Harmonic gauges on Riemann surfaces and stable bundles
Using the harmonic theory developed by Takegoshi for representation of relative cohomology and the framework of computation of curvature of direct image bundles by Berndtsson, we prove that the higher direct images by a smooth morphism of the relative canonical bundle twisted by a semi-positive vector bundle are locally free and semi-positively curved, when endowed with a suitable Hodge type metric.
Let be a -adic local field with residue field such that and be a -adic representation of . Then, by using the theory of -adic differential modules, we show that is a Hodge-Tate (resp. de Rham) representation of if and only if is a Hodge-Tate (resp. de Rham) representation of where is a certain -adic local field with residue field the smallest perfect field containing .