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In mathematics, Dolgachev surfaces are certain simply connected elliptic surfaces, introduced by . They can be used to give examples of an infinite family of homeomorphic simply connected compact 4-manifolds no two of which are diffeomorphic. ==Properties== The blowup ''X''0 of the projective plane in 9 points can be realized as an elliptic fibration all of whose fibers are irreducible. A Dolgachev surface ''X''''q'' is given by applying logarithmic transformations of orders 2 and ''q'' to two smooth fibers for some ''q'' ≥ 3. The Dolgachev surfaces are simply connected and the bilinear form on the second cohomology group is odd of signature (1, 9) (so it is the unimodular lattice I1,9). The geometric genus ''p''''g'' is 0 and the Kodaira dimension is 1. found the first examples of homeomorphic but not diffeomorphic 4-manifolds ''X''0 and ''X''3. More generally the surfaces ''X''''q'' and ''X''''r'' are always homeomorphic, but are not diffeomorphic unless ''q'' = ''r''. showed that the Dolgachev surface ''X''3 has a handlebody decomposition without 1- and 3-handles. 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「Dolgachev surface」の詳細全文を読む スポンサード リンク
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