On the fundamental group-scheme

We show that the fundamental group-scheme of a separably rationally connected variety defined over an algebraically closed field is trivial. Let X be a geometrically irreducible smooth projective variety defined over a finite field k admitting a k-rational point. Let {En,σn}_n?0 be a flat principal G-bundle over X, where G is a reductive linear algebraic group defined over k. We show that there is a positive integer a such that the principal G-bundle is isomorphic to E₀, where FX is the absolute Frobenius morphism of X. From this it follows that E₀ is given by a representation of the fundamental group-scheme of X in G.     

Barsotti–Tate groups and p-adic representations of the fundamental group scheme

On a scheme S over a base scheme B we study the category of locally constant BT groups, i.e. groups over S that are twists, in the flat topology, of BT groups defined over B. These groups generalize p-adic local systems and can be interpreted as integral p-adic representations of the fundamental group scheme of S/B (classifying finite flat torsors on the base scheme) when such a group exists. We generalize to these coefficients the Katz correspondence for p-adic local systems and show that they are closely related to the maximal nilpotent quotient of the fundamental group scheme.     

Globally F-regular and log Fano varieties

We prove that every globally F-regular variety is log Fano. In other words, if a prime characteristic variety X is globally F-regular, then it admits an effective Q-divisor Δ such that −KX−Δ is ample and (X,Δ) has controlled (Kawamata log terminal, in fact globally F-regular) singularities. A weak form of this result can be viewed as a prime characteristic analog of de Fernex and Hacon's new point of view on Kawamata log terminal singularities in the non-Q-Gorenstein case. We also prove a converse statement in characteristic zero: every log Fano variety has globally F-regular type. Our techniques apply also to F-split varieties, which we show to satisfy a “log Calabi-Yau” condition. We also prove a Kawamata-Viehweg vanishing theorem for globally F-regular pairs.     

The action of the etale fundamental group scheme on the connected component of the essentially finite one

We follow the pattern in [14, Section 4] to define an action of the etale fundamental group scheme on the local component of the essentially finite fundamental group scheme of Nori. We show that the associated representation is faithful when X is a curve of genus ⩾2.     

Monodromy for principal bundles

Given a strongly semistable principal bundle E_G over a curve, in Biswas et al. (2006) [4], a group-scheme for it was constructed, which was named as the monodromy group-scheme. Here we extend the construction of the monodromy group-scheme to principal bundles over higher dimensional varieties.     

On the stable principal Higgs sheaves

Let G be a connected reductive linear algebraic group defined over C with Lie algebra g. Let be a stable principal Higgs G-sheaf on a compact connected Kähler manifold. We consider all holomorphic sections of the adjoint vector bundle ad(EG) of EG that commute with the Higgs field φ. These correspond to the infinitesimal automorphisms of the principal Higgs G-sheaf. Any element of the center of g gives such a section. We prove that all the sections are given by the center of g.     

Uniqueness in the characteristic Cauchy problem of the Klein–Gordon equation and tame restrictions of generalized functions

We show that every tempered distribution, which is a solution of the (homogenous) Klein–Gordon equation, admits a “tame” restriction to the characteristic (hyper)surface {x ⁰ + x ⁿ = 0} in (1 + n)-dimensional Minkowski space and is uniquely determined by this restriction. The restriction belongs to the space which we have introduced in (Ullrich in J. Math. Phys. 45, 2004). Moreover, we show that every element of appears as the “tame” restriction of a solution of the (homogeneous) Klein–Gordon equation.     

On rational connectedness of globally F-regular threefolds

In this paper, we show that projective globally F  -regular threefolds, defined over an algebraically closed field of characteristic p≥11$p\ge 11$, are rationally chain connected.     

Stability of homogeneous principal bundles with a classical groupas the structure group

Let G be a connected semisimple linear algebraic group defined over an algebraically closed field k and P ⊂ G a parabolic subgroup without any simple factor. Let H be a connected reductive linear algebraic group defined over the field k such that all the simple quotients of H are of classical type. Take any homomorphism π : P → H such that the image of p is not contained in any proper parabolic subgroup of H. Consider the corresponding principal H-bundle E_P(H) = (G × H)/P over G/P. We prove that E_P (H) is strongly stable with respect to any polarization on G/P.     

Gated sets in metric spaces

The concept of a gated subset in a metric space is studied, and it is shown that properties of disjoint pairs of gated subsets can be used to investigate projections in Tits buildings.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

On the asymptotic behavior of the energy for the wave equations with time depending coefficients

We consider the asymptotic behavior of the total energy of solutions to the Cauchy problem for wave equations with time dependent propagation speed. The main purpose of this paper is that the asymptotic behavior of the total energy is dominated by the following properties of the coefficient: order of the differentiability, behavior of the derivatives as t → ∞ and stabilization of the amplitude described by an integral. Moreover, the optimality of these properties are ensured by actual examples. Supported by Grants-in-Aid for Young Scientists (B) (No.16740098), The Ministry of Education, Culture, Sports, Science and Technology.     

Holonomy group scheme of an integral curve

Let Y be a projective variety over a field k (of arbitrary characteristic). Assume that the normalization X of Y is such that is normal, being the algebraic closure of k. We define a notion of strong semistability for vector bundles on Y. We show that a vector bundle on Y is strongly semistable if and only if its pull back to X is strongly semistable and hence it is a tensor category. In case , we show that strongly semistable vector bundles on Y form a neutral Tannakian category. We define the holonomy group scheme of Y to be the Tannakian group scheme for this category. For a strongly semistable principal G‐bundle , we construct a holonomy group scheme. We show that if Y is an integral complex nodal curve, then the holonomy group of a strongly semistable vector bundle on Y is the Zariski closure of the (topological) fundamental group of Y.     

Greedy F-colorings of graphs

Let G=(V,E) be a connected graph. For a symmetric, integer-valued function δ on V×V, where K is an integer constant, N₀ is the set of nonnegative integers, and Z is the set of integers, we define a C-mapping by F(u,v,m)=δ(u,v)+m−K. A coloring c of G is an F-coloring if F(u,v,|c(u)−c(v)|)?0 for every two distinct vertices u and v of G. The maximum color assigned by c to a vertex of G is the value of c, and the F-chromatic number F(G) is the minimum value among all F-colorings of G. For an ordering of the vertices of G, a greedy F-coloring c of s is defined by (1) c(v₁)=1 and (2) for each i with 1?i<n, c(v_i+1) is the smallest positive integer p such that F(v_j,v_i+1,|c(v_j)−p|)?0, for each j with 1?j?i. The greedy F-chromatic number gF(s) of s is the maximum color assigned by c to a vertex of G. The greedy F-chromatic number of G is gF(G)=min{gF(s)} over all orderings s of V. The Grundy F-chromatic number is GF(G)=max{gF(s)} over all orderings s of V. It is shown that gF(G)=F(G) for every graph G and every F-coloring defined on G. The parameters gF(G) and GF(G) are studied and compared for a special case of the C-mapping F on a connected graph G, where δ(u,v) is the distance between u and v and .     

On the vertex group of n-manifolds

In this paper we obtain the decomposition of the vertex group of n-manifolds, extending the one given by Kauffman and Lins for dimension 3 and solving the related conjecture. The result is obtained in the more general category of gems: the vertex group of a gem , representing an n-manifold M, is the free product of n copies of the fundamental group of M and a free group F of rank N–n, where N is the number of n-residues of . In particular, for crystallizations FZ and consequently the vertex group is an invariant of M.     

Zariski closure and the dimension of the Gaussian low of the product of random matrices. I

Summary We prove the Central Limit Theorem for products of i.i.d. random matrices. The main aim is to find the dimension of the corresponding Gaussian law. It turns out that ifG is the Zariski closure of a group generated by the support of the distribution of our matrices, and ifG is semi-simple, then the dimension of the Gaussian law is equal to the dimension of the diagonal part of Cartan decomposition ofG.In this article we present a detailed exposition of results announced in [GGu]. For reasons explained in the introduction, this part is devoted to the case ofSL(m, ) group. The general semi-simple Lie group will be considered in the second part of the work.The central limit theorem for products of independent random matrices is our main topic, and the results obtained complete to a large extent the general picture of the subject.The proofs rely on methods from two theories. One is the theory of asymptotic behaviour of products of random matrices itself. As usual, the existence of distinct Lyapunov exponents is the most important fact here. The other is the theory of algebraic groups. We want to point out that algebraic language and methods play a very important role in this paper.In fact, this mixture of methods has already been used for the study of Lyapunov exponents in [GM1, GM2, GR3]. We believe that it is impossible to avoid the algebraic approach if one aims to obtain complete and effective answers to natural problems arising in the theory of products of random matrices.In order also to present the general picture of the subject we describe several results which are well known. Some of these can be proven for stationary sequences of matrices, others are true also for infinite dimensional operators (see e.g. [BL, O, GM2, L, R]). But our main concern is with independent matrices, in which case very precise and constructive statements can be obtained.     

On Waring's problem for many forms and Grassmann defective varieties

This paper is devoted to the classical Waring type problem for several algebraic forms. Its geometric translation in terms of Grassmann defectivity for projective varieties yields the best possible solution whenever the number of polynomials is greater than the number of variables. In particular, together with a classical theorem of Alessandro Terracini, this result gives a complete answer to Waring's problem for several ternary forms.