Perverse coherent sheaves and the geometry of special pieces in the unipotent variety

Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U be an open subset whose complement has codimension at least 2. We extend the Deligne-Bezrukavnikov theory of perverse coherent sheaves by showing that a coherent middle extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities than has previously been known. We also introduce a derived category version of the coherent middle extension functor. Under suitable hypotheses, we introduce a construction (called "S2-extension") in terms of perverse coherent sheaves of algebras on X that takes a finite morphism to U and extends it in a canonical way to a finite morphism to X. In particular, this construction gives a canonical "S2-ification" of appropriate X. The construction also has applications to the "Macaulayfication" problem, and it is particularly well-behaved when X is Gorenstein. Our main goal, however, is to address a conjecture of Lusztig on the geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown in the exceptional groups and in positive characteristic) by the action of a certain finite group. We use S2-extension to give a uniform construction of the desired variety.Comment: 30 pages; minor corrections and addition

Life at a Fur Trading Post in British Columbia a Century Ago.

"To sum up life in a fur trading post in British Columbia a century ago did not differ materially from that at a fur post anywhere in the widespread domain of the Hudson's Bay Company.

The Dixon-Meares Controversy- Containing, Remarks on the Voyages of John Meares, by George Dixon, An Answer to Mr. George Dixon, by John Meares, and Further Remarks on the Voyages of John Meares, by George Dixon

"The controversy sheds light on the early days of the Maritime fur trade. Meares was an important figure in the trade and his Voyages were widely read and are still to be found in libraries. Dixon has shown that Meares was incapable of telling a consistent story. The three pamphlets are, therefore, a necessary commentary on the Voyages.

Canada on the Pacific: 1866-1925

"Canada is now finding her place on the Pacific. Her aims are peaceable. Her armaments are negligible. She seeks to advance her cause not by war but through avenues of trade… In common with all other English-speaking nations on the 'Pacific Rim' she is forced to solve the vexed problem of Asiatic immigration. But she is attempting to provide a solution which will not injure the feelings of the Asiatics. The 'Gentlemen's Agreement' with Japan has worked well. Chinese exclusion since 1923 seems to have produced no very hard feelings. The British Indian question, though not satisfactorily settled, is dormant...

Loading of a surface-electrode ion trap from a remote, precooled source

We demonstrate loading of ions into a surface-electrode trap (SET) from a remote, laser-cooled source of neutral atoms. We first cool and load $\sim$ $10^6$ neutral $^{88}$Sr atoms into a magneto-optical trap from an oven that has no line of sight with the SET. The cold atoms are then pushed with a resonant laser into the trap region where they are subsequently photoionized and trapped in an SET operated at a cryogenic temperature of 4.6 K. We present studies of the loading process and show that our technique achieves ion loading into a shallow (15 meV depth) trap at rates as high as 125 ions/s while drastically reducing the amount of metal deposition on the trap surface as compared with direct loading from a hot vapor. Furthermore, we note that due to multiple stages of isotopic filtering in our loading process, this technique has the potential for enhanced isotopic selectivity over other loading methods. Rapid loading from a clean, isotopically pure, and precooled source may enable scalable quantum information processing with trapped ions in large, low-depth surface trap arrays that are not amenable to loading from a hot atomic beam

Inelastic collisions of ultra-cold heteronuclear molecules in an optical trap

Ultra-cold RbCs molecules in high-lying vibrational levels of the a$^3\Sigma^+$ ground electronic state are confined in an optical trap. Inelastic collision rates of these molecules with both Rb and Cs atoms are determined for individual vibrational levels, across an order of magnitude of binding energies. A simple model for the collision process is shown to accurately reproduce the observed scattering rates

Staggered sheaves on partial flag varieties

Staggered t-structures are a class of t-structures on derived categories of equivariant coherent sheaves. In this Note, we show that the derived category of coherent sheaves on a partial flag variety, equivariant for a Borel subgroup, admits a staggered t-structure with the property that all objects in its heart have finite length. As a consequence, we obtain a basis for its equivariant K-theory consisting of simple staggered sheaves. To cite this article: P.N. Achar, D.S. Sage, C. R. Acad. Sci. Paris, Ser. I 347 (2009). © 2009 Académie des sciences