Nearby cycles and semipositivity in positive characteristic

Abstract

We study restriction of logarithmic Higgs bundles to the boundary divisor and we construct the corresponding nearby-cycles functor in positive characteristic. As applications we prove some strong semipositivity theorems for analogs of complex polarized variations of Hodge structures and their generalizations. This implies, e.g., semipositivity for the relative canonical divisor of a semistable reduction in positive characteristic and it gives some new strong results generalizing semipositivity even for complex varieties.

On irreducible symplectic varieties of K3[]-type in positive characteristic

“…In the characteristic zero case, the above results allow us to deal with varieties with more general singularities than usually considered. As in [La6] our results imply also various semipositivity theorems (also for polarizable complex variations of Hodge structure on complex projective varieties with quotient singularities in codimension 2). We leave formulation of the corresponding results to the interested reader.…”

“…The main problem in proof of Theorem 0.1 is to show that if E satisfies (3) then it is locally free. In case X is projective, analogous results (see [Si2,Theorem 2], [La3,Theorem 4.1], [La5,Theorem 11], [La6,Theorem 2.2] and [FL, Theorem A and B]) are always proven using another characterization of numerically flat vector bundles as some semistable coherent torsion free O X -modules with the same Hilbert polynomial as the trivial bundle of the same rank. Then one can use induction on the dimension of the variety.…”

“…This was further generalized in [SYZ] to deal with projective representations of the étale fundamental group. Somewhat different point ofview, not restricted to bundles with vanishing Chern classes, was used in [La5] and [La6] to establish various results like Bogomolov's inequality for semistable Higgs sheaves, the Miyaoka-Yau inequality and strong semipositivity theorems.…”

Bogomolov's inequality and Higgs sheaves on normal varieties in positive characteristic

“…Note that if (X , D) is almost F-liftable and D = D i , where D i ⊂ X are prime divisors then (D i , ( j =i D j ) ∩ D i ) is also almost F-liftable. This observation follows from the corresponding fact for F-liftable log smooth pairs and it was explicitly stated before [La6,Lemma 3.7]; see also [AWZ2,Lemma 3.2] for a simple proof.…”

“…M. Sheng and K. Zuo in [LSZ2] and their existence was proven in [LSZ2] and [La3]. Canonical Higgs-de Rham sequences in the above sense first appeared in the proof of [La6,Lemma 3.10]. They are better suited to dealing with normal varieties as one cannot define suitable Chern classes for torsion free sheaves on normal varieties.…”