5,903 research outputs found
o-minimal GAGA and a conjecture of Griffiths
We prove a conjecture of Griffiths on the quasi-projectivity of images of
period maps using algebraization results arising from o-minimal geometry.
Specifically, we first develop a theory of analytic spaces and coherent sheaves
that are definable with respect to a given o-minimal structure, and prove a
GAGA-type theorem algebraizing definable coherent sheaves on complex algebraic
spaces. We then combine this with algebraization theorems of Artin to show that
proper definable images of complex algebraic spaces are algebraic. Applying
this to period maps, we conclude that the images of period maps are
quasi-projective and that the restriction of the Griffiths bundle is ample.Comment: Comments welcome! v2: minor change
Vortex filament solutions of the Navier-Stokes equations
We consider solutions of the Navier-Stokes equations in with vortex
filament initial data of arbitrary circulation, that is, initial vorticity
given by a divergence-free vector-valued measure of arbitrary mass supported on
a smooth curve. First, we prove global well-posedness for perturbations of the
Oseen vortex column in scaling-critical spaces. Second, we prove local
well-posedness (in a sense to be made precise) when the filament is a smooth,
closed, non-self-intersecting curve. Besides their physical interest, these
results are the first to give well-posedness in a neighborhood of large
self-similar solutions of Navier-Stokes, as well as solutions which are
locally approximately self-similar.Comment: 89 page
Extent of stacking disorder in diamond
Hexagonal diamond has been predicted computationally to display extraordinary
physical properties including a hardness that exceeds cubic diamond. However, a
recent electron microscopy study has shown that so-called hexagonal diamond
samples are in fact not discrete materials but faulted and twinned cubic
diamond. We now provide a quantitative analysis of cubic and hexagonal stacking
in diamond samples by analysing X-ray diffraction data with the DIFFaX software
package. The highest fractions of hexagonal stacking we find in materials which
were previously referred to as hexagonal diamond are below 60%. The remainder
of the stacking sequences are cubic. We show that the cubic and hexagonal
sequences are interlaced in a complex way and that naturally occurring
Lonsdaleite is not a simple phase mixture of cubic and hexagonal diamond.
Instead, it is structurally best described as stacking disordered diamond. The
future experimental challenge will be to prepare diamond samples beyond 60%
hexagonality and towards the so far elusive 'perfect' hexagonal diamond
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