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We study the isometry group of the Grothendieck group equipped with a bilinear asymmetric Euler form. We prove several properties of this group; in particular, we show that it is isomorphic to the direct product of by the free Abelian group of rank . We also explicitly calculate its generators for . [ABSTRACT FROM AUTHOR]

We investigate group actions on the category of coherent sheaves over weighted projective lines. We show that the equivariant category with respect to certain finite group action is equivalent to the category of coherent sheaves over a weighted projective curve, where the genus of the underlying smooth projective curve and the weight data are given by explicit formulas. Moreover, we obtain a trichotomy result for all such equivariant relations. This equivariant approach provides an effective way to investigate smooth projective curves via weighted projective lines. As an application, we give a description for the category of coherent sheaves over a hyperelliptic curve of arbitrary genus. Links to smooth projective curves of genus 2 and also to Arnold's exceptional unimodal singularities are also established. [ABSTRACT FROM AUTHOR]

We give a perhaps less known version of Hartogs' extension for connected, non-compact complex manifolds M such that H c 1 (M , O) = 0 . Then we prove extensions/non-extensions for similar questions related to holomorphic vector/line bundles over complements of Stein compact sets in Stein manifolds. [ABSTRACT FROM AUTHOR]