|
In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result contributing to the Riemann–Roch problem for complex algebraic varieties of all dimensions. It was the first successful generalisation of the classical Riemann–Roch theorem on Riemann surfaces to all higher dimensions, and paved the way to the Grothendieck–Hirzebruch–Riemann–Roch theorem proved about three years later. ==Statement of Hirzebruch–Riemann–Roch theorem== The Hirzebruch–Riemann–Roch theorem applies to any holomorphic vector bundle ''E'' on a compact complex manifold ''X'', to calculate the holomorphic Euler characteristic of ''E'' in sheaf cohomology, namely the alternating sum : of the dimensions as complex vector spaces. Hirzebruch's theorem states that χ(''X'', ''E'') is computable in terms of the Chern classes ''C''''j''(''E'') of ''E'', and the Todd polynomials ''T''''j'' in the Chern classes of the holomorphic tangent bundle of ''X''. These all lie in the cohomology ring of ''X''; by use of the fundamental class (or, in other words, integration over ''X'') we can obtain numbers from classes in ''H''2''n''(''X''). The Hirzebruch formula asserts that : taken over all relevant ''j'' (so 0 ≤ ''j'' ≤ ''n''), using the Chern character ch(''E'') in cohomology. In other words, the cross products are formed in cohomology ring of all the 'matching' degrees that add up to 2''n'', where to 'massage' the ''C''''j''(''E'') a formal manipulation is done, setting : and the total Chern class : Formulated differently the theorem gives the equality : where ''td(X)'' is the Todd class of the tangent bundle of ''X''. Significant special cases are when ''E'' is a complex line bundle, and when ''X'' is an algebraic surface (Noether's formula). Weil's Riemann–Roch theorem for vector bundles on curves, and the Riemann–Roch theorem for algebraic surfaces (see below), are included in its scope. The formula also expresses in a precise way the vague notion that the Todd classes are in some sense reciprocals of characteristic classes. 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「Hirzebruch–Riemann–Roch theorem」の詳細全文を読む スポンサード リンク
|