SEMI-POSITIVITY AND FROBENIUS CRYSTALS On Semi-Positivity and Filtered Frobenius Crystals

The purpose of this paper is to prove that certain subquotients of the MF∇-objects of Faltings ([1]) are semi-positive. In particular, we obtain an algebraic proof of a result (generalizing those of [5], [9]) on the semi-positivity of the higher direct images of certain kinds of sheaves for semistable families of algebraic varieties. Our Main Theorem, proven in §3, is as follows: Theorem 3.4: Let f : (X,E) → (S,D) be a semistable family of varieties of relative dimension d, with S a smooth, proper scheme over a field L of characteristic zero. Let (A,∇A, F i(A)) be a globally crystalline filtered vector bundle with connection on (X,E). Then for any nonnegative integer α, the coherent sheaf of OS-modules Rf∗(ωX/S ⊗OX (A/F 1(A))∨)