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In algebraic geometry, a branch of mathematics, Serre duality is a duality present on non-singular projective algebraic varieties ''V'' of dimension ''n'' (and in greater generality for vector bundles and further, for coherent sheaves). It shows that a cohomology group ''H''''i'' is the dual space of another one, ''H''''n''−''i''. In the case for holomorphic vector bundle ''E'' over a smooth compact complex manifold ''V'', the statement is in the form: ::: in which ''V'' is not necessarily projective. ==Algebraic curve== The case of algebraic curves was already implicit in the Riemann-Roch theorem. For a curve ''C'' the coherent groups ''H''''i'' vanish for ''i'' > 1; but ''H''1 does enter implicitly. In fact, the basic relation of the theorem involves ''l''(''D'') and ''l''(''K''−''D''), where ''D'' is a divisor and ''K'' is a divisor of the canonical class. After Serre we recognise ''l''(''K''−''D'') as the dimension of ''H''1(''D''), where now ''D'' means the line bundle determined by the divisor ''D''. That is, Serre duality in this case relates groups ''H''1(''D'') and ''H''0(''KD'' *), and we are reading off dimensions (notation: ''K'' is the canonical line bundle, ''D'' * is the dual line bundle, and juxtaposition is the tensor product of line bundles). In this formulation the Riemann-Roch theorem can be viewed as a computation of the Euler characteristic of a sheaf :''h''0(''D'') − ''h''1(''D''), in terms of the genus of the curve, which is :''h''1(''C'',''O''''C''), and the degree of ''D''. It is this expression that can be generalised to higher dimensions. Serre duality of curves is therefore something very classical; but it has an interesting light to cast. For example, in Riemann surface theory, the deformation theory of complex structures is studied classically by means of quadratic differentials (namely sections of ''L''(''K''2)). The deformation theory of Kunihiko Kodaira and D. C. Spencer identifies deformations via ''H''1(''T''), where ''T'' is the tangent bundle sheaf ''K'' *. The duality shows why these approaches coincide. 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「Serre duality」の詳細全文を読む スポンサード リンク
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