5,519 research outputs found
Phase space distribution of Gabor expansions
We present an example of a complete and minimal Gabor system consisting of
time-frequency shifts of a Gaussian, localized at the coordinate axes in the
time-frequency plane (phase space). Asymptotically, the number of
time-frequency shifts contained in a disk centered at the origin is only 2/pi
times the number of points from the von Neumann lattice found in the same disk.
Requiring a certain regular distribution in phase space, we show that our
system has minimal density among all complete and minimal systems of
time-frequency shifts of a Gaussian.Comment: 9 pages, submitted to publicatio
Deligne pairings and families of rank one local systems on algebraic curves
For smooth families of projective algebraic curves, we extend the notion of
intersection pairing of metrized line bundles to a pairing on line bundles with
flat relative connections. In this setting, we prove the existence of a
canonical and functorial "intersection" connection on the Deligne pairing. A
relationship is found with the holomorphic extension of analytic torsion, and
in the case of trivial fibrations we show that the Deligne isomorphism is flat
with respect to the connections we construct. Finally, we give an application
to the construction of a meromorphic connection on the hyperholomorphic line
bundle over the twistor space of rank one flat connections on a Riemann
surface.Comment: 48 pp. 1 figur
BCOV invariants of Calabi--Yau manifolds and degenerations of Hodge structures
Calabi--Yau manifolds have risen to prominence in algebraic geometry, in part
because of mirror symmetry and enumerative geometry. After
Bershadsky--Cecotti--Ooguri--Vafa (BCOV), it is expected that genus 1 curve
counting on a Calabi--Yau manifold is related to a conjectured invariant, only
depending on the complex structure of the mirror, and built from Ray--Singer
holomorphic analytic torsions. To this end, extending work of
Fang--Lu--Yoshikawa in dimension 3, we introduce and study the BCOV invariant
of Calabi--Yau manifolds of arbitrary dimension. To determine it, knowledge of
its behaviour at the boundary of moduli spaces is imperative. We address this
problem by proving precise asymptotics along one-parameter degenerations, in
terms of topological data and intersection theory. Central to the approach are
new results on degenerations of metrics on Hodge bundles, combined with
information on the singularities of Quillen metrics in our previous work.Comment: Minor revision. Mainly restructure of the text, minor improvements
and corrections. Added information about subdominant terms of -norm
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