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Abstract
After reviewing some “fundamental group schemes” that can be attached to a variety by means of Tannaka duality, we consider the example of the Higgs fundamental group scheme, surveying its main properties and relations with the other fundamental groups, and giving some examples.
The paper aims to deduce the relation between the category of topology and algebra from viewpoint of geometry and dynamical system. We introduce and define a dynamical manifold as a manifold associated with a time parameter. We obtain the induced chain of topological dynamics on the fundamental group from the chain of dynamical maps on a dynamical manifold. For many adjunctions in this context, we deduce the limit topological dynamics and conditional topological dynamics on the fundamental group. We use the category of commutative diagrams as chains of dynamical manifolds to deduce the chains on fundamental groups. Also, we describe how the manifold changes in a dynamical system from the view of the fundamental group.
Let G be a semisimple linear algebraic group defined over an algebraically closed field k . Fix a smooth projective curve X defined over k , and also fix a closed point . Given any strongly semistable principal G -bundle over X , we construct an affine algebraic group scheme defined over k , which we call the monodromy of . The monodromy group scheme is a subgroup scheme of the fiber over x of the adjoint bundle for . We also construct a reduction of structure group of the principal G -bundle to its monodromy group scheme. The construction of this reduction of structure group involves a choice of a closed point of over x . An application of the monodromy group scheme is given. We prove the existence of strongly stable principal G -bundles with monodromy G
A finitely generated group Γ is called representation rigid (briefly, rigid) if for every n , Γ has only finitely many classes of simple ℂ representations in dimension n . Examples include higher rank S -arithmetic groups. By Margulis super rigidity, the latter have a stronger property: they are representation super rigid ; i.e., their proalgebraic completion is finite dimensional. We construct examples of nonlinear rigid groups which are not super rigid, and which exhibit every possible type of infinite dimensionality. Whether linear representation rigid groups are super rigid remains an open question. Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/42819/1/10711_2004_Article_381723.pdf
These are the lecture notes for a short course on tensor categories. The coverage in these notes is relatively non-technical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something interesting in them. Once the basic definitions are given, the focus is mainly on k-linear categories with finite dimensional hom-spaces. Connections with quantum groups and low dimensional topology are pointed out, but these notes have no pretension to cover the latter subjects at any depth. Essentially, these notes should be considered as annotations to the extensive bibliography. Comment: 57 pages. Notes for a three-hour lecture course. Extensively revised, updated and extended. To appear in Revista de la Uni\'on Matem\'atica Argentina
This is a survey paper on Bruzzo’s Conjecture, which characterizes semistable Higgs bundles with vanishing discriminant in terms of their behaviour when restricted to curves.
Relying on a notion of "numerical effectiveness" for Higgs bundles, we show that the category of "numerically flat" Higgs vector bundles on a smooth projective variety X is a Tannakian category. We introduce the associated group scheme, that we call the "Higgs fundamental group scheme of X," and show that its properties are related to a conjecture about the vanishing of the Chern classes of numerically flat Higgs vector bundles.
We determine some classes of varieties X - that include the varieties with
numerically effective tangent bundle - satisfying the following property: if E
is a Higgs bundle such that f*E is semistable for any morphism f from a smooth
projective curve to X, then E is slope semistable and the discriminant of E is
zero in H4(X,R). We also characterize some classes of varieties such that the
underlying vector bundle of a slope semistable Higgs bundle is always slope
semistable.
In the first section it is shown how to introduce on an abstract category operations of tensor products and duals having properties similar to the familiar operations on the category Vec
k of finite-dimensional vector spaces over a field k. What complicates this is the necessity of including enough constraints so that, whenever an obvious isomorphism (e.g.,
exists in Vec
k, a unique isomorphism is constrained to exist also in the abstract setting.
Generalizing a result of Miyaoka, we prove that the semistability of a vector bundle E on a smooth projective curve over a field of characteristic zero is equivalent to the nefness of any of certain divisorial classes θs, λs in the Grassmannians Grs(E) of locally-free quotients of E and in the projective bundles PQs, respectively (here 0<s<rkE and Qs is the universal quotient bundle on Grs(E)). The result is extended to Higgs bundles. In that case a necessary and sufficient condition for semistability is that all classes λs are nef. We also extend this result to higher-dimensional complex projective varieties by showing that the nefness of the classes λs is equivalent to the semistability of the bundle E together with the vanishing of the characteristic class .
We study Miyaoka-type semistability criteria for principal Higgs G-bundles E on complex projective manifolds of any dimension. We prove that E has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of G, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class.In a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principal (Higgs) bundles, discussing their main properties. For (non-Higgs) principal bundles, we show that a numerically flat principal bundle admits a reduction to a Levi factor which has a flat Hermitian–Yang–Mills connection, and, as a consequence, that the cohomology ring of a numerically flat principal bundle with coefficients in R is trivial. To our knowledge this notion of numerical effectiveness is new even in the case of (non-Higgs) principal bundles.
We use coverings by smooth projective varieties then apply nonabelian Hodge techniques to study the topology of proper Deligne-Mumford stacks as well as more general simplicial varieties.
The S-fundamental group scheme is the group scheme corresponding to the
Tannaka category of numerically flat vector bundles. We use determinant line
bundles to prove that the S-fundamental group of a product of two complete
varieties is a product of their S-fundamental groups as conjectured by V. Mehta
and the author. We also compute the abelian part of the S-fundamental group
scheme and the S-fundamental group scheme of an abelian variety or a variety
with trivial etale fundamental group.
We introduce a new fundamental group scheme for varieties defined over an algebraically closed field of positive characteristic and we use it to study generalization of some of C. Simpson's results to positive characteristic. We also study the properties of this group and we prove Lefschetz type theorems. Comment: 37 pages; final version will appear in Ann. Inst. Fourier 61 (2011)
After providing a suitable definition of numerical effectiveness for Higgs bundles, and a related notion of numerical flatness, in this paper we prove, together with some side results, that all Chern classes of a Higgs-numerically flat Higgs bundle vanish, and that a Higgs bundle is Higgs-numerically flat if and only if it is has a filtration whose quotients are flat stable Higgs bundles. We also study the relation between these numerical properties of Higgs bundles and (semi)stability.