Kodaira–Saito vanishing via Higgs bundles in positive characteristic

Abstract

Abstract. The goal of this paper is to give a new proof of a special case of the Kodaira-Saito vanishing theorem for a variation of Hodge structure on the complement of a divisor with normal crossings. The proof does not use the theory of mixed Hodge modules, but instead reduces it to a more general vanishing theorem for semistable nilpotent Higgs bundles, which is then proved by using some facts about Higgs bundles in positive characteristic.In 1990, Saito [S1, prop 2.33] gave a far reaching generalization of Kodaira's vanishing theorem using his theory of mixed Hodge modules. A number of interesting applications have been found in recent years; we refer to Popa's survey [P2] for a discussion of these, and to [P1, Sl] for a discussion of the theorem itself. We would like to explain the easiest, but still important, case of the theorem where the mixed Hodge module is of the form Rj * V (in standard "non perverse" notation), where V is a polarized variation of pure Hodge structure on the complement j : U → X of a divisor with simple normal crossings D. Let us also assume that V has unipotent local monodromies around components of D. By Deligne [D], the flat vector bundle O U ⊗ V has a canonical extension V such that the original connection extends to a logarithmic connection ∇ : V → Ω 1 X (log D) ⊗ V with nilpotent residues. By a theorem of Schmid [Sc], V has a filtration F by subbundles extending the Hodge filtration. This induces a filtration on the de Rham complexwhich, for us, starts in degree 0. Saito's theorem tell us that if L is an ample line bundle and i > dim X, thenMore generally, this holds when V is replaced by an admissible variation of mixed Hodge structure.The first goal of this paper is to give a short proof of this special case by reduction to characteristic p > 0. Illusie [I] had previously given a proof by such a reduction, when V arises geometrically from a semistable map of varieties. Our proof, however, is different and it works even for nongeometric cases (and this seems important for certain applications, e.g. to Shimura varieties [Su]). We first replace the variation of Hodge structure V by the vector bundle E = Gr F (V ), together with the so called Higgs field θ = Gr F (∇). By work of Simpson, this pair is semistable. In addition, the rational Chern classes c i (E) = 0, and θ is nilpotent. In fact, our main result is Partially supported by the NSF .

Nearby cycles and semipositivity in positive characteristic

“…In this last case Theorem 2.2 allows to compute higher Chern classes of twisted preperiodic Higgs bundles. Let us also remark that a special case of the above result was implicitly used in proof of [Ar,Theorem 3] (see Remark 2.13).…”

“…Remark 2.13. In proof of [Ar,Lemma 4.4] and [Ar,Lemma 4.5] (needed for [Ar,Theorem 3]) the author implicitly uses that B(E, θ ) is locally free if (E, θ ) is locally free. More precisely, he applies [Ar,Lemma 4.3] to B(E, θ ) and this fails if B(E, θ ) is not locally free.…”

“…In proof of [Ar,Lemma 4.4] and [Ar,Lemma 4.5] (needed for [Ar,Theorem 3]) the author implicitly uses that B(E, θ ) is locally free if (E, θ ) is locally free. More precisely, he applies [Ar,Lemma 4.3] to B(E, θ ) and this fails if B(E, θ ) is not locally free. It is easy to find examples for which (E, θ ) is semistable, E is locally free but B(E, θ ) is not even reflexive.…”

“…It is easy to find examples for which (E, θ ) is semistable, E is locally free but B(E, θ ) is not even reflexive. In particular, in both [Ar,Lemma 4.4] and [Ar,Lemma 4.5] one needs to assume that (E, θ ) is semistable with vanishing Chern classes and then use our Corollary 2.9. Note also that at the time of writing [Ar], Corollary 2.9 was not claimed in the logarithmic case that was used there.…”

“…A notable exception is Arapura's proof of [Ar,Theorem 2] that uses reduction to positive characteristic. However, his proof uses vanishing theorems and it does not give any semipositivity results in positive characteristic.…”