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We show that for smooth and proper varieties over local fields with no non-trivial vector fields, good reduction descends over purely inseparable extensions. We use this to extend the Neron-Ogg-Shafarevich criterion for K3 surfaces to the equicharacteristic $p>0$ case., 15 pages, comments welcome!

In this paper we prove a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, showing that for$k$a perfect field of characteristic$p$, a rational (logarithmic) line bundle on the special fibre of a semistable scheme over$k\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7}$lifts to the total space if and only if its first Chern class does. The proof is elementary, using standard properties of the logarithmic de Rham–Witt complex. As a corollary, we deduce similar algebraicity lifting results for cohomology classes on varieties over global function fields. Finally, we give a counter-example to show that the variational Tate conjecture for divisors cannot hold with$\mathbb{Q}_{p}$-coefficients.

In this article we study various forms of $\ell$-independence (including the case $\ell =p$) for the cohomology and fundamental groups of varieties over finite fields and equicharacteristic local fields. Our first result is a strong form of $\ell$-independence for the unipotent fundamental group of smooth and projective varieties over finite fields. By then proving a certain ‘spreading out’ result we are able to deduce a much weaker form of $\ell$-independence for unipotent fundamental groups over equicharacteristic local fields, at least in the semistable case. In a similar vein, we can also use this to deduce $\ell$-independence results for the cohomology of smooth and proper varieties over equicharacteristic local fields from the well-known results on $\ell$-independence for smooth and proper varieties over finite fields. As another consequence of this ‘spreading out’ result we are able to deduce the existence of a Clemens–Schmid exact sequence for formal semistable families. Finally, by deforming to characteristic $p$, we show a similar weak version of $\ell$-independence for the unipotent fundamental group of a semistable curve in mixed characteristic.

We correct the proof of the main $\ell$-independence result of the above-mentioned paper by showing that for any smooth and proper variety over an equicharacteristic local field, there exists a globally defined such variety with the same ($p$-adic and $\ell$-adic) cohomology.

The naive analogue of the N\'eron-Ogg-Shafarevich criterion is false for K3 surfaces, that is, there exist K3 surfaces over Henselian, discretely valued fields $K$, with unramified $\ell$-adic \'etale cohomology groups, but which do not admit good reduction over $K$. Assuming potential semi-stable reduction, we show how to correct this by proving that a K3 surface has good reduction if and only if $H^2_{\mathrm{\acute{e}t}}(X_{\overline{K}},\mathbb{Q}_\ell)$ is unramified, and the associated Galois representation over the residue field coincides with the second cohomology of a certain "canonical reduction" of $X$. We also prove the corresponding results for $p$-adic \'etale cohomology., Comment: 52 pages, completely rewritten with significantly stronger main results

We construct the (filtered) Ogus realisation of Voevodsky motives over a number field $K$. This realisation extends the functor defined on $1$-motives by Andreatta, Barbieri-Viale and Bertapelle. As an illustration we note that the analogue of the Tate conjecture holds for K3 surfaces., Comment: final accepted version

By combing Poincare duality for convergent higher direct images of a smooth and proper morphism, together with a theorem of Kedlaya on contagion of overconvergence and de Jong's alterations, we show that these higher direct images are overconvergent when the coefficients are by reduction to the projective case.

In this short note we explain the proof that proper surjective and faithfully flat maps are morphisms of effective descent for overconvergent isocrystals. We then show how to deduce the folklore theorem that for an arbitrary variety over a perfect field of characteristic $p$, the Frobenius pull-back functor is an equivalence on the overconvergent category., 11 pages, comments welcome. Updated to include a reference to the proper case handled by Shiho

In this article we prove exactness of the homotopy sequence of overconvergent $p$-adic fundamental groups for a smooth and projective morphism in characteristic $p$. We do so by first proving a corresponding result for rigid analytic varieties in characteristic $0$, following dos Santos in the algebraic case. In characteristic $p$, we then proceed by a series of reductions to the case of a liftable family of curves, where we can apply the rigid analytic result., 31 pages, comments very welcome!

In this article we give a survey of the various forms of Berthelot's conjecture and some of the implications between them. By proving some comparison results between pushforwards of overconvergent isocrystals and those of arithmetic $\mathcal{D}$-modules, we manage to deduce some cases of the conjecture from Caro's results on the stability of overcoherence under pushforward via a smooth and proper morphism of varieties. In particular, we show that Ogus' convergent pushforward of an overconvergent $F$-isocrystal under a smooth and projective morphism is overconvergent., Comment: 17 pages. Final version, published in J. Number Theory

The main result of this chapter is that \(\mathscr {E}_K^\dagger \)-valued rigid cohomology \(H^i_\mathrm {rig}(X/\mathscr {E}_K^\dagger ,\mathscr {E})\) is finite dimensional for any smooth scheme \(X/\mathscr {E}_K^\dagger \) and any \(\mathscr {E}\in F\text {-}\mathrm {Isoc}^\dagger (X/\mathscr {E}_K^\dagger )\), and moreover the base change of these vector spaces to \(\mathscr {E}_K\) coincides with classical rigid cohomology. After introducing the appropriate notion of a dagger algebra in our context, the key point is to prove a relative version of the p-adic local monodromy theorem for \((\varphi ,\nabla )\)-modules over Rbba rings attached to these dagger algebras, which we do by descending the corresponding result from affinoid algebras over \(\mathscr {E}_K\). Once certain other properties of \(\mathscr {E}_K^\dagger \)-valued rigid cohomology have been established, such as excision, a Gysin isomorphism&c. have been established, the eventual proof of finite dimensionality for smooth varieties proceeds in the usual way. Base change is proved simultaneously, and this then allows us to deduce results such as a Kunneth formula from their counterparts over \(\mathscr {E}_K\).

In this article I define and study the overconvergent rigid fundamental group of a variety over an equicharacteristic local field. This is a non-abelian $(\varphi,\nabla)$-module over the bounded Robba ring $\mathcal{E}_K^\dagger$, whose underlying unipotent group (after base changing to the Amice ring $\mathcal{E}_K$) is exactly the classical rigid fundamental group. I then use this to prove an equicharacteristic, $p$-adic analogue of Oda's theorem that a semistable curve over a $p$-adic field has good reduction iff the Galois action on its $\ell$-adic unipotent fundamental group is unramified., Comment: 34 pages, comments very welcome!

In this chapter we introduce a refinement of rigid cohomology for varieties over Laurent series fields \(k(\!(t)\!)\) in characteristic p, taking values in vector spaces over the bounded Robba ring \(\mathscr {E}_K^\dagger \). We achieve this by considering a more general kind of ‘frame’ \((X,Y,\mathfrak {P})\) than in the classical theory, obtained by compactifying our varieties X as schemes over \(k[\![t ]\!]\) rather than just over \(k(\!(t)\!)\). With this definition in place we may transport almost all of Berthelot’s original constructions and results word for word into our setting, showing that these \(H^i_\mathrm {rig}(X/\mathscr {E}_K^\dagger )\) cohomology groups are well defined (i.e. independent of the choice of such a frame). We also introduce categories of coefficients \(F\text {-}{\mathrm {Isoc}}^\dagger (X/\mathscr {E}_K^\dagger )\) for this cohomology theory.

In this chapter we discuss some arithmetic applications of the theory of \(\mathscr {E}_K^\dagger \)-valued rigid cohomology. The first step is to introduce a more refined category of coefficients \(F\text {-}\mathrm {Isoc}^\dagger (X/K)\), consisting of isocrystals relative to K, and show that for \(\mathscr {E}\in F\text {-}\mathrm {Isoc}^\dagger (X/K)\) the cohomology groups \(H^i_\mathrm {rig}(X/\mathscr {E}_K^\dagger ,\mathscr {E})\) come with the extra structure of a \((\varphi ,\nabla )\)-module over \(\mathscr {E}_K^\dagger \) (a p-adic analogue of a Galois representation). By showing a comparison result with Hyodo–Kato cohomology we are then able to deduce the analogue of the weight-monodromy conjecture in this context, by reducing to the global case handled by Crew. We also use a construction of Marmora to formulate a conjecture on \(\ell \)-independence comparing these p-adic cohomology groups to the \(\ell \)-adic ones Open image in new window for \(\ell \ne p\), and prove this conjecture in dimension 1.

In this article we study combinatorial degenerations of minimal surfaces of Kodaira dimension 0 over local fields, and in particular show that the `type' of the degeneration can be read off from the monodromy operator acting on a suitable cohomology group. This can be viewed as an arithmetic analogue of results of Persson and Kulikov on degenerations of complex surfaces, and extends various particular cases studied by Matsumoto, Liedtke/Matsumoto and Hern\'andez-Mada. We also study `maximally unipotent' degenerations of Calabi--Yau threefolds, following Koll\'ar/Xu, showing in this case that the dual intersection graph is a 3-sphere., Comment: 27 pages. Final version, published in Algebra & Number Theory

In this chapter we introduce a version of Le Stum’s overconvergent site for \({\mathscr {E}}_{K}^{\dagger }\)-valued cohomology, and show that \(H^i_\mathrm {rig}(X/{\mathscr {E}}_{K}^{\dagger },\mathscr {E})\) can be computed as the cohomology of this site. This then allows us to prove that cohomological descent holds for both fppf and proper hypercovers, again by adapting the proofs in the classical case. By using de Jong’s theorem on alteration, we may then deduce finite dimensionality of \(H^i_\mathrm {rig}(X/{\mathscr {E}}_{K}^{\dagger },\mathscr {E})\) in general, extending the case of smooth schemes in the previous chapter. We also introduce a version of \({\mathscr {E}}_{K}^{\dagger }\)-valued rigid cohomology with compact support, although can only prove the required finiteness results under strong assumptions on the coefficients. Under these assumption, we also deduce a version of Poincare duality from the classical case over \(\mathscr {E}_K\).

In this paper we define a rigid rational homotopy type, associated to any variety $X$ over a perfect field $k$ of positive characteristic. We prove comparison theorems with previous definitions in the smooth and proper, and log-smooth and proper case. Using these, we can show that if $k$ is a finite field, then the Frobenius structure on the higher rational homotopy groups is mixed. We also define a relative rigid rational homotopy type, and use it to define a homotopy obstruction for the existence of sections., Comment: 30 pages. Final version, published in Proceedings of the LMS

In this paper we define a relative rigid fundamental group, which associates to a section $p$ of a smooth and proper morphism $f:X\rightarrow S$ in characteristic $p$, a Hopf algebra in the ind-category of overconvergent $F$-isocrystals on $S$. We prove a base change property, which says that the fibres of this object are the Hopf algebras of the rigid fundamental groups of the fibres of $f$. We explain how to use this theory to define period maps as Kim does for varieties over number fields, and show in certain cases that the targets of these maps can be interpreted as varieties., Comment: 38 pages. Final version, published in Rend. Sem. Math. Univ. Padova