Errata Corrige to Theorems A and B for Dagger Quasi-Stein Spaces
We give a counter-example to Theorem B for dagger quasi-Stein spaces.
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arXiv subjects
Christopher Lazda
Publications and source records attributed to Christopher Lazda.
Rigidification of arithmetic $\mathscr{D}$-modules and an overconvergent Riemann-Hilbert correspondence
In this article, I define triangulated categories of constructible isocrystals on varieties over a perfect field of positive characteristic, in which Le Stum's abelian category of constructible isocrystals sits as the heart of a natural t-structure. I then prove a Riemann-Hilbert correspondence, showing that, for objects admitting some (unspecified) Frobenius, this triangulated category is equivalent to the triangulated category of overholonomic $\mathscr{D}^\dagger$-modules in the sense of Caro. I also show that the cohomological functors $f^!$, $f_+$ and $\otimes$ defined for $\mathscr{D}^\dagger$-modules have natural interpretations on the constructible side of this correspondence. Finally, I use this to prove that, for any variety $X$ admitting an immersion into a smooth and proper formal scheme, rigid cohomology (with lisse coefficients) agrees with cohomology defined using arithmetic $\mathscr{D}$-modules.
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A comparison between compactly supported rigid and $\pmb{\mathscr{D}}$-module cohomology
The goal of this article is to prove a comparison theorem between rigid cohomology and cohomology computed using the theory of arithmetic $\mathscr{D}$-modules. To do this, we construct a specialisation functor from Le Stum's category of constructible isocrystals to the derived category of arithmetic $\mathscr{D}$-modules. For objects `of Frobenius type', we show that the essential image of this functor consists of overholonomic $\mathscr{D}^\dagger$-modules, and lies inside the heart of the dual constructible t-structure. We use this to give a more global construction of Caro's specialisation functor $\mathrm{sp}_+$ for overconvergent isocrystals, which enables us to prove the comparison theorem for compactly supported cohomology.
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Reduction of Kummer surfaces modulo 2 in the non-supersingular case
We obtain necessary and sufficient conditions for the good reduction of Kummer surfaces attached to abelian surfaces with non-supersingular reduction when the residue field is perfect of characteristic 2. In this case, good reduction with an algebraic space model is equivalent to good reduction with a scheme model, which we explicitly construct.
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Proper pushforwards on analytic adic spaces
We construct proper pushforwards for partially proper morphisms of analytic adic spaces. This generalises the theory due to van der Put in the case of rigid analytic varieties over a non-Archimedean field. For morphisms which are smooth and partially proper in the sense of Kiehl, we furthermore construct the trace map and duality pairing.
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Good reduction of K3 Surfaces in equicharacteristic p
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.
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The filtered Ogus realisation of motives
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.
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Local acyclicity in $p$-adic cohomology
We prove an analogue for $p$-adic coefficients of the Deligne--Laumon theorem on local acyclicity for curves. That is, for an overconvergent $F$-isocrystal $E$ on a relative curve $f:U\rightarrow S$ admitting a good compactification, we show that the cohomology sheaves of $\mathbf{R}f_!E$ are overconvergent isocrystals if and only if $E$ has constant Swan conductor at infinity.
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A note on effective descent for overconvergent isocrystals
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.
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A homotopy exact sequence for overconvergent isocrystals
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.
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Cycle classes in overconvergent rigid cohomology and a semistable Lefschetz $(1,1)$ theorem
In this article we prove a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, stating that a rational (logarithmic) line bundle on the special fibre of a semistable scheme over $k [\![ t ]\!]$ 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.
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A N\'eron-Ogg-Shafarevich criterion for K3 surfaces
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.
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Around $\ell$-independence
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 semistable varieties 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.
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Fundamental groups and good reduction criteria for curves over positive characteristic local fields
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.
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Combinatorial degenerations of surfaces and Calabi--Yau threefolds
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.
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Incarnations of Berthelot's conjecture
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.
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Rigid cohomology over Laurent series field I: First definitions and basic properties
This is the first in a series of papers in which we construct and study a new $p$-adic cohomology theory for varieties over Laurent series fields $k(\!(t)\!)$ in characteristic $p$. This will be a version of rigid cohomology, taking values in the bounded Robba ring $\mathcal{E}_K^\dagger$, and in this paper, we give the basic definitions and constructions. The cohomology theory we define can be viewed as a relative version of Berthelot's rigid cohomology, and is constructed by compactifying $k(\!(t)\!)$-varieties as schemes over $k[\![ t]\!]$ rather than over $k(\!(t)\!)$. We reprove the foundational results necessary in our new context to show that the theory is well defined and functorial, and we also introduce a category of `twisted' coefficients. In latter papers we will show some basic structural properties of this theory, as well as discussing some arithmetic applications including the weight monodromy conjecture and independence of $\ell$ results for equicharacteristic local fields.
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Rigid cohomology over Laurent series fields II: Finiteness and Poincaré duality for smooth curves
In this paper we prove that the $\mathcal{E}^\dagger_K$-valued cohomology, introduced in [9] is finite dimensional for smooth curves over Laurent series fields $k((t))$ in positive characteristic, and forms an $\mathcal{E}^\dagger_K$-lattice inside `classical' $\mathcal{E}_K$-valued rigid cohomology. We do so by proving a suitable version of the p-adic local monodromy theory over $\mathcal{E}^\dagger_K$, and then using an étale pushforward for smooth curves to reduce to the case of $\mathbb{A}^1$. We then introduce $\mathcal{E}^\dagger_K$-valued cohomology with compact supports, and again prove that for smooth curves, this is finite dimensional and forms an $\mathcal{E}^\dagger_K$-lattice in $\mathcal{E}_K$-valued cohomology with compact supports. Finally, we prove Poincaré duality for smooth curves, but with restrictions on the coefficients.
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