An Oka principle for equivariant isomorphisms

Let G be a reductive complex Lie group acting holomorphically on normal Stein spaces X and Y, which are locally G-biholomorphic over a common categorical quotient Q. When is there a global G-biholomorphism X → Y? If the actions of G on X and Y are what we, with justification, call generic, we prove that the obstruction to solving this local-to-global problem is topological and provide sufficient conditions for it to vanish. Our main tool is the equivariant version of Grauert's Oka principle due to Heinzner and Kutzschebauch. We prove that X and Y are G-biholomorphic if X is K-contractible, where K is a maximal compact subgroup of G, or if X and Y are smooth and there is a G-diffeomorphism ψ : X → Y over Q, which is holomorphic when restricted to each fibre of the quotient map X → Q. We prove a similar theorem when ψ is only a G-homeomorphism, but with an assumption about its action on G-finite functions. When G is abelian, we obtain stronger theorems. Our results can be interpreted as instances of the Oka principle for sections of the sheaf of G-biholomorphisms from X to Y over Q. This sheaf can be badly singular, even for a low-dimensional representation of SL2(ℂ). Our work is in part motivated by the linearisation problem for actions on ℂn. It follows from one of our main results that a holomorphic G-action on ℂn, which is locally G-biholomorphic over a common quotient to a generic linear action, is linearisable.

Swath sonar bathymetry during POLARSTERN cruise ANT-XV/2 (PS47) with links to multibeam raw data files

Multibeam data were measured during R/V Polarstern cruise ANT-XV/2 along track lines of approximately 10200 NM total length during transits, surveys and partly during stationary work, mainly in the Scotia Sea and the Weddell Sea. Areal multibeam surveys were performed in the vicinity of the South Shetland trench, the Bransfield Basin, the South Sandwich trench, and off the Ekstrom Ice Shelf for time periods of three to eight days. The multibeam sonar system Hydrosweep DS-2 was operated using 59 beams and 90° aperture angle, in some shallow areas 120°. The refraction correction was achieved utilizing sound velocity profiles sampled during the cruise, and by the system's own cross fan calibration. The quality of data might be reduced during bad weather periods or adverse sea ice conditions. The dataset contains raw data that are not processed and thus may contain errors and blunders in depth and position.