This paper is devoted to studying the centre of the multi-parameter quantum group $U_{q,G}(\mathfrak{g})$ introduced by Okado and Yamane, where $\mathfrak{g}$ is a complex simple Lie algebra, and all parameters lie in general position. We mainly establish the Harish-Chandra theorem, proving that the Harish-Chandra homomorphism is an isomorphism; in particular, we determine the centre $Z(U_{q,G}) \cong (U^{0}_{\flat})^{W}$ is isomorphic to a polynomial algebra or a quotient algebra of a polynomial algebra. The same result holds for the $(U^{0}_{\flat})^{W}$ of the two-parameter quantum group $U_{r,s}(\mathfrak{g})$.
Given a complex variety X, suppose there is a degeneration X of it to a union of toric varieties glued along toric divisors. Then, one can use tropical curves to study holomorphic curves on X through degenerate curves on the central fiber X0 of X. In such studies, the case corresponding to so-called non-superabundant tropical curves is relatively well understood. This is the case where the curve satisfies a version of transversality. On the other hand, in the case corresponding to superabundant tropical curves, where transversality fails, not much is known. The first important step for such a study is describing the obstruction to deforming degenerate curves on X0 to generic fibers of X. In this paper, we present a general formula describing such an obstruction. In the study of superabundant tropical curves, allowing higher-valent vertices (i.e., those with valency greater than three) is essential, although there have been few studies of them in this context. Our formula covers such cases.
Let M be a surface with a Riemannian metric and UM the unit tangent bundle over M with the canonical contact sub-Riemannian structure D C T(UM). In this paper, the complete local classification of singularities, under the Legendre projection UM-+ M, is given for sub-Riemannian geodesics of (UM, D). Legendre singularities of sub-Riemannian geodesics are classified completely also for another Legendre projection from UM to the space of Riemannian geodesics on M. The duality on Legendre singularities is observed related to the pendulum motion.
Let k0 be an algebraic number field of finite degree, So be a finite set of primes and Ls, be the field obtained by adjoining to ko all primitive q-th roots of unity, where q runs over all primes not belonging to So. We shall consider, for an odd prime 1, the maximal unramified pro-1 abelian extension of Lso and investigate the structure of this Galois group with certain cyclotomic action.
For a resolution space (X-, E) of a normal complex surface singularity (X, o), the fundamental cycle Z(E) and maximal ideal cycle M-E are important geometric objects associated to (X, o), which satisfy M-E (sic) Z(E). In 1966, M. Artin proved that M-E = Z(E) for all resolutions of all rational singularities. However, for non-rational singularities, it is a delicate problem whether M-E = Z(E) or not. Any normal surface double point (i.e., multiplicity two) is a hypersurface singularity defined by z(2) = f(x, y). For such singularities, we prove that M-E > Z(E) holds on the minimal resolution if and only if f has a canonical decomposition f = f ([L])f ([c]) f([0]) in C{x, y} called "Laufer decomposition". Moreover, we give a numerical procedure to determine whether M-E = Z(E) or not on the minimal resolution from the embedded topology of the branch curve singularity ({f = 0}, o).
We give a description of shape theory using finite topological To-spaces (finite partially ordered sets). This description serves as a first step towards developing computational methods in shape theory for future works. Additionally, we introduce the notion of core for inverse sequences of finite topological spaces and prove some properties.
In this paper, we make some progress about a boundary behavior of the almost-complete Kahler-Einstein metric of negative Ricci curvature on a quasi-projective manifold with semiample log-canonical bundle. First its volume growth near the boundary is investigated in terms of the Kodaira dimension of the boundary, and then we characterize the boundary to be of general type via the volume growth. Moreover the volume growth is determined in the case of a Calabi-Yau boundary. We also affirmatively solve a modified version of the conjecture suggested previously by the author about the residue of the Kahler-Einstein metric if the boundary is a smooth finite quotient of an abelian variety.
We define a totally geodesic immersion of irreducible type from a symmetric space of compact type into a Grassmannian and classify such immersions. Any totally geodesic immersion is related to a homogeneous vector bundle with the canonical connection and the eigenspaces of the Laplace operator acting on the space of sections of the bundle.
Consider a d-dimensional branching Brownian motion starting with a single particle at the origin and let nt be the number of particles at time t whose ancestral lines have remained up to t within a ball of radius r(t) centered at the origin, where r(t) increases sublinearly with t. We obtain a full limit large-deviation result as time tends to infinity on the probability that nt is atypically small. A phase transition is identified, at which the nature of the optimal strategy to realize the aforementioned large-deviation event changes, and the Lyapunov exponent giving the decay rate of the associated large-deviation probability is continuous. As a corollary, we also obtain a kind of law of large numbers for nt under the stronger assumption that r(t) increases subdiffusively with t.
We state several questions, and prove some partial results, about the Chow ring A^∗(X) of complete intersections in projective space. For one thing, we prove that if X is a general Calabi-Yau hypersurface, the intersection product A^2(X)· A^i(X) is one-dimensional, for any i>0. We also show that quintic threefolds have a multiplicative Chow-Künneth (MCK) decomposition. We wonder whether all Calabi-Yau hypersurfaces might have an MCK decomposition, and prove this is the case conditional to a conjecture of Voisin.
This note proves orbifold versions of Kobayashi's theorem. The main result asserts that a compact Kähler orbifold with non-negative Ricci curvature, along with certain conditions regarding singularities, is simply connected.
We propose new invariants in equivariant birational geometry, combining equivariant intermediate Jacobians and the Burnside formalism, for smooth rationally connected threefolds with actions of finite groups.
We show that the canonical actions of the Thompson group V and its generalizations on the Cantor set are not strongly ergodic. This implies that the associated crossed product von Neumann algebras are not full. This also yields a non-embedding result for the Thompson groups.
In this paper, we define a new concept of Noetherian commutative rings which stands between Gorenstein and Cohen-Macaulay properties. We show that this new property keep hold under common operations of commutative rings such as localization, polynomial extension and under mild assumptions, flat extension, tensor product, Segre product and so on. We show that for Schubert cycles, the Ehrhart rings of cycle graphs and perfect graphs, this new concept is close to Gorenstein property.
Let k be an algebraically closed uncountable field of characteristic zero. Let R be a complete local hypersurface over k. Denote by CM(R) the category of maximal Cohen-Macaulay R-modules and by D^sg(R) the singularity category of R. Denote by CM_0(R) the full category of CM(R) consisting of modules that are locally free on the punctured spectrum of R, and by D^sg_0(R) the full subcategory of D^sg(R) consisting of objects that are locally zero on the punctured spectrum of R. In this paper, under the assumption that R has finite or countable CM-representation type, we completely classify the extension-closed subcategories of CM_0(R) in dimension at most two, and the extension-closed subcategories of D^sg_0(R) in arbitrary dimension.
Chern's conjecture states that a closed minimal hypersurface in the euclidean sphere is isoparametric if it has constant scalar curvature. When the number g of distinct principal curvatures is greater than three, few satisfactory results have been known. We attack the conjecture in the Dupin hypersurface case. Our results are: A closed proper Dupin hypersurface with constant mean curvature is isoparametric (i) if g=3, (ii) if g=4 and has constant scalar curvature, or (iii) if g=4 and has constant Lie curvature, and (iv) if g=6 and has constant Lie curvatures. These cover all the non-trivial cases for a closed proper Dupin to be isoparametric since g can take only values 1,2,3,4,6. The originality of the proof is a use of topology and geometry, which reduces assumptions needed in the algebraic argument.
The concept of concrete regularity structure gives the algebraic backbone of the operations involved in the local expansions used in the regularity structure approach to singular stochastic partial differential equations. The spaces and the details of the structures depend on each equation. We introduce here a parameter-dependent universal algebraic regularity structure that can host all the regularity structures used in the study of singular stochastic partial differential equations. This is done by using the correspondence between the notions of model on a regularity structure and the notion of paracontrolled system. We prove that the iterated paraproducts that form the fundamental bricks of paracontrolled systems have some local expansion properties that are governed by this universal structure.
The notion of the Yau sequence was introduced by Tomaru, as an attempt to extend Yau's elliptic sequence for (weakly) elliptic singularities to normal surface singularities of higher fundamental genera. In this paper, we obtain the canonical cycle using the Yau cycle for certain surface singularities of degree two. Furthermore, we obtain a formula of arithmetic genera. We also give some properties about the classification of weighted dual graphs of certain surface singularities of degree two.
