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In algebraic geometry, a supersingular K3 surface is a K3 surface over a field ''k'' of characteristic ''p'' > 0 such that the slopes of Frobenius on the crystalline cohomology ''H''2(''X'',''W''(''k'')) are all equal to 1.〔M. Artin and B. Mazur. Ann. Sci. École Normale Supérieure 10 (1977), 87-131. P. 90.〕 These have also been called Artin supersingular K3 surfaces. Supersingular K3 surfaces can be considered the most special and interesting of all K3 surfaces. ==Definitions and main results== More generally, a smooth projective variety ''X'' over a field of characteristic ''p'' > 0 is called supersingular if all slopes of Frobenius on the crystalline cohomology ''H''a(''X'',''W''(''k'')) are equal to ''a''/2, for all ''a''. In particular, this gives the standard notion of a supersingular abelian variety. For a variety ''X'' over a finite field ''F''''q'', it is equivalent to say that the eigenvalues of Frobenius on the l-adic cohomology ''H''a(''X'',''Q''''l'') are equal to ''q''''a''/2 times roots of unity. It follows that any variety in positive characteristic whose ''l''-adic cohomology is generated by algebraic cycles is supersingular. A K3 surface whose ''l''-adic cohomology is generated by algebraic cycles is sometimes called a Shioda supersingular K3 surface. Since the second Betti number of a K3 surface is always 22, this property means that the surface has 22 independent elements in its Picard group (ρ = 22). From what we have said, a K3 surface with Picard number 22 must be supersingular. Conversely, the Tate conjecture would imply that every supersingular K3 surface over an algebraically closed field has Picard number 22. This is now known in every characteristic ''p'' except 2, since the Tate conjecture was proved for all K3 surfaces in characteristic ''p'' at least 3 by Nygaard-Ogus (1985), , , and . To see that K3 surfaces with Picard number 22 exist only in positive characteristic, one can use Hodge theory to prove that the Picard number of a K3 surface in characteristic zero is at most 20. In fact the Hodge diamond for any complex K3 surface is the same (see classification), and the middle row reads 1, 20, 1. In other words, ''h''2,0 and ''h''0,2 both take the value 1, with ''h''1,1 = 20. Therefore, the dimension of the space spanned by the algebraic cycles is at most 20 in characteristic zero; surfaces with this maximum value are sometimes called singular K3 surfaces. Another phenomenon which can only occur in positive characteristic is that a K3 surface may be unirational. Michael Artin observed that every unirational K3 surface over an algebraically closed field must have Picard number 22. (In particular, a unirational K3 surface must be supersingular.) Conversely, Artin conjectured that every K3 surface with Picard number 22 must be unirational.〔M. Artin. Ann. Sci. École Normale Supérieure 7 (1974), 543-567. P. 552.〕 This is now known in every characteristic except 3. Artin's conjecture was proved in characteristic 2 by , and in every characteristic ''p'' at least 5 by . Another proof for ''p'' at least 5 has been given by . 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「Supersingular K3 surface」の詳細全文を読む スポンサード リンク
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