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Hironaka's example
In geometry, Hironaka's example is a non-Kähler complex manifold that is a deformation of Kähler manifolds found by . Hironaka's example can be used to show that several other plausible statements holding for smooth varieties of dimension at most 2 fail for smooth varieties of dimension at least 3.
==Hironaka's example==
Take two smooth curves ''C'' and ''D'' in a smooth projective 3-fold ''P'', intersecting in two points ''c'' and ''d'' that are nodes for the reducible curve ''C''∪''D''. For some applications these should be chosen so that there is a fixed-point-free automorphism exchanging the curves ''C'' and ''D'' and also exchanging the points ''c'' and ''d''. Hironaka's example ''V'' is obtained by blowing up the curves ''C'' and ''D'', with ''C'' blown up first at the point ''c'' and ''D'' blown up first at the point ''d''. Then ''V'' has two smooth rational curves ''L'' and ''M'' lying over ''c'' and ''d'' such that ''L''+''M'' is algebraically equivalent to 0, so ''V'' cannot be projective.
For an explicit example of this configuration, take ''t'' to be a point of order 2 in an elliptic curve ''E'', take ''P'' to be ''E''×''E''/(''t'')×''E''/(''t''), take ''C'' and ''D'' to be the sets of points of the form (''x'',''x'',0) and (''x'',0,''x''), so that ''c'' and ''d'' are the points (0,0,0) and (''t'',0,0), and take the involution σ to be the one taking (''x'',''y'',''z'') to (''x'' + ''t'', ''z'',''y'').
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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