Jakob Stix
The geometric condition of T. Saito for trivial action of the wild monodromy of a smooth proper curve over the generic point of a trait is transformed to the condition of logarithmic smooth reduction. The proof emphasizes methods and results from logarithmic geometry. It applies to quasi-projective smooth curves with etale boundary divisor.
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Let X be a smooth variety over a field of positive characteristic, and let E be an overconvergent isocrystal on X. We establish a criterion for the existence of a "canonical logarithmic extension" of E to a good compactification of X. In the process, we construct a category of overconvergent log-isocrystals and discuss its basic properties. In subsequent work, we will use these results to show that a canonical logarithmic extension always exists after pulling back E along a suitable cover of X.
We study duality theorems for the relative logarithmic de Rham-Witt sheaves on semi-stable schemes $X$ over a local ring $\mathbb{F}_q[[t]]$, where $\mathbb{F}_q$ is a finite field. As an application, we obtain a new filtration on the maximal abelian quotient $π^{\text{ab}}_1(U)$ of the étale fundamental groups $π_1(U)$ of an open subscheme $U \subseteq X$, which gives a measure of ramification along a divisor $D$ with normal crossing and $\text{Supp}(D) \subseteq X-U$. This filtration coincides with the Brylinski-Kato-Matsuda filtration in the relative dimension zero case.
In this paper, we give a criterion for the existence of logarithmic embeddings -- which was first introduced by Steenbrink -- for general normal crossing varieties. Using this criterion, we also give a new proof of the theorem of Kawamata--Namikawa which states a criterion for the existence of the log structures of semistable type.
$\mathbb{Z}_p$-lattices in semistable Galois representations
该研究证明对数棱柱F-晶体范畴与半稳定Galois表示中的ℤₚ-格范畴等价，并借助Breuil–Kisin模的对数版本用线性代数数据描述这类伽罗瓦表示，为数论中伽罗瓦表示研究提供新的范畴等价工具。
A log-motivic cohomology for semistable varieties and its $p$-adic deformation theory
本文提出了一种半稳定概形的log-动机上同调理论，并探讨了其p进变形理论，通过研究Hyodo-Kato类与Hodge滤波的关系，推广了Bloch-Esnault-Kerz的结论。
For a proper, smooth scheme $X$ over a $p$-adic field $K$, we show that any proper, flat, semistable $\mathcal{O}_K$-model $\mathcal{X}$ of $X$ whose logarithmic de Rham cohomology is torsion free determines the same $\mathcal{O}_K$-lattice inside $H^i_{dR}(X/K)$ and, moreover, that this lattice is functorial in $X$. For this, we extend the results of Bhatt--Morrow--Scholze on the construction and the analysis of an $A_{inf}$-valued cohomology theory of $p$-adic formal, proper, smooth $\mathcal{O}_{\overline{K}}$-schemes $\mathfrak{X}$ to the semistable case. The relation of the $A_{inf}$-coho…
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 conject…
Let $X\rightarrow C$ be a totally real semistable degeneration over a smooth real curve $C$ with degenerate fiber $X_0$. Assuming that the irreducible components of $X_0$ are simple from a cohomological point of view, we give a bound for the individual Betti numbers of a real smooth fiber near $0$ in terms of the complex geometry of the degeneration. This generalizes previous work of Renaudineau-Shaw, obtained via combinatorial techniques, for tropical degenerations of hypersurfaces in smooth toric varieties. The main new ingredient is the use of real logarithmic geometry, which allows to work…
We introduce a valuation-theoretic approach to the problem of semistable reduction (i.e., existence of logarithmic extensions on suitable covers) of overconvergent isocrystals with Frobenius structure. The key tool is the quasicompactness of the Riemann-Zariski space associated to the function field of a variety.
We complete our proof that given an overconvergent F-isocrystal on a variety over a field of positive characteristic, one can pull back along a suitable generically finite cover to obtain an isocrystal which extends, with logarithmic singularities and nilpotent residues, to some complete variety. We also establish an analogue for F-isocrystals overconvergent inside a partial compactification. By previous results, this reduces to solving a local problem in a neighborhood of a valuation of height 1 and residual transcendence degree 0. We do this by studying the variation of some numerical invari…
Logarithmic $A_{\mathrm{inf}}$-cohomology, Part II
本文为充分对数光滑的 $p$-进形式概形建立带系数的对数 $A_{\mathrm{inf}}$-上同调理论，证明与 étale、de Rham、晶体上同调的比较同构，并由此给出半稳定局部系统 $C_{\mathrm{st}}$ 猜想的新证明。
We study the $E_1$-degeneration of the logarithmic Hodge to de Rham spectral sequence of the special fiber of a semistable family over a discrete valuation ring. On the one hand, we prove that the $E_1$-degeneration property is invariant under admissible blow-ups. Assuming functorial resolution of singularities over $\mathbb{Z}$, this implies that the $E_1$-degeneration property depends only on the generic fiber. On the other hand, we show by explicit examples that the decomposability of the logarithmic de Rham complex is not invariant under admissible blow-ups, which answer negatively an open…
We study various aspects on nontrivial logarithmic co-Higgs structure associated to unstable bundles on algebraic curves. We check several criteria for (non-)existence of nontrivial logarithmic co-Higgs structures and describe their parameter spaces. We also investigate the Segre invariants of these structures and see their non-simplicity. In the end we also study the higher dimensional case, specially when the tangent bundle is not semistable.
This paper deals with connections on $p$-adic analytic curves, in the sense of Berkovich. The curves must be compact but the connections are allowed to have a finite number of meromorphic singularities on them. For any choice of a semistable formal model of the curve, we define an intrinsic notion of normalized radius of convergence as a function on the curve, with values in $(0,1]$. For a sufficiently refined choice of the semistable model, we prove continuity and logarithmic concavity of that function. We characterize \emph{Robba connections}, that is connections whose sheaf of solutions is …
Net logarithmic tangent sheaves of complete intersections
本文定义了网对数切层作为对数切层的推广，适用于光滑完全交簇及其子簇，并研究了光滑三次曲面上此类切层在模空间中的轨迹。
We develop a semistability algorithm for vector bundles which are given as a kernel of a surjective morphism between splitting bundles on the projective space over an algebraically closed field K. This class of bundles is a generalization of syzygy bundles. We show how to implement this algorithm in a computer algebra system. Further we give applications, mainly concerning the computation of Tannaka dual groups of stable vector bundles of degree 0 on the projective space and on certain smooth complete intersection curves. We also use our algorithm to close an open case left in a recent work of…
Let $f:\Cal C\to S$ be a flat family of curves over a smooth curve $S$ such that $f$ is smooth over $S_0=S\ssm\{s_0\}$ and $f^{-1}(s_0)=\Cal C_0$ is irreducible with one node. We have an associated family $\Cal M_{S_0}\to S_0$ of moduli spaces of semistable vector bundles and the relative theta line bundle $Θ_{S_0}$. We are interested in the problem: to find suitable degeneration $\Cal M_S$ of moduli spaces and extension $Θ_S$ of theta line bundles such that the direct image of $Θ_S$ is a vector bundle on $S$ with a logarithmic projective connection. In this paper, we figured out the condition…
We view the moduli space of semistable sheaves on a K3 surface as a global quotient stack, and compute its cotangent complex in terms of the universal sheaf on the Quot scheme. Relevant facts on the classical and reduced Atiyah classes are reviewed. We also define the notion of a symplectic stack, and show that it includes all moduli stacks of semistable sheaves on K3 surfaces.
The non-abelian Hodge correspondence for extensions on quasi-projective curves
本文为拟射影曲线上的对数平坦丛与对数希格斯丛建立精确等价，推广了温和辛普森-望月对应至幂零留数情形。
We prove that the cycle-valued logarithmic Gromov--Witten theory of a product of simple normal crossings pairs $X\times Y$ decomposes into a product of pieces coming from $X$ and $Y$, provided that the decomposition is considered over a blowup of the moduli space of curves. The result is established by applying the toroidal semistable reduction theorem to the stabilization morphism on the stack of maps to the Artin fans. Since products of smooth pairs are naturally simple normal crossings pairs, the result provides a direct avenue of access to questions in logarithmic Gromov--Witten theory via…
We show that a particular subfunctor of the relative logarithmic Picard functor for families of aligned, log semistable curves over a regular base scheme and smooth over an open dense subscheme of the base is representable by a smooth algebraic space which is slightly more separated than the classical relative Picard functor. We show the existence of its maximal separated quotient and give a new functorial interpretation in terms of logarithmic geometry for the Néron model of the relative Picard functor of the smooth locus when we restrict to transversal pull backs from spectra of discrete val…
Counting sheaves on curves
本研究计算复射影曲线上半稳定凝聚层的Joyce计数不变量，通过正则化和得到显式表达式，为半稳定层模空间上同调配对问题提供新视角，并将互素情形的经典结果推广到非互素情形。
Clusters and semistable models of hyperelliptic curves in the wild case
本文研究了超椭圆曲线在剩余特征p=2时的半稳定模型，提出了相对稳定模型的计算方法，并揭示了特殊纤维分支与根的簇之间的对应关系。
Logarithmic Non-Abelian Hodge Theory for curves in prime characteristic
本研究在素特征曲线框架下提出对数非阿贝尔Hodge定理Log-p-NAHT，建立对数G-联络与对数G-Higgs丛模空间的联系，推导其半稳定版本及几何、上同调推论，含GL_r情形的相交上同调嵌入结论。
Let $X$ be a smooth projective curve over a field of characteristic zero and let $D$ be a non-empty set of rational points of $X$. We calculate the motivic classes of moduli stacks of semistable parabolic bundles with connections on $(X,D)$ and motivic classes of moduli stacks of semistable parabolic Higgs bundles on $(X,D)$. As a by-product we give a criteria for non-emptiness of these moduli stacks, which can be viewed as a version of the Deligne-Simpson problem.
The structure of the moduli of gauged maps from a smooth curve
本文将Harder-Narasimhan理论扩展至规范映射模栈，构造了$Θ$-分层结构，并据此给出了任意线性表示下K理论规范Gromov-Witten不变量的公式。
These lecture notes provide an introduction to logarithmic geometry with a view towards recent applications in the desingularization theory.
Geometry of the Donaldson-Friedman Pushout: Twistor degenerations and instanton charge
本文研究了Donaldson-Friedman构造中出现的奇异中心纤维，将其视为两个吹胀的Twistor空间沿例外四面体的Ferrand推出。通过显式代数几何模型，描述了与连通和构造相关的Twistor退化。并推导了半稳定光滑的组件特化公式，以及表面在双定位处的刚性粘合约束。
The moduli stack of enriched structures and a logarithmic compactification
本文从对数观点定义丰富曲线及其模栈，通过光滑对数爆破构造简单模紧化，肯定回答[BH19]中的两个问题，并推广至$r$-丰富曲线。
Let $R$ be a strictly local ring complete for a discrete valuation, with fraction field $K$ and residue field of characteristic $p > 0$. Let $X$ be a smooth, proper variety over $K$. Nicaise conjectured that the rational volume of $X$ is equal to the trace of the tame monodromy operator on $\ell$-adic cohomology if $X$ is cohomologically tame. He proved this equality if $X$ is a curve. We study his conjecture from the point of view of logarithmic geometry, and prove it for a class of varieties in any dimension: those having logarithmic good reduction.
Many papers are devoted to study logarithmic sheaves associated to reduced divisors, in particular logarithmic bundles associated to plane curves since forty years in differential and algebraic topology or geometry. An interesting family of these curves are the so-called free ones for which the associated logarithmic sheaf is the direct sum of two line bundles. When the curve is a finite set of distinct lines (i.e. a line arrangement), Terao conjectured thirty years ago that its freeness depends only on its combinatorics. A lot of efforts were done to prove it but at this time it is only prove…
On logarithmic paper some real algebraic curves look like smoothed broken lines. Moreover, the broken lines can be obtained as limits of those curves. The corresponding deformation can be viewed as a quantization, in which the broken line is a classical object and the curves are quantum. This generalizes to a new connection between algebraic geometry and the geometry of polyhedra, which is more straight-forward than the other known connections and gives a new insight into constructions used in the topology of real algebraic varieties.
In a 1993 article, G. Faltings gave a new construction of the moduli space $U$ of semistable vector bundles on a smooth curve $X$, avoiding geometric invariant theory. Roughly speaking, Faltings showed that the normalisation $B$ of the ring $A$ of theta functions (associated with vector bundles on $X$) suffices to realize $U$ as a projective variety. Describing Faltings' work, C.S. Seshadri asked how close $A$ is to $B$. In this article, we address this question from a geometric point of view. We consider the rational map, $π: U @>>> Proj(A)$, and show that, not only is $π$ defined everywhere,…
In this thesis, we use logarithmic methods to study motivic objects. Let R be a complete discrete valuation ring with perfect residue field k, and denote by K its fraction field. We give in chapter 2 a new construction of the motivic Serre invariant of a smooth K-variety and extend it additively to arbitrary K-varieties. The main advantage of this construction is to rely only on resolution of singularities and not on a characteristic zero assumption, as did previous results. As an application, we give a conditional positive answer to Serre's question on the existence of rational fixed points o…
A tale of two moduli spaces: logarithmic and multi-scale differentials
本文证明了对数微分与多尺度微分在全局留数条件下等价，并建立了它们的（粗糙）模栈同构，同时通过显式爆破描述了橡胶空间和多尺度空间，提出了扭曲Hodge丛中的细化双分岔圈公式。
Any arrangement of hyperplanes in general position in $P^n$ can be regarded as a divisor with normal crossing. We study the bundles of logarithmic 1-forms corresponding to such divisors` from the point of view of classification of vector bundles on $P^n$. It turns out that all such bundles are stable. The study of jumping lines of these bundles gives a unified treatment of several classical constructions associating a curve to a collection of points in $P^n$. The main result of the paper is "Torelli theorem" which says that the collection of hyperplanes can be recovered from the isomorphism cl…
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