The result is a natural extension of maximal torus theory.

On Frobenius manifolds from Gromov–Witten theory of orbifold projective lines with $r$ orbifold points

Abstract: We prove that the Frobenius structure constructed from the Gromov–Witten theory for an orbifold projective line with at most $r$ orbifold points is uniquely determined by the WDVV equations with certain natural initial conditions.

Modules of bilinear differential operators over the orthosymplectic superalgebra $\mathfrak{osp}(1|2)$

Abstract: Let $\frak{F}_\lambda, \lambda\in \mathbb{C}$, be the space of tensor densities of degree $\lambda$ on the supercircle $S^{1|1}$. We consider the superspace $\mathfrak{D}_{\lambda_1,\lambda_2,\mu}$ of bilinear differential operators from $\frak{F}_{\lambda_1}\otimes\frak{F}_{\lambda_2}$ to $\frak{F}_{\mu}$ as a module over the orthosymplectic superalgebra $\mathfrak{osp}(1|2)$. We prove the existence and the uniqueness of a canonical conformally equivariant symbol map from $\mathfrak{D}_{\lambda_1,\lambda_2,\mu}^k$ to the corresponding space of symbols. An explicit expression of the associated quantization map is also given.

Three consecutive approximation coefficients: asymptotic frequencies in semi-regular cases

Abstract: Denote by $p_n/q_n, n=1,2,3,\ldots,$ the sequence of continued fraction convergents of a real irrational number $x$. Define the sequence of approximation coefficients by $\theta_n(x):=q_n\left|q_nx-p_n\right|, n=1,2,3,\ldots$. In the case of regular continued fractions the six possible patterns of three consecutive approximation coefficients, such as $\theta_{n-1}<\theta_n<\theta_{n+1}$, occur for almost all $x$ with only two different asymptotic frequencies. In this paper it is shown how these asymptotic frequencies can be determined for two other semi-regular cases. It appears that the optimal continued fraction has a similar distribution of only two asymptotic frequencies, albeit with different values. The six different values that are found in the case of the nearest integer continued fraction will show to be closely related to those of the optimal continued fraction.

Wavelets in weighted norm spaces

Abstract: We give a complete characterization of the classes of weight functions for which the higher rank Haar wavelet systems are unconditional bases in weighted norm Lebesgue spaces. Particulary it follows that higher rank Haar wavelets are unconditional bases in the weighted norm spaces with weights which have strong zeros at some points. This shows that the class of weight functions for which higher rank Haar wavelets are unconditional bases is much richer than it was supposed.

A control theorem for the torsion Selmer pointed set

Abstract: Minhyong Kim defined the Selmer variety associated with a curve $X$ over a number field, which is a non-abelian analogue of the ${\mathbb Q}_p$-Selmer group of the Jacobian variety of $X$. In this paper, we define a torsion analogue of the Selmer variety. Recall that Mazur's control theorem describes the behavior of the torsion Selmer groups of an abelian variety with good ordinary reduction at $p$ in the cyclotomic tower of number fields. We give a non-abelian analogue of Mazur's control theorem by replacing the torsion Selmer group by a torsion analogue of the Selmer variety.

Stochastic calculus for Markov processes associated with semi-Dirichlet forms

Abstract: We present a new Fukushima type decomposition in the framework of semi-Dirichlet forms. This generalizes the result of Ma, Sun and Wang [17, Theorem 1.4] by removing the condition (S). We also extend Nakao's integral to semi-Dirichlet forms and derive Ito's formula related to it.

The isometry groups of compact manifolds with almost negative Ricci curvature

Abstract: We estimate the order of isometry groups of compact Riemannian manifolds which have negative Ricci curvature except for small portions, in terms of geometric quantities.

Double lines on quadric hypersurfaces

Abstract: We study double line structures in projective spaces and quadric hypersurfaces, and investigate the geometry of irreducible components of Hilbert scheme of curves and moduli of stable sheaves of pure dimension 1 on a smooth quadric threefold.

A coupling of Brownian motions in the $\mathcal{L}_0$-geometry

Abstract: Under a complete Ricci flow, we construct a coupling of two Brownian motions such that their $\mathcal{L}_0$-distance is a supermartingale. This recovers a result of Lott [J. Lott, Optimal transport and Perelman's reduced volume, Calc. Var. Partial Differential Equations 36 (2009), no. 1, 49–84.] on the monotonicity of $\mathcal{L}_0$-distance between heat distributions.

Large deviations for continuous additive functionals of symmetric Markov processes

Abstract: Let $X$ be a locally compact separable metric space and $m$ a positive Radon measure on $X$ with full topological support. Let ${\bf{M}}=(P_x,X_t)$ be an $m$-symmetric Markov process on $X$. Let $(\mathcal{E},\mathcal{D}(\mathcal{E}))$ be the Dirichlet form on $L^2(X;m)$ generated by ${\bf{M}}$. Let $\mu$ be a positive Radon measure in the Green-tight Kato class and $A^\mu_t$ the positive continuous additive functional in the Revuz correspondence to $\mu$. Under certain conditions, we establish the large deviation principle for positive continuous additive functionals $A^\mu_t$ of symmetric Markov processes.

Pseudo-Hermitian manifolds with automorphism group of maximal dimension

Abstract: This paper concerns a local characterization of 5-dimensional pseudo-Hermitian manifolds with maximal automorphism group in the case the underlying almost CR structures are not integrable. We also present examples of globally homogeneous model of maximal dimensional automorphism group.