|
In mathematics, the irregularity of a complex surface ''X'' is the Hodge number ''h''0,1= dim ''H''1(''O''''X''), usually denoted by ''q'' . The irregularity of an algebraic surface is sometimes defined to be this Hodge number, and sometimes defined to be the dimension of the Picard variety , which is the same in characteristic 0 but can be smaller in positive characteristic. The name "irregularity" comes from the fact that for the first surfaces investigated in detail, the smooth complex surfaces in P3, the irregularity happens to vanish. The irregularity then appeared as a new "correction" term measuring the difference ''p''''g'' − ''p''''a'' of the geometric genus and the arithmetic genus of more complicated surfaces. Surfaces are sometimes called regular or irregular depending on whether or not the irregularity vanishes. For a complex analytic manifold ''X'' in general dimension the Hodge number ''h''0,1 = dim ''H''1(''O''''X'') is called irregularity ''q''. ==Complex surfaces== For non-singular complex projective (or Kähler) surfaces the following numbers are all equal: *The irregularity *The dimension of the Albanese variety *The dimension of the Picard variety *The Hodge number ''h''0,1 = dim(''H''1(Ω0)) *The Hodge number ''h''1,0 = dim(''H''0(Ω1)) *The difference ''p''''g'' − ''p''''a'' of the geometric genus and the arithmetic genus. For surfaces in positive characteristic, or for non-Kähler complex surfaces, the numbers above need not all be equal. proved that for complex projective surfaces the dimension of the Picard variety is equal to the Hodge number ''h''0,1, and the same is true for all compact Kähler surfaces. The irregularity of smooth compact Kähler surfaces is invariant under bimeromorphic transformations. For general compact complex surfaces the two Hodge numbers ''h''1,0 and ''h''0,1 need not be equal, but ''h''0,1 is either ''h''1,0 or ''h''1,0+1, and is equal to ''h''1,0 for compact Kähler surfaces. 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「irregularity of a surface」の詳細全文を読む スポンサード リンク
|