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In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry. Let ''M'' be a complex manifold, and write ''O''''M'' for the sheaf of holomorphic functions on ''M''. Let ''O''''M'' * be the subsheaf consisting of the non-vanishing holomorphic functions. These are both sheaves of abelian groups. The exponential function gives a sheaf homomorphism : because for a holomorphic function ''f'', exp(''f'') is a non-vanishing holomorphic function, and exp(''f'' + ''g'') = exp(''f'')exp(''g''). Its kernel is the sheaf 2π''i''Z of locally constant functions on ''M'' taking the values 2π''in'', with ''n'' an integer. The exponential sheaf sequence is therefore : The exponential mapping here is not always a surjective map on sections; this can be seen for example when ''M'' is a punctured disk in the complex plane. The exponential map ''is'' surjective on the stalks: Given a germ ''g'' of an holomorphic function at a point ''P'' such that ''g''(''P'') ≠ 0, one can take the logarithm of ''g'' in a neighborhood of ''P''. The long exact sequence of sheaf cohomology shows that we have an exact sequence : for any open set ''U'' of ''M''. Here ''H''0 means simply the sections over ''U'', and the sheaf cohomology ''H''1(2π''i''Z|''U'') is the singular cohomology of ''U''. The connecting homomorphism is therefore a generalized winding number and measures the failure of ''U'' to be contractible. In other words, there is a potential topological obstruction to taking a ''global'' logarithm of a non-vanishing holomorphic function, something that is always ''locally'' possible. A further consequence of the sequence is the exactness of : Here ''H''1(''O''''M'' *) can be identified with the Picard group of holomorphic line bundles on ''M''. The connecting homomorphism sends a line bundle to its first Chern class. ==References== * , see especially p. 37 and p. 139 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「exponential sheaf sequence」の詳細全文を読む スポンサード リンク
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