74,415 research outputs found
Unitary grassmannians
We study projective homogeneous varieties under an action of a projective
unitary group (of outer type). We are especially interested in the case of
(unitary) grassmannians of totally isotropic subspaces of a hermitian form over
a field, the main result saying that these grassmannians are 2-incompressible
if the hermitian form is generic. Applications to orthogonal grassmannians are
provided.Comment: 25 page
Non-archimedean amoebas and tropical varieties
We study the non-archimedean counterpart to the complex amoeba of an
algebraic variety, and show that it coincides with a polyhedral set defined by
Bieri and Groves using valuations. For hypersurfaces this set is also the
tropical variety of the defining polynomial. Using non-archimedean analysis and
a recent result of Conrad we prove that the amoeba of an irreducible variety is
connected. We introduce the notion of an adelic amoeba for varieties over
global fields, and establish a form of the local-global principle for them.
This principle is used to explain the calculation of the nonexpansive set for a
related dynamical system.Comment: 19 pages, 2 figures. Added AIM preprint numbe
On compatibility between isogenies and polarisations of abelian varieties
We discuss the notion of polarised isogenies of abelian varieties, that is,
isogenies which are compatible with given principal polarisations. This is
motivated by problems of unlikely intersections in Shimura varieties. Our aim
is to show that certain questions about polarised isogenies can be reduced to
questions about unpolarised isogenies or vice versa.
Our main theorem concerns abelian varieties B which are isogenous to a fixed
abelian variety A. It establishes the existence of a polarised isogeny A to B
whose degree is polynomially bounded in n, if there exist both an unpolarised
isogeny A to B of degree n and a polarised isogeny A to B of unknown degree. As
a further result, we prove that given any two principally polarised abelian
varieties related by an unpolarised isogeny, there exists a polarised isogeny
between their fourth powers.
The proofs of both theorems involve calculations in the endomorphism algebras
of the abelian varieties, using the Albert classification of these endomorphism
algebras and the classification of Hermitian forms over division algebras
The boundary manifold of a complex line arrangement
We study the topology of the boundary manifold of a line arrangement in CP^2,
with emphasis on the fundamental group G and associated invariants. We
determine the Alexander polynomial Delta(G), and more generally, the twisted
Alexander polynomial associated to the abelianization of G and an arbitrary
complex representation. We give an explicit description of the unit ball in the
Alexander norm, and use it to analyze certain Bieri-Neumann-Strebel invariants
of G. From the Alexander polynomial, we also obtain a complete description of
the first characteristic variety of G. Comparing this with the corresponding
resonance variety of the cohomology ring of G enables us to characterize those
arrangements for which the boundary manifold is formal.Comment: This is the version published by Geometry & Topology Monographs on 22
February 200
Big Line Bundles over Arithmetic Varieties
We prove a Hilbert-Samuel type result of arithmetic big line bundles in
Arakelov geometry, which is an analogue of a classical theorem of Siu. An
application of this result gives equidistribution of small points over
algebraic dynamical systems, following the work of Szpiro-Ullmo-Zhang. We also
generalize Chambert-Loir's non-archimedean equidistribution
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