8,233 research outputs found
Symmetric spaces of higher rank do not admit differentiable compactifications
Any nonpositively curved symmetric space admits a topological
compactification, namely the Hadamard compactification. For rank one spaces,
this topological compactification can be endowed with a differentiable
structure such that the action of the isometry group is differentiable.
Moreover, the restriction of the action on the boundary leads to a flat model
for some geometry (conformal, CR or quaternionic CR depending of the space).
One can ask whether such a differentiable compactification exists for higher
rank spaces, hopefully leading to some knew geometry to explore. In this paper
we answer negatively.Comment: 13 pages, to appear in Mathematische Annale
Discrete isometry groups of symmetric spaces
This survey is based on a series of lectures that we gave at MSRI in Spring
2015 and on a series of papers, mostly written jointly with Joan Porti. Our
goal here is to:
1. Describe a class of discrete subgroups of higher rank
semisimple Lie groups, which exhibit some "rank 1 behavior".
2. Give different characterizations of the subclass of Anosov subgroups,
which generalize convex-cocompact subgroups of rank 1 Lie groups, in terms of
various equivalent dynamical and geometric properties (such as asymptotically
embedded, RCA, Morse, URU).
3. Discuss the topological dynamics of discrete subgroups on flag
manifolds associated to and Finsler compactifications of associated
symmetric spaces . Find domains of proper discontinuity and use them to
construct natural bordifications and compactifications of the locally symmetric
spaces .Comment: 77 page
Critical Wilson Lines in Toroidal Compactifications of Heteroric Strings
Critical values of Wilson lines and general background fields for toroidal
compactifications of heterotic string theories are constructed systematically
using Dynkin diagrams.Comment: 32 pages, LATE
Modular subvarieties of arithmetic quotients of bounded symmetric domains
Arithmetic quotients are quotients of bounded symmetric domains by arithmetic
groups, and modular subvarieties of arithmetic quotients are themselves
arithmetic quotients of lower dimension which live on arithmetic quotients, by
an embedding induced from an inclusion of groups of hermitian type. We show the
existence of such modular subvarieties, drawing on earlier work of the author.
If is a fixed arithmetic subgroup, maximal in some sense, then we
introduce the notion of ``-integral symmetric'' subgroups, which in
turn defines a notion of ``integral modular subvarieties'', and we show that
there are finitely many such on an (isotropic, i.e, non-compact) arithmetic
variety.Comment: 48 pages, also available at http://www.mathematik.uni-kl.de/~wwwagag/
LaTeX (e-mail: [email protected]
Compactifications of moduli spaces inspired by mirror symmetry
We study moduli spaces of nonlinear sigma-models on Calabi-Yau manifolds,
using the one-loop semiclassical approximation. The data being parameterized
includes a choice of complex structure on the manifold, as well as some ``extra
structure'' described by means of classes in H^2. The expectation that this
moduli space is well-behaved in these ``extra structure'' directions leads us
to formulate a simple and compelling conjecture about the action of the
automorphism group on the K\"ahler cone. If true, it allows one to apply
Looijenga's ``semi-toric'' technique to construct a partial compactification of
the moduli space. We explore the implications which this construction has
concerning the properties of the moduli space of complex structures on a
``mirror partner'' of the original Calabi-Yau manifold. We also discuss how a
similarity which might have been noticed between certain work of Mumford and of
Mori from the 1970's produces (with hindsight) evidence for mirror symmetry
which was available in 1979. [The author is willing to mail hardcopy preprints
upon request.]Comment: 25 pp., LaTeX 2.09 with AmS-Font
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