algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, a supersingular K3 surface is a

K3 surface In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected alg ...

over a field ''k'' of characteristic ''p'' > 0 such that the slopes of Frobenius on the

crystalline cohomology In mathematics, crystalline cohomology is a Weil cohomology theory for schemes ''X'' over a base field ''k''. Its values ''H'n''(''X''/''W'') are modules over the ring ''W'' of Witt vectors over ''k''. It was introduced by and developed by . ...

''H''²(''X'',''W''(''k'')) are all equal to 1. 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 In mathematics, a supersingular variety is (usually) a smooth projective variety in nonzero characteristic such that for all ''n'' the slopes of the Newton polygon of the ''n''th crystalline cohomology are all ''n''/2 . For special classes of ...

. 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 In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension ...

''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 cycle In mathematics, an algebraic cycle on an algebraic variety ''V'' is a formal linear combination of subvarieties of ''V''. These are the part of the