Frobenius and the Hodge Spectral Sequence

Abstract

Let X k be a smooth proper scheme over a perfect field of characteristic p and let n be a natural number. A fundamental theorem of Barry Mazur relates the Hodge numbers of X k to the action of Frobenius on the crystalline cohomology H cris (X W) of X over the Witt ring W of k, which can be viewed as a linear map 8: F*W H n cris (X W) H n cris (X W). If H n cris (X W) is torsion free, then since W is a discrete valuation ring, the source and target of 8 admit (unrelated) bases with respect to which the matrix of 8 is diagonal. For i, j # Z with i+ j=n, the (i, j ) th Hodge number h j (8) of 8 is defined to be the number of diagonal terms in this matrix whose p-adic ordinal is i. Mazur's theorem [1, 8.26] asserts that if the crystalline cohomology is torsion free and the Hodge spectral sequence of X k degenerates at E1 , then these ``Frobenius'' Hodge numbers coincide with the ``geometric'' Hodge numbers: