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The Hodge conjecture is a major unsolved problem in algebraic geometry that relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they are sums of Poincaré duals of the homology classes of subvarieties. It was formulated by the Scottish mathematician William Vallance Douglas Hodge as a result of a work in between 1930 and 1940 to enrich the description of de Rham cohomology to include extra structure that is present in the case of complex algebraic varieties. It received little attention before Hodge presented it in an address during the 1950 International Congress of Mathematicians, held in Cambridge, Massachusetts, U.S. The Hodge conjecture is one of the Clay Mathematics Institute's Millennium Prize Problems, with a prize of $1,000,000 for whoever can prove or disprove the Hodge conjecture. == Motivation == Let ''X'' be a compact complex manifold of complex dimension ''n''. Then ''X'' is an orientable smooth manifold of real dimension 2''n'', so its cohomology groups lie in degrees zero through 2''n''. Assume ''X'' is a Kähler manifold, so that there is a decomposition on its cohomology with complex coefficients: : where ''Hp, q''(''X'') is the subgroup of cohomology classes which are represented by harmonic forms of type (''p'', ''q''). That is, these are the cohomology classes represented by differential forms which, in some choice of local coordinates ''z''1, ..., ''zn'', can be written as a harmonic function times : (See Hodge theory for more details.) Taking wedge products of these harmonic representatives corresponds to the cup product in cohomology, so the cup product is compatible with the Hodge decomposition: : Since ''X'' is a compact oriented manifold, ''X'' has a fundamental class. Let ''Z'' be a complex submanifold of ''X'' of dimension ''k'', and let ''i'' : ''Z'' → ''X'' be the inclusion map. Choose a differential form α of type (''p'', ''q''). We can integrate α over ''Z'': : To evaluate this integral, choose a point of ''Z'' and call it 0. Around 0, we can choose local coordinates ''z''1, ..., ''zn'' on ''X'' such that ''Z'' is just ''z''''k'' + 1 = ... = ''zn'' = 0. If ''p'' > ''k'', then α must contain some ''dzi'' where ''zi'' pulls back to zero on ''Z''. The same is true if ''q'' > ''k''. Consequently, this integral is zero if (''p'', ''q'') ≠ (''k'', ''k''). More abstractly, the integral can be written as the cap product of the homology class of ''Z'' and the cohomology class represented by α. By Poincaré duality, the homology class of ''Z'' is dual to a cohomology class which we will call (), and the cap product can be computed by taking the cup product of () and α and capping with the fundamental class of ''X''. Because () is a cohomology class, it has a Hodge decomposition. By the computation we did above, if we cup this class with any class of type (''p'', ''q'') ≠ (''k'', ''k''), then we get zero. Because ''H''2''n''(''X'', C) = ''Hn, n''(''X''), we conclude that () must lie in ''Hn-k, n-k''(''X'', C). Loosely speaking, the Hodge conjecture asks: :''Which cohomology classes in ''Hk, k''(''X'') come from complex subvarieties ''Z''?'' 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「Hodge conjecture」の詳細全文を読む スポンサード リンク
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