On p-adic differential equations on semistable varieties (PhD thesis)

Abstract

Let V be a complete discrete valuation ring of mixed characteristic (0, p) with uniformizer π, K be the fraction field and k be the residue field. We study p-adic differential equations on a semistable variety over V .

We consider a proper semistable variety X over V and a relative normal crossing divisor D, such that, locally for the étale topology, X is étale over the space SpecV [x1, . . . , xn, y1, . . . ym]/(x1 · · · xr − π) , with r ≤ n, and D is defined by the equation y1 · · · ys = 0, with s ≤ m. We consider on X the open U defined by the complement of the divisor D and we call UK and Uk the generic fiber and the special fiber respectively. In an analogous way we call DK, XK and Dk, Xk the generic and the special fiber of D, X.

In the geometric situation described, we investigate the relations between algebraic differential equations on XK and analytic differential equations on the rigid analytic space associated to Xˆ, the completion of X along its special fiber. More precisely we consider the category, denoted by MIC(UK/K) reg,unip, of couples (E, ∇) with E a sheaf of coherent OUK -modules and ∇ an integrable connection regular along DK which admit an extension to a couple (Eext , ∇ext), with Eext a sheaf of locally free OXK -modules and ∇ext an integrable connection on XK that has logarithmic singularities along DK and nilpotent residue. As for the rigid side we look at I † ((Uk, Xk)/V ) lf,log,unip, the category of locally free overconvergent log isocrystals on the log pair given by ((Uk, M ⊕ MD),(Xk, M ⊕ MD)/(SpfV, N)) with unipotent monodromy along Dk, where M ⊕ MD is the log structure on Xk induced by the log structure on X defined by the divisor D ∪ Xk. The main result is the existence and the full faithfulness of an algebrization functor I † ((Uk, Xk)/V ) log,lf,unip → MIC(UK/K) reg,unip . The steps of the proof are as follows. We start from the category of locally free log overconvergent isocrystals on ((Uk, M⊕ MD),(Xk, M ⊕ MD)/(SpfV, N)) with unipotent monodromy and we prove that the restriction functor j † : Iconv((X, Mˆ ⊕ MD)/(V, N))lf,res=nilp −→ I † ((Uk, Xk)/V ) log,lf,unip is an equivalence of categories. We denote by Iconv((X, Mˆ ⊕ MD)/(V, N))lf,res=nilp the category of locally free log convergent isocrystals on the log convergent site ((X, Mˆ ⊕ MD)/(V, N))conv with nilpotent residue. On the other hand we have a fully faithful functor ˜i : Iconv((X, Mˆ ⊕ MD)/(Spf(V ), N))lf,res=nilp −→ MIC((XK, MD)/K) lf,res=nilp 6 between locally free log convergent isocrystals with nilpotent residue and locally free OXK -modules with integrable connection on XK, logarithmic singularities on DK and nilpotent residue. Thanks to the theorem of algebraic logarithmic extension of Andrè and Baldassarri we can conclude that there is an equivalence of categories between MIC((XK, MD)/K) lf,res=nilp → MIC(UK/K) reg,unip .