17,001 research outputs found
The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves
Let K be an algebraically closed field which is complete with respect to a
nontrivial, non-Archimedean valuation and let \Lambda be its value group. Given
a smooth, proper, connected K-curve X and a skeleton \Gamma of the Berkovich
analytification X^\an, there are two natural real tori which one can consider:
the tropical Jacobian Jac(\Gamma) and the skeleton of the Berkovich
analytification Jac(X)^\an. We show that the skeleton of the Jacobian is
canonically isomorphic to the Jacobian of the skeleton as principally polarized
tropical abelian varieties. In addition, we show that the tropicalization of a
classical Abel-Jacobi map is a tropical Abel-Jacobi map. As a consequence of
these results, we deduce that \Lambda-rational principal divisors on \Gamma, in
the sense of tropical geometry, are exactly the retractions of principal
divisors on X. We actually prove a more precise result which says that,
although zeros and poles of divisors can cancel under the retraction map, in
order to lift a \Lambda-rational principal divisor on \Gamma to a principal
divisor on X it is never necessary to add more than g extra zeros and g extra
poles. Our results imply that a continuous function F:\Gamma -> R is the
restriction to \Gamma of -log|f| for some nonzero meromorphic function f on X
if and only if F is a \Lambda-rational tropical meromorphic function, and we
use this fact to prove that there is a rational map f : X --> P^3 whose
tropicalization, when restricted to \Gamma, is an isometry onto its image.Comment: 21 pages, 1 figur
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The Feasibility of a Fully Miniaturized Magneto-Optical Trap for Portable Ultracold Quantum Technology
Experiments using laser cooled atoms and ions show real promise for practical
applications in quantum- enhanced metrology, timing, navigation, and sensing as
well as exotic roles in quantum computing, networking and simulation. The heart
of many of these experiments has been translated to microfabricated platforms
known as atom chips whose construction readily lend themselves to integration
with larger systems and future mass production. To truly make the jump from
laboratory demonstrations to practical, rugged devices, the complex surrounding
infrastructure (including vacuum systems, optics, and lasers) also needs to be
miniatur- ized and integrated. In this paper we explore the feasibility of
applying this approach to the Magneto-Optical Trap; incorporating the vacuum
system, atom source and optical geometry into a permanently sealed micro- litre
system capable of maintaining mbar for more than 1000 days of
operation with passive pumping alone. We demonstrate such an engineering
challenge is achievable using recent advances in semiconductor microfabrication
techniques and materialsComment: 23 pages, 10 figure
On the structure of nonarchimedean analytic curves
Let K be an algebraically closed, complete nonarchimedean field and let X be
a smooth K-curve. In this paper we elaborate on several aspects of the
structure of the Berkovich analytic space X^an. We define semistable vertex
sets of X^an and their associated skeleta, which are essentially finite metric
graphs embedded in X^an. We prove a folklore theorem which states that
semistable vertex sets of X are in natural bijective correspondence with
semistable models of X, thus showing that our notion of skeleton coincides with
the standard definition of Berkovich. We use the skeletal theory to define a
canonical metric on H(X^an) := X^an - X(K), and we give a proof of Thuillier's
nonarchimedean Poincar\'e-Lelong formula in this language using results of
Bosch and L\"utkebohmert.Comment: 23 pages. This an expanded version of section 5 of arXiv:1104.0320
which appears in the conference proceedings "Tropical and Non-Archimedean
Geometry
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