Dan Abramovich, Qile Chen
We make an observation which enables one to deduce the existence of an algebraic stack of log maps for all generalized Deligne--Faltings log structures (in particular simple normal crossings divisor) from the simplest case with log structures given by a Cartier divisor (essentially the smooth divisor case).
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We introduce a new compactification of the space of relative stable maps. This new method uses logarithmic geoemtry in the sense of Kato-Fontaine-Illusie rather than the expanded degeneration. The underlying structure of our log stable maps is stable in the usual sense.
Faltings showed that "arithmetic Kodaira--Spencer classes" satisfying a certain compatibility axiom cannot exist. By modifying his definitions slightly, we show that the Deligne--Illusie classes satisfy what could be considered an "arithmetic Kodaira--Spencer" compatibility condition. Afterwards we discuss a "wittfinitesimal Torelli problem" and its relation to CM Jacobians.
Logarithmic Crystalline Representations
本文提出对数Fontaine-Faltings模块和对数晶体质谱的构造方法，扩展了Faltings在1989年关于étale上同调与晶体质谱比较定理的研究。
We extend the formalism of "log spaces" of arXiv:1507.06752 to topoi equipped with a sheaf of monoids, and discuss Deligne--Faltings structures and root stacks in this context.
The evaluation stack for minimal logarithmic stable maps is constructed, parameterizing families of standard log points in the target log scheme. This construction provides the ingredients necessary to define appropriate evaluation maps for minimal log stable maps and establish the logarithmic Gromov-Witten theory of a log-smooth Deligne- Faltings log scheme.
In a beautiful paper Deligne and Illusie proved the degeneration of the Hodge-to-de Rham spectral sequence using positive characteristic methods. In a recent paper Arinkin, Căldăraru and the author of this paper gave a geometric interpretation of the problem of Deligne-Illusie showing that the triviality of a certain line bundle on a derived scheme implies the the Deligne-Illusie result. In the present paper we generalize these ideas to logarithmic schemes and using the theory of twisted derived intersection of logarithmic schemes we obtain the Hodge theorem for the logarithmic de Rham complex…
Let ${\cal L}$ be a variation of Hodge structures on the complement $X^{*}$ of a normal crossing divisor (NCD) $ Y$ in a smooth analytic variety $X$ and let $ j: X^{*} = X - Y \to X $ denotes the open embedding. The purpose of this paper is to describe the weight filtration $W$ on a combinatorial logarithmic complex computing the (higher) direct image ${\bf j}_{*}{\cal L} $, underlying a mixed Hodge complex when $X$ is proper, proving in this way the results in the note [14] generalizing the constant coefficients case. When a morphism $f: X \to D$ to a complex disc is given with $Y = f^{-1}(0)…
Logarithmic Quasimaps
本文利用对数几何构造了任意亏格下相对于简单正规交叉除子的拟映射的Deligne-Mumford栈模空间，并证明其具有预期维数的虚拟基本类，所得数值不变量与Battistella-Nabijou理论一致。
Logarithmic geometry and Frobenius
本文在对数几何框架下，借助Deligne系统理论构造了有限域上带权滤的$\ell$-进层阿贝尔范畴，并研究其高阶直接像的渐近行为，揭示其与单值权猜想的紧密联系。
We generalize the logarithmic decomposition theorem of Deligne-Illusie to a filtered version. There are two applications. The easier one provides a mod $p$ proof for a vanishing theorem in characteristic zero. The deeper one gives rise to a positive characteristic analogue of a theorem of Deligne on the mixed Hodge structure attached to complex algebraic varieties.
Let $K$ be a field which is complete for a discrete valuation. We prove a logarithmic version of the Néron-Ogg-Shafarevich criterion: if $A$ is an abelian variety over $K$ which is cohomologically tame, then $A$ has good reduction in the logarithmic setting, i.e. there exists a projective, log smooth model of $A$ over $\mathcal{O}_K$. This implies in particular the existence of a projective, regular model of $A$, generalizing a result of Künnemann. The proof combines a deep theorem of Gabber with the theory of degenerations of abelian varieties developed by Mumford, Faltings-Chai et al.
We prove lower bound and finiteness properties for arakelovian heights with respect to pre-log-log hermitian ample line bundles. These heights were introduced by Burgos, Kramer and Kühn, in their extension of the arithmetic intersection theory of Gillet and Soulé, aimed to deal with hermitian vector bundles equipped with metrics admitting suitable logarithmic singularities. Our results generalize the corresponding properties for the heights of Bost-Gillet-Soulé, as well as the properties established by Faltings for heights of points attached to hermitian line bundles whose metrics have logarit…
In this note, we prove the logarithmic $p$-adic comparison theorem for open rigid analytic varieties. We prove that a smooth rigid analytic variety with a strict simple normal crossing divisor is locally $K(π,1)$ (in a certain sense) with respect to $\mathbb{F}_p$-local systems and ramified coverings along the divisor. We follow Scholze's method to produce a pro-version of the Faltings site and use this site to prove a primitive comparison theorem in our setting. After introducing period sheaves in our setting, we prove aforesaid comparison theorem.
We present a new method to solve certain $\bar{\partial}$-equations for logarithmic differential forms by using harmonic integral theory for currents on Kahler manifolds. The result can be considered as a $\bar{\partial}$-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic forms, give geometric and simpler proofs of Deligne's degeneracy theorem for the logarithmic Hodge to de Rham spectral sequences at $E_1$-level, as well as certain injectivity theorem on compact Kahler manifolds. Furthermore, for a family of logarithmic deformatio…
On the Hodge theory of toroidal embeddings and corresponding vanishings
本文在环面嵌入背景下建立了Deligne对数比较定理与Hodge-de Rham谱序列的$E_1$-退化，并分别证明了环面簇与环面簇的Kawamata-Viehweg消失定理和Bott消失定理。
In this article, we will give the Deligne 1-motives up to isogeny corresponding to the $\mathbb{Q}$-limiting mixed Hodge structures of semi-stable degenerations of curves, by using logarithmic structures and Steenbrink's cohomological mixed Hodge complexes associated to semi-stable degenerations of curves.
We construct a logarithmic model of connections on smooth quasi-projective $n$-dimensional geometrically irreducible varieties defined over an algebraically closed field of characteristic $0$. It consists of a good compactification of the variety together with $(n+1)$ lattices on it which are stabilized by log differential operators, and compute algebraically de Rham cohomology. The construction is derived from the existence of good Deligne-Malgrange lattices, a theorem of Kedlaya and Mochizuki which consists first in eliminating the turning points. Moreover, we show that a logarithmic model o…
Regular logarithmic connections
本文引入复光滑对数概形上的正则可积联络概念，构造其与解析化联络范畴的等价关系，并结合典范扩张与良好紧致化给出基于Kato--Nakayama空间的拓扑描述。
Toroidal Compactifications and Stacky Cohomology of Mumford--Tate Domains
本文构造并研究了Mumford-Tate域的log-toric Hodge堆栈，通过将Kato-Usui部分紧化由neat算术群商得，实现了Hodge理论与堆栈边界数据的分离。
Mixed Hodge Structures on Generalized Theta Divisors and Graph Motives
本文研究一般化θ除子上的混合Hodge结构，并探讨其与图动机的可能联系。
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