A generalization of Higgs bundles to higher dimensional varieties

Let X be a smooth n-dimensional projective variety defined over and let L be a line bundle on X. In this paper we shall construct a moduli space parametrizing -cohomology L-twisted Higgs pairs, i.e., pairs where E is a vector bundle on X and . If we take , the canonical line bundle on X, the variety is canonically identified with the cotangent bundle of the smooth locus of the moduli space of stable vector bundles on X and, as such, it has a canonical symplectic structure. We prove that, in the general case, in correspondence to the choice of a non-zero section , one can define, in a natural way, a Poisson structure on . We also analyze the relations between this Poisson structure on and the canonical symplectic structure of the cotangent bundle to the smooth locus of the moduli space of parabolic bundles over X, with parabolic structure over the divisor D defined by the section s. These results generalize to the higher dimensional case similar results proved in [Bo1] in the case of curves. Received November 4, 1997; in final form May 28, 1998     

On genera of smooth curves in higher dimensional varieties

We prove that for any smooth projective variety of dimension , there exists an integer , such that for any integer , there exists a smooth curve in with .

     

Character varieties of higher dimensional representations and splittings of 3-manifolds

Geometriae Dedicata - In 1983 Culler and Shalen established a way to construct essential surfaces in a 3-manifold from ideal points of the $$\mathrm {SL}_2$$ -character variety associated to the...     

KMS weights on higher rank buildings

We extend some of the results of Carey-Marcolli-Rennie on modular index invariants of Mumford curves to the case of higher rank buildings. We discuss notions of KMS weights on buildings, that generalize the construction of graph weights over graph C*-algebras.     

Decomposable subspaces,linear sections of Grassmann varieties,and higher weights of Grassmann codes

We consider the question of determining the maximum number of points on sections of Grassmannians over finite fields by linear subvarieties of the Plücker projective space of a fixed codimension. This corresponds to a known open problem of determining the complete weight hierarchy of linear error correcting codes associated to Grassmann varieties. We recover most of the known results as well as prove some new results. A basic tool used is a characterization of decomposable subspaces of exterior powers, that is, subspaces in which every nonzero element is decomposable. Also, we use a generalization of the Griesmer–Wei bound that is proved here for arbitrary linear codes.     

On the birational unboundedness of higher dimensional {\mathbb{Q}}-Fano varieties

We show that the family of (-factorial and log terminal) -Fano n-folds with Picard number one is birationally unbounded for n ≥ 6. T. Okada is partially supported by JSPS Research Fellowships for Young Scientists.     

Higher dimensional Enriques varieties and automorphisms of generalized Kummer varieties

We define Enriques varieties as a higher dimensional generalization of Enriques surfaces and construct examples by using fixed point free automorphisms on generalized Kummer varieties. We also classify all automorphisms of generalized Kummer varieties that come from an automorphism of the underlying abelian surface.     

Near varieties of idempotent generated completely simple semigroups

Algebra universalis - Completely simple semigroups form a variety if we consider them as algebras with multiplication and inversion. A near variety of idempotent generated completely simple...     

Generalized bifurcation of higher dimensional tori

From the structural stability viewpoint vector fields on M which are tangent to the boundary of M are analysed. A class of vector fields is characterized as structurally stable; this class corresponds to the class of Morse-Smale vector fields on closed manifolds, studied by Palis, Peixoto Smale, and others. In this class, new phenomena occur as saddle connections along the boundary of M which are persistent by small perturbations. Thus, the techniques introduced by J. Palis (Topology8 (1969)), which inspired the proof of the stability of a vector field in such a class, were substantially changed. Such modifications represent a main difficulty in extending our results to higher dimensions.     

Three dimensional linear motion transformation in a higher order dimensional space

The objective of the work presented in this paper is an attempt at solving and transforming of the known from the classical mechanics three dimensional – single mass mathematical and mechanical vibration models in a higher order dimensional space with any virtual sectional curvature – positive or negative, constant or variable. The object of the investigation is a class of three dimensional surfaces. The aims of the work presented in this paper are to illustrate the performance of the common algorithm in three dimensional linear motion transformation, that means to transform 3D space in a higher order dimensional space and a comparison is derived on the behavior of the common algorithm depending on the surface properties. A characterization of the Riemannian Manifolds is performed by means of curvature operators in the three dimensional solution. The computer codes Mathematica and MATLAB are used in the numerical simulation. The system motion is investigated in a 3-D qualitative aspect in time and frequency domain. The application can be in topology when geodesists make snap shots of the surface profile, then the curved lines can be analyzed and transformed in the desired space dimension. Any kind of a trajectory of motion can be transformed successfully in a higher order dimensional space and vice verse by means of applying of the common algorithm.     

Maximal pattern complexity of higher dimensional words

This paper studies the pattern complexity of n-dimensional words. We show that an n-recurrent but not n-periodic word has pattern complexity at least 2k, which generalizes the result of [T. Kamae, H. Rao, Y.-M. Xue, Maximal pattern complexity of two dimension words, Theoret. Comput. Sci. 359 (1-3) (2006) 15-27] on two-dimensional words. Analytic directions of a word are defined and its topological properties play a crucial role in the proof.Accordingly n-dimensional pattern Sturmian words are defined. Irrational rotation words are proved to be pattern Sturmian. A new class of higher dimensional words, the simple Toeplitz words, are introduced. We show that they are also pattern Sturmian words.