73 research outputs found
Smooth representations and sheaves
The paper is concerned with `geometrization' of smooth (i.e. with open
stabilizers) representations of the automorphism group of universal domains,
and with the properties of `geometric' representations of such groups. As an
application, we calculate the cohomology groups of several classes of smooth
representations of the automorphism group of an algebraically closed extension
of infinite transcendence degree of an algebraically closed field.Comment: 20 pages, final versio
Higher dimensional abelian Chern-Simons theories and their link invariants
The role played by Deligne-Beilinson cohomology in establishing the relation
between Chern-Simons theory and link invariants in dimensions higher than three
is investigated. Deligne-Beilinson cohomology classes provide a natural abelian
Chern-Simons action, non trivial only in dimensions , whose parameter
is quantized. The generalized Wilson -loops are observables of the
theory and their charges are quantized. The Chern-Simons action is then used to
compute invariants for links of -loops, first on closed
-manifolds through a novel geometric computation, then on
through an unconventional field theoretic computation.Comment: 40 page
Semipurity of tempered Deligne cohomology
In this paper we define the formal and tempered Deligne cohomology groups,
that are obtained by applying the Deligne complex functor to the complexes of
formal differential forms and tempered currents respectively. We then prove the
existence of a duality between them, a vanishing theorem for the former and a
semipurity property for the latter. The motivation of these results comes from
the study of covariant arithmetic Chow groups. The semi-purity property of
tempered Deligne cohomology implies, in particular, that several definitions of
covariant arithmetic Chow groups agree for projective arithmetic varieties
Remarks on hard Lefschetz conjectures on Chow groups
We propose two conjectures of Hard Lefschetz type on Chow groups and prove
them for some special cases. For abelian varieties, we shall show they are
equivalent to well-known conjectures of Beauville and Murre.Comment: to appear in Sciences in China, Ser. A Mathematic
Cohomological Hasse principle and motivic cohomology for arithmetic schemes
In 1985 Kazuya Kato formulated a fascinating framework of conjectures which
generalizes the Hasse principle for the Brauer group of a global field to the
so-called cohomological Hasse principle for an arithmetic scheme. In this paper
we prove the prime-to-characteristic part of the cohomological Hasse principle.
We also explain its implications on finiteness of motivic cohomology and
special values of zeta functions.Comment: 47 pages, final versio
A Class of Topological Actions
We review definitions of generalized parallel transports in terms of
Cheeger-Simons differential characters. Integration formulae are given in terms
of Deligne-Beilinson cohomology classes. These representations of parallel
transport can be extended to situations involving distributions as is
appropriate in the context of quantized fields.Comment: 41 pages, no figure
Galois sections for abelianized fundamental groups
Given a smooth projective curve of genus at least 2 over a number field
, Grothendieck's Section Conjecture predicts that the canonical projection
from the \'etale fundamental group of onto the absolute Galois group of
has a section if and only if the curve has a rational point. We show that there
exist curves where the above map has a section over each completion of but
not over . In the appendix Victor Flynn gives explicit examples in genus 2.
Our result is a consequence of a more general investigation of the existence
of sections for the projection of the \'etale fundamental group `with
abelianized geometric part' onto the Galois group. We give a criterion for the
existence of sections in arbitrary dimension and over arbitrary perfect fields,
and then study the case of curves over local and global fields more closely. We
also point out the relation to the elementary obstruction of
Colliot-Th\'el\`ene and Sansuc.Comment: This is the published version, except for a characteristic 0
assumption added in Section 5 which was unfortunately omitted there. Thanks
to O. Wittenberg for noticing i
Noncommutative Geometry in the Framework of Differential Graded Categories
In this survey article we discuss a framework of noncommutative geometry with
differential graded categories as models for spaces. We outline a construction
of the category of noncommutative spaces and also include a discussion on
noncommutative motives. We propose a motivic measure with values in a motivic
ring. This enables us to introduce certain zeta functions of noncommutative
spaces.Comment: 19 pages. Minor corrections and one reference added; to appear in the
proceedings volume of AGAQ Istanbul, 200
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