390 research outputs found
Fibre tilings
Generalizing an earlier notion of secondary polytopes, Billera and Sturmfels introduced the important concept of fibre polytopes, and showed how they were related to certain kinds of subdivision induced by the projection of one polytope onto another. There are two obvious ways in which this concept can be extended: first, to possibly unbounded polyhedra, and second, by making the definition a categorical one. In the course of these investigations, it became clear that the whole subject fitted even more naturally into the context of finite tilings which admit strong duals. In turn, this new approach provides more unified and perspicuous explanations of many previously known but apparently quite disparate results
Toric moment mappings and Riemannian structures
Coadjoint orbits for the group SO(6) parametrize Riemannian G-reductions in
six dimensions, and we use this correspondence to interpret symplectic
fibrations between these orbits, and to analyse moment polytopes associated to
the standard Hamiltonian torus action on the coadjoint orbits. The theory is
then applied to describe so-called intrinsic torsion varieties of Riemannian
structures on the Iwasawa manifold.Comment: 25 pages, 14 figures; Geometriae Dedicata 2012, Toric moment mappings
and Riemannian structures, available at
http://www.springerlink.com/content/yn86k22mv18p8ku2
Log Hodge groups on a toric Calabi-Yau degeneration
We give a spectral sequence to compute the logarithmic Hodge groups on a
hypersurface type toric log Calabi-Yau space, compute its E_1 term explicitly
in terms of tropical degeneration data and Jacobian rings and prove its
degeneration at E_2 under mild assumptions. We prove the basechange of the
affine Hodge groups and deduce it for the logarithmic Hodge groups in low
dimensions. As an application, we prove a mirror symmetry duality in dimension
two and four involving the usual Hodge numbers, the stringy Hodge numbers and
the affine Hodge numbers.Comment: 49 pages, 3 figure
Non-archimedean canonical measures on abelian varieties
For a closed d-dimensional subvariety X of an abelian variety A and a
canonically metrized line bundle L on A, Chambert-Loir has introduced measures
on the Berkovich analytic space associated to A with
respect to the discrete valuation of the ground field. In this paper, we give
an explicit description of these canonical measures in terms of convex
geometry. We use a generalization of the tropicalization related to the Raynaud
extension of A and Mumford's construction. The results have applications to the
equidistribution of small points.Comment: Thorough revision according to the comments of the referee. To appear
in Compositi
Arithmetic geometry of toric varieties. Metrics, measures and heights
We show that the height of a toric variety with respect to a toric metrized
line bundle can be expressed as the integral over a polytope of a certain
adelic family of concave functions. To state and prove this result, we study
the Arakelov geometry of toric varieties. In particular, we consider models
over a discrete valuation ring, metrized line bundles, and their associated
measures and heights. We show that these notions can be translated in terms of
convex analysis, and are closely related to objects like polyhedral complexes,
concave functions, real Monge-Amp\`ere measures, and Legendre-Fenchel duality.
We also present a closed formula for the integral over a polytope of a function
of one variable composed with a linear form. This allows us to compute the
height of toric varieties with respect to some interesting metrics arising from
polytopes. We also compute the height of toric projective curves with respect
to the Fubini-Study metric, and of some toric bundles.Comment: Revised version, 230 pages, 3 figure
- …