Lattices of Algebraic Cycles on Fermat Varieties in Positive Characteristics

Let X be the Fermat hypersurface of dimension 2 m and of degree q +1 defined over an algebraically closed field of characteristic p >0, where q is a power of p , and let NL m ( X ) be the free abelian group of numerical equivalence classes of linear subspaces of dimension m contained in X . By the intersection form, we regard NL m ( X ) as a lattice. Investigating the configuration of these linear subspaces, we show that the rank of NL m ( X ) is equal to the 2 m th Betti number of X , that the intersection form multiplied by (−1) m is positive definite on the primitive part of NL m ( X ), and that the discriminant of NL m ( X ) is a power of p . Let L m ( X ) be the primitive part of NL m ( X ) equipped with the intersection form multiplied by (−1) m . In the case p = q =2, the lattice L m ( X ) is described in terms of certain codes associated with the unitary geometry over F 2 . Since L 1 ( X ) is isomorphic to the root lattice of type E 6 , the series of lattices L m ( X ) can be considered as a generalization of E 6 . The lattice L 2 ( X ) is isomorphic to the laminated lattice of rank 22. This isomorphism explains Conway's identification ·222 ≅ PSU(6,2) geometrically. The lattice L 3 ( X ) is of discriminant 2 16 · 3, minimal norm 8, and kissing number 109421928. 2000 Mathematics Subject Classification : 14C25, 11H31, 51D25.