Log-isocristaux surconvergents et holonomie

Abstract

Abstract Let 𝒱 be a complete discrete valuation ring of unequal characteristic with perfect residue field. Let $\X $ be a separated smooth formal 𝒱-scheme, 𝒵 be a normal crossing divisor of $\X $ , $\X ^\#:= (\X , \ZZ )$ be the induced formal log-scheme over 𝒱 and $u: \X ^\# \rightarrow \X $ be the canonical morphism. Let X and Z be the special fibers of $\X $ and 𝒵, T be a divisor of X and ℰ be a log-isocrystal on $\X ^\#$ overconvergent along T , that is, a coherent left $\D ^\dag _{\X ^\#} (\hdag T) _{\Q }$ -module, locally projective of finite type over $ \O _{\X } (\hdag T) _{\Q }$ . We check the relative duality isomorphism: $u_{T,+} (\E ) \riso u_{T,!} (\E (\ZZ ))$ . We prove the isomorphism $u_{T,+} (\E ) \riso \D ^\dag _{\X } (\hdag T) _{\Q } \otimes _{\D ^\dag _{\X ^\#} (\hdag T) _{\Q }} \E (\ZZ )$ , which implies their holonomicity as $\D ^\dag _{\X } (\hdag T) _{\Q }$ -modules. We obtain the canonical morphism ρ ℰ : u T ,+ (ℰ)→ℰ( † Z ). When ℰ is moreover an isocrystal on $\X $ overconvergent along T , we prove that ρ ℰ is an isomorphism.