Daniel Caro
Up to a translation in the language of arithmetic $\D$-modules, we prove a conjecture of Berthelot on the preservation of the overconvergence under the direct image by a smooth proper morphism of varieties over a perfect field of characteristic $p>0$.
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This article is the first one of a series of three articles devoted to direct images of isocrystals: here we consider isocrystals without Frobenius structure; in the second one (resp. the third one), we will introduce a Frobenius structure in the convergent (resp. overconvergent) context. For a liftable proper smooth morphism we establish the overconvergence of direct images, owing to a base change theorem for a proper morphism between rigid analytic spaces. This result partially answers a conjecture of Berthelot on the overconvergence of direct images under a proper smooth morphism.
This article is the third one of a series of three articles devoted to direct images of isocrystals: here we consider overconvergent isocrystals with Frobenius structure. For a liftable proper smooth morphism we establish the overconvergence of direct images, owing to the first article and the existence of lifts of Frobenius. This result partially answers a conjecture of Berthelot on the overconvergence of direct images of overconvergent F-isocrystals under a proper smooth morphism.
Let $k$ be a perfect field of characteristic $p>0$, $\mathcal{V}$ a complete discrete valuation ring with residue field $k$ and field of fractions $K$ of characteristic 0, and $S$ a separated $k$-scheme of finite type. When $S$ is smooth over $k$, we partially prove here a conjecture of Berthelot about the overconvergence of the higher direct images of the structure sheaf under a proper smooth morphism $f:X\to S$; when $k$ is perfect and $\mathcal{V}$ is tamely ramified such direct images are always convergent, not only for the structure sheaf but also for (almost) every convergent $F$-isocrys…
In this paper, we develop the theory of relative log convergent cohomology. We prove the coherence of relative log convergent cohomology in certain case by using the comparison theorem between relative log convergent cohomlogy and relative log crystalline cohomology, and we relates relative log convergent cohomology to relative rigid cohomology to show the validity of Berthelot's conjecture on the coherence and the overconvergence of relative rigid cohomology for proper smooth families when they admit nice proper log smooth compactification to which the coefficient extends logarithmically.
We prove equality of the various $p$-adic period morphisms for smooth, not necessarily proper, schemes. We start with showing that the $K$-theoretical uniqueness criterium we had found for proper smooth schemes extends to proper finite simplicial schemes in the good reduction case and to cohomology with compact support in the semistable reduction case. It yields the equality of the period morphisms for cohomology with compact support defined using the syntomic, almost étale, and motivic constructions. We continue with showing that the $h$-cohomology period morphism agrees with the syntomic and…
We develop properties of unramified, étale and smooth morphisms between Berkovich spaces over $\mathbb{Z}$. We prove that they satisfy properties analogous to those of morphisms of schemes and we provide analytification criteria. Our results hold for any valued field, rings of integers of a number field and discrete valuation rings. Those cases are treated by a unified way.
We prove that, for certain extensions of valued fields which admit a sensible theory of ramification groups, there exist canonical towers that correspond to the break-points of their Herbrand function. In particular, each of the intermediate field extensions in the tower has a Herbrand function with only one break-point and there is at most one extension with trivial Herbrand function. We apply the result to the setting of finite morphisms of Berkovich curves where we prove the existence of canonical local and global factorization of such morphisms according to their metric properties. Finally…
We explain a correct proof of the decomposition theorem for direct images of constant Hodge modules by proper Kähler morphisms of complex manifolds. We also give some examples showing certain difficulty in the non-constant Hodge module case.
We generalize the adjunction between the functors $Rf_*$ and $f^!$ of derived categories of quasi-coherent sheaves for proper morphisms $f\colon X \to Y$ of Noetherian schemes to the following situation: Let $f$ be a finite type morphism and let $Z' \subseteq X$ and $Z \subseteq Y$ be closed subsets such that $f$ restricts to a proper morphism $f'\colon Z'\to Z$ of $f$. Then the functor $Rf_*$ is left adjoint to $RΓ_{Z'}f^!$ when considered as functors between complexes supported on $Z'$ or $Z$.
Functorial flatification of proper morphisms
本研究针对正常态射的平坦化问题，借鉴Hironaka算法思路提出通过吹胀的函子平坦化算法，特征零下可通过光滑中心吹胀实现，还给出了特征零下基于Kummer吹胀的函子平展化算法。
Kernels of Arithmetic Jet Spaces and Frobenius Morphism
本文研究了算术jet空间的核与Frobenius映射，揭示了其与Witt向量结构的联系。
A note on quasi-perfect morphisms
本文研究了诺特环上准完美映射的局部性质，证明了准完美性可通过目标空间的étale局部环及其完备化和（严格）赫恩塞利化检测，得出准完美映射的 locus 是 zariski 开集。
Proper pushforwards on analytic adic spaces
本文为解析仿射空间的部分固有态射构造固有推前映射，推广了van der Put的相关理论，还针对Kiehl意义下光滑且部分固有的态射构造了迹映射与对偶配对。
We make a very detailed analysis of the numerical properties of effective divisors whose support is contained in the exceptional locus of a birational morphism of smooth projective surfaces. As an application we extend Miyaoka's inequality on the number of canonical singularities on a projective normal surface with non-negative Kodaira dimension to the non minimal case, obtaining a slightly better result than known extensions by Megyesi and Langer.
Let $X$ be a projective and smooth variety over an algebraically closed field $k$. Let $f:Y\rightarrow X$ be a proper and surjective morphism of $k$-varieties. Assuming that $f$ is separable, we prove that the Tannakian category associated to the vector bundles $E$ on $X$ such that $f^*E$ is trivial is equivalent to the category of representations of a finite and etale group scheme. We give a counterexample to this conclusion in the absence of separability.
We propose here a transcendantal proof of the coherence of the higher direct images of a coherent sheaf by a proper morphism of algebraic varieties, which does not use Chow's lemma nor any projective method. The main tool here are comparison with coherent cohomology of a (Berkovich) analytic space over a trivially valued field, and Kiehls' theorem about the proper morphisms between rigid-analytic spaces.
Let X be a smooth projective variety defined over an algebraically closed field k. Nori constructed a category of vector bundles on X, called essentially finite vector bundles, which is reminiscent of the category of representations of the fundamental group (in characteristic zero). In fact, this category is equivalent to the category of representations of a pro--finite group scheme which controls all finite torsors. We show that essentially finite vector bundles coincide with those which become trivial after being pulled back by some proper and surjective morphism to X.
Pullback of principal bundles along proper morphisms
本文研究主丛沿紧合满态射拉回的行为，涵盖特征0代数闭域上的概形与复解析空间两类情形。
Suppose that $f: Y\to X$ is a proper, dominant, tamely ramified morphism of algebraic surfaces, over a perfect field. We show that it is possible to perform sequences of monoidal transforms $Y'\to Y$ and $X'\to X$ to obtain an induced morphism $Y'\to Y$ which is a monomial morphism.
Stein degree of proper morphisms
本文探讨了proper morphisms的Stein度数概念，介绍了其在birational geometry中的应用，并提出了关于Stein度数的有界性问题。
Given a proper morphism X -> S, we show that a large class of objects in the derived category of X naturally form an Artin stack locally of finite presentation over S. This class includes S-flat coherent sheaves and, more generally, contains the collection of all S-flat objects which can appear in the heart of a reasonable sheaf of t-structures on X. In this sense, this is the Mother of all Moduli Spaces (of sheaves). The proof proceeds by studying the finite presentation properties, deformation theory, and Grothendieck existence theorem for objects in the derived category, and then applying A…
We continue our study on infinitesimal lifting properties of maps between locally noetherian formal schemes started in math.AG/0604241. In this paper, we focus on some properties which arise specifically in the formal context. In this vein, we make a detailed study of the relationship between the infinitesimal lifting properties of a morphism of formal schemes and those of the corresponding maps of usual schemes associated to the directed systems that define the corresponding formal schemes. Among our main results, we obtain the characterization of completion morphisms as pseudo-closed immersi…
We classify morphisms from proper varieties to Brauer-Severi varieties, which generalizes the classical correspondence between morphisms to projective space and globally generated invertible sheaves. As an application, we study del Pezzo surfaces of large degree with a view towards Brauer-Severi varieties, and recover classical results on rational points, the Hasse principle, and weak approximation.
Let $f : X \longrightarrow Y$ be a proper and local complete intersection morphism of schemes. We prove that $\mathbb{R}f_{*}$ preserves perfect complexes, without any projectivity or noetherian assumptions. This provides a different proof of a theorem by Neeman and Lipman based on techniques from derived algebraic geometry to proceed a reduction to the noetherian case.
Finiteness of cohomology for pro-locally proper maps
本文为概形有限空间引入固有态射概念，通过经典概形结论与下降论证证明Grothendieck有限性定理的类比，并推广了使该定理成立的概形态射类。
We extend the Cox-Hu-Keel construction of the Cox rings to any proper birational morphisms of normal noetherian schemes. It allows the representation of any proper birational morphism by a map of schemes with mild singularities with torus actions. In a particular case, the notion generalizes the combinatorial construction of Satriano and the recent construction of multiple weighted blow-ups on Artin-stacks by Abramovich-Quek. The latter can be viewed as an extension of stack theoretic blow-ups by Abramovich, Temkin, and Wlodarczyk, a similar construction of McQuillan and the author's recent co…
A criterion for extending morphisms from open subsets of smooth fibrations of algebraic varieties
本文研究代数簇光滑纤维化中开子集上的态射扩张问题，给出一个类似 Zariski 主定理的充分条件，以保证到真态射目标的 $S$-态射可整体扩张。
Rigidity of modular morphisms via Fujita decomposition
本文利用曲线族Hodge丛的Fujita分解性质，证明了Torelli、Prym、Spin-Torelli态射及射影曲线模叠间覆盖映射的刚性，即它们不可形变。
A number theoretic characterization of $E$-smooth and (FRS) morphisms: estimates on the number of $\mathbb{Z}/p^{k}\mathbb{Z}$-points
该研究利用动机积分理论，给出了(FRS)态射纤维上$\u2124/p^{k}\u2124$-点数量的一致估计，证明其与(FRS)性质等价，还定义了细化的$E$-光滑态射并给出对应估计，同时改进了$\u03b5$-射流平坦态射的相关结果。
The Hitchin morphism is a map from the moduli space of Higgs bundles $\mathscr{M}_X$ to the Hitchin base $\mathscr{B}_X$, where $X$ is a smooth projective variety. When $X$ has dimension at least two, this morphism is not surjective in general. Recently, Chen-Ngô introduced a closed subscheme $\mathscr{A}_X$ of $\mathscr{B}_X$, which is called the space of spectral data. They proved that the Hitchin morphism factors through $\mathscr{A}_X$ and conjectured that $\mathscr{A}_X$ is the image of the Hitchin morphism. We prove that when $X$ is a smooth projective surface, this conjecture is true fo…
We prove formal GAGA for good moduli space morphisms under an assumption of "enough vector bundles" (which holds for instance for quotient stacks). This supports the philosophy that though they are non-separated, good moduli space morphisms largely behave like proper morphisms.
To any finite local embedding of Deligne--Mumford stacks $g: Y\to X$ we associate an étale, universally closed morphism $F_{Y/X}\to X$ such that for the complement $Y^2_X$ of the image of the diagonal $Y \to Y\times_XY$, the stack $F_{Y^2_X/Y}$ admits a canonical closed embedding in $F_{Y/X}$, and $F_{Y/X}\times_XY$ is a disjoint union of copies of $F_{Y^2_X/Y}$. The stack $F_{Y/X}$ has a natural functorial presentation, and the morphism $F_{Y/X}\to X$ commutes with base-change. The image of $Y^2_X$ in $Y$ is the locus of points where the morphism $Y \to g(Y)$ is not smooth. Thus for many prac…
Relations between some kinds of formal and standard smoothness, for morphisms of schemes, are clarified in surprisingly simple and direct ways, bypassing much of the customarily employed machinery. Even the deep local-to-global property of formal smoothness has a fairly elementary proof, under mild additional hypotheses.
Perverse filtration for generalized Kummer varieties of fibered surfaces
本文研究2维群概形纤维化的广义Kummer簇导出的真态射，证明其关联的反常层滤具有乘性。
Unipotent morphisms
本文引入代数叠的幂单态射理论，证明向量丛的局部到整体原理，并应用于Gabber定理的幂单类比及正特征下DM叠的分辨性质。
We study $F$-signature under proper birational morphisms $π: Y \to X$, showing that $F$-signature strictly increases for small morphisms or if $ K_Y \geq π^*K_X$. In certain cases, we can even show that the $F$-signature of $Y$ is at least twice as that of $X$. We also provide examples of $F$-signature dropping and Hilbert-Kunz multiplicity increasing under birational maps without these hypotheses.
Hodge modules and Kähler morphisms
本文证明了沿真凯勒态射的带整结构霍奇模分解定理，推广了 Saito 的射影情形结果，方法基于周期映射紧化与 Cattani-Kaplan-Schmid 的 L²-上同调定理。
We show that the weakly étale morphisms, used to define the pro-étale site of a scheme, are characterized by a lifting property similar to the one which characterizes formally étale morphisms. In order to prove this, we prove a theorem called Henselian descent which is a "Henselized version" of the fact that a scheme defines a sheaf for the fpqc topology. Finally, we show that weakly étale algebras over regular rings arising in geometry are ind-étale and that weakly étale algebras do not always lift along surjective ring homomorphisms.
Deformations of Canonical Bundle for Smooth Weakly Kähler Morphisms
本文研究光滑弱Kähler态射的典范丛形变，证明在中心纤维维数为4且其典范丛nef时，邻近纤维的典范丛也保持nef性质。
This article concerns properties of mixed $\ell$-adic complexes on varieties over finite fields, related to the action of the Frobenius automorphism. We establish a fiberwise criterion for the semisimplicity and Frobenius semisimplicity of the direct image complex under a proper morphism of varieties over a finite field. We conjecture that the direct image of the intersection complex on the domain is always semisimple and Frobenius semisimple; this conjecture would imply that a strong form of the decomposition theorem of Beilinson-Bernstein-Deligne-Gabber is valid over finite fields. We prove …
Injectivity theorems for higher direct images under proper Kähler morphisms on snc spaces
该研究针对snc空间上的真Kähler态射，将Fujino内射性定理推广到相对情形，利用调和积分等技术解决非紧性问题并得到应用。
On Proper Descent of Smooth Affine Surfaces with Finite Homotopy Rank-Sum
本文研究了光滑复仿射曲面在有限可缩映射下的同调性质下降问题，提出有限同调秩和性质并证明其有效性，同时分类了被复代数2-扭矩支配的曲面。
Let $S$ be a finite dimensional noetherian scheme. For any proper morphism between smooth $S$-schemes, we prove a Riemann-Roch formula relating higher algebraic $K$-theory and motivic cohomology, thus with no projective hypothesis neither on the schemes nor on the morphism. We also prove, without projective assumptions, an arithmetic Riemann-Roch theorem involving Arakelov's higher $K$-theory and motivic cohomology as well as an analogue result for the relative cohomology of a morphism. These results are obtained as corollaries of a motivic statement that is valid for morphisms between oriente…
The linear Shafarevich conjecture for quasiprojective varieties and algebraicity of Shafarevich morphisms
本文证明了非固有版本的 Shafarevich 猜想，即具有忠实复表示的正规复代数簇泛复叠是全纯凸空间的解析Zariski开子集，并证明了任意复局部系统对应的 Shafarevich 态射的代数性。
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