A p-ADIC PROOF OF HODGE SYMMETRY FOR THREEFOLDS

The Hodge theorem also asserts that these two spaces are complex conjugates and hence the equality of the two dimensions. In this note we give a p-adic proof of (1.1) when X/C is a smooth projective threefold. Our approach is based on the following observation: one first notes a purely p-adic assertion that (1.1) holds when Hodge numbers are replaced by more delicate p-adic invariants introduced by Ekedahl [3] called Hodge-Witt numbers. These invariants take in to account the torsion in the slope spectral sequence as well as the slopes of Frobenius in the crystalline cohomology of the variety. Hodge-Witt symmetry was proved by Ekedahl using his delicate duality theorem (we note here that we do not use Ekedahl’s duality in the proof given below; we have replaced this by an elementary assertion h W = h 2,0 W ) as well as the crystalline Hard-Lefschetz theorem (when X is a smooth projective threefold over a perfect field of characteristic p). Once Hodge-Witt symmetry is proved one appeals to another result of Ekedahl (see [5]) which guarantees the equality of Hodge-Witt numbers and the Hodge numbers under suitable circumstances. That these required conditions are met when X is a smooth projective variety over complex numbers is a simple consequence of Deligne-Illusie criterion (see [2]) for degeneration of Hodge-de Rham spectral sequence in characteristic p. The restriction on dimension in Theorem 3.1 arises because Hodge-Witt symmetry is not known. Ekedahl gave a necessary and sufficient condition for Hodge-Witt symmetry to hold (in any dimension). This condition is given in terms of a certain equality of domino numbers for the slope spectral sequence. While this condition is not known to hold in dimension bigger than three, in Remark 3.2 and Scholium 3.3 we give one set of hypothesis under which these condition are satisfied. Further Ekedahl’s conditions also hold, for instance, if X is a smooth, projective HodgeWitt variety over a perfect field of characteristic p > 0. Hence we deduce that if a smooth projective variety X/C has Hodge-Witt or ordinary reduction at an infinite set of primes then Hodge symmetry holds in characteristic zero. We would like to thank Luc Illusie for correspondence and comments; we would also like to thank Minhyong Kim for discussions on Hodge symmetry and also to Jim Carlson whose lectures on mixed Hodge theory at the University of Arizona revived our interest in algebraic proofs of Hodge symmetry. We are grateful to