Tensor products in p-adic Hodge theory

Abstract

There is a classical relation between the p-adic absolute value of the eigenvalues of Frobenius on crystalline cohomology and Hodge numbers, for a variety in characteristic p: “the Newton polygon lies on or above the Hodge polygon” [14], [1]. For a variety in characteristic p with a lift to characteristic 0, Fontaine conjectured and Faltings proved a more precise statement: there is an inequality which relates the slope of Frobenius on any Frobenius-invariant subspace of the crystalline cohomology to the Hodge filtration, restricted to that subspace [7], [4]. A vector space over a p-adic field together with a σ-linear endomorphism and a filtration which satisfies this inequality is called a weakly admissible filtered isocrystal (see section 1 for the precise definition). The category of such objects is one possible p-adic analogue of the category of Hodge structures: in particular, it is an abelian category. We give a new proof of Faltings’s theorem that the tensor product of weakly admissible filtered isocrystals over a p-adic field is weakly admissible [5]. By a similar argument, we also prove a characterization of weakly admissible filtered isocrystals with G-structure in terms of geometric invariant theory, which was conjectured by Rapoport and Zink [19]. Before Faltings, Laffaille [12] had proved the tensor product theorem in the case of filtered isocrystals over an unramified extension of Qp. Faltings’s proof works by reducing this problem of σ-linear algebra to a different problem of pure linear algebra, the problem of showing that the tensor product of two vector spaces, each equipped with a finite “semistable” set of filtrations, is semistable. The latter problem is solved by constructing suitable integral lattices (in [5]) or hermitian metrics (in [20]) on vector spaces with a semistable set of filtrations, just as one can prove that the tensor product of semistable bundles on an algebraic curve is semistable using Narasimhan-Seshadri’s hermitian metrics ([6], [16]). In this paper, we can avoid the reduction from filtered isocrystals to filtered vector spaces. The point is that Ramanan and Ramanathan’s algebraic proof [17] that the tensor product of semistable vector bundles is semistable can be modified to apply directly to filtered isocrystals. We have an inequality to prove for a class of linear subspaces S of a tensor product V ⊗W . The inequality is obvious for sufficiently general subspaces S and also if S is a very special subspace, say if S is a decomposable subspace S1 ⊗ S2 ⊂ V ⊗W . But it is not clear how to prove the inequality we want if S is somewhere in the middle. The solution, following Ramanan and Ramanathan, is to use geometric invariant theory to give a sharp dichotomy between “general” subspaces and “special” subspaces of V ⊗W , in such a way that we get useful information in either case.