Hélène Esnault, Eckart Viehweg
We show that hypergeometric differential equations, unitary and Gauss-Manin connections give rise to de Rham cohomology sheaves which do not admit a Bloch-Ogus resolution. The latter is in contrast to Panin's theorem asserting that corresponding étale cohomology sheaves do fulfill Bloch-Ogus theory.
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We will prove a Kodaira-Nakano type of vanishing theorem for the logarithmic de Rham complex of unitary local system. We will then study the weight filtration on the logarithmic de Rham complex, and prove a stronger statement for the associated graded complex.
Dimensional vanishing of the saturated de Rham-Witt complex
本文研究了饱和de Rham-Witt复形在奇异方案上的维度消逝性质，证明其在存在奇点时仍保持良好的消逝性质，优于传统上同调理论。
We establish a positive characteristic analogue of intersection cohomology theory for variations of Hodge structure. It includes: a) the de Rham-Higgs comparison theorem for the intersection de Rham complex; b) the $E_1$-degeneration theorem for the intersection de Rham complex of a periodic de Rham bundle; c) the Kodaira-Saito vanishing theorem for the intersection cohomology groups of a periodic Higgs bundle. These results generalize the decomposition theorem of Deligne-Illusie and the de Rham-Higgs theorem of Ogus-Vologodsky, the $E_1$-degneration theorem of Deligne-Illusie, Illusie, Faltin…
Structure theorem for log de Rham-Witt sheaves with vanishing
本文证明了对数 de Rham-Witt 层沿有效 Cartier 除子消没的结构定理，回答了两位学者的问题，并为后续 Milnor K-理论研究奠定基础。
Given a smooth formal scheme over the ring of integers of a mixed-characteristic perfectoid field, we study its $p$-adic vanishing cycles via de Rham--Witt and $q$-de Rham complexes.
We generalize Illusie's result to prove the decomposition of the de Rham complex with smooth horizontal coefficients for a semistable $S$-morphism $f:X\ra Y$ which is liftable over $\Z/p^2\Z$. As an application, we prove the Kollár vanishing theorem in positive characteristic for a semistable $S$-morphism $f:X\ra Y$ which is liftable over $\Z/p^2\Z$, where all concerned horizontal divisors are smooth over $Y$.
In this paper we introduce the word "fresco" to denote a $[λ]-$primitive monogenic geometric (a,b)-module. The study of this "basic object" (generalized Brieskorn module with one generator) which corresponds to the minimal filtered (regular) differential equation satisfied by a relative de Rham cohomology class, began in [B.09] where the first structure theorems are proved. Then in [B.10] we introduced the notion of theme which corresponds in the $[λ]-$primitive case to frescos having a unique Jordan-H{ö}lder sequence. Themes correspond to asymptotic expansion of a given vanishing period, so t…
In this paper, we define a certain Hodge-theoretic structure for an arbitrary variety X over the complex number field by using the theory of mixed Hodge module due to Morihiko Saito. We call it an arithmetic Hodge structure of X. It is shown that extension groups of arithmetic Hodge structure do not vanish even for degree $\geq2$. Moreover, we define higher Abel-Jacobi maps from Bloch's higher Chow groups of X to these extension groups. These maps essentially involve the classical Abel-Jacobi maps by Weil and Griffiths, and Mumford's infinitesimal invariants of 0-cycles on surfaces.
On an arbitrary toric variety, we introduce the logarithmic double complex, which is essentially the same as the algebraic de Rham complex in the nonsingular case, but which behaves much better in the singular case. Over the field of complex numbers, we prove the toric analog of the algebraic de Rham theorem which Grothendieck formulated and proved for general nonsingular algebraic varieties re-interpreting an earlier work of Hodge-Atiyah. Namely, for a finite simplicial fan which need not be complete, the complex cohomology groups of the corresponding toric variety as an analytic space coinci…
A differential form vanishing on the tangent space at smooth points of a reduced embedded analytic germ is called conormal. For proving that a conormal one--form of a hypersurface vanishes at its singularities we state a Bertini--type theorem.
We describe methods for calculation of polytopes of quasiadjunction for plane curve singularities which are invariants giving a Hodge theoretical refinement of the zero sets of multivariable Alexander polynomials. In particular we identify some hyperplanes on which all polynomials in multivariable Bernstein ideal vanish.
Gentle Introduction of Unitary Cohomology Vanishing
本文基于李红山的工作，以导论形式介绍酉上同调消失定理，探讨复流形上酉局部系统的性质，分析典范扩张、谱序列、Higgs丛等内容，证明相关谱序列E1退化，深化对酉局部系统上同调性质的理解。
In this paper we introduce the word "fresco" to denote a \ $[λ]-$primitive monogenic geometric (a,b)-module. The study of this "basic object" (generalized Brieskorn module with one generator) which corresponds to the minimal filtered (regular) differential equation satisfied by a relative de Rham cohomology class, began in [B.09] where the first structure theorems are proved. Then in [B.10] we introduced the notion of theme which corresponds in the \ $[λ]-$primitive case to frescos having a unique Jordan-H{ö}lder sequence. Themes correspond to asymptotic expansion of a given vanishing period, …
On the De Rham Cohomology of Zero-Dimensional Schemes
本文研究零维概形的de Rham上同调，证明了基于Macaulay基的消没定理，并分析了Artinian局部环的上同调维数，通过Galois分裂计算了fat point概形的上同调。
Cohomology of the differential fundamental group of algebraic curves
本文研究特征零域上光滑射影曲线的微分基本群，通过Tannakian对偶定义，探讨带联络向量丛的de Rham上同调与对应表示的群上同调的关系，并得到消没与非消没结果。
We extend the results of Deligne and Illusie on liftings modulo $p^2$ and decompositions of the de Rham complex in several ways. We show that for a smooth scheme $X$ over a perfect field $k$ of characteristic $p>0$, the truncations of the de Rham complex in $\max(p-1, 2)$ consecutive degrees can be reconstructed as objects of the derived category in terms of its truncation in degrees at most one (or, equivalently, in terms the obstruction class to lifting modulo $p^2$). Consequently, these truncations are decomposable if $X$ admits a lifting to $W_2(k)$, in which case the first nonzero differe…
Non-decomposability of the de Rham complex and non-semisimplicity of the Sen operator
本文研究了特征 $p$ 下 de Rham 复形的分解障碍，证明了存在 Hodge-to-de Rham 谱序列不退化的可提升光滑射影簇，并基于交换代数与 Steenrod 运算揭示了 Sen 算子的非半单性及 $E_{\infty}$-代数形式性条件。
In this paper we study the cohomology of the de Rham complex of sheaves of reflexive differential forms on a normal complex space. First, we prove that the complex is exact in degree one under suitable conditions on the underlying topological space, but that exactness in general depends on the complex structure. Second, we show exactness in high degrees for holomorphically contractible spaces under mild assumptions on the nature of singularities, e.g. klt singularities. Subsequently, the exactness of the de Rham complex of reflexive differential forms is related to the Lipman-Zariski conjectur…
On the D-module of an isolated singularity
本文研究复超曲面孤立奇点处由1/f幂生成的解析D-模族，并通过补集de Rham上同调的极点阶滤过进行刻画。
Consider a Maslov zero Lagrangian submanifold diffeomorphic to a Lie group on which an anti-symplectic involution acts by the inverse map of the group. We show that the Fukaya $A_\infty$ endomorphism algebra of such a Lagrangian is quasi-isomorphic to its de Rham cohomology tensored with the Novikov field. In particular, it is unobstructed, formal, and its Floer and de Rham cohomologies coincide. Our result implies that the smooth fibers of a large class of singular Lagrangian fibrations are unobstructed and their Floer and de Rham cohomologies coincide. This is a step in the SYZ and family Fl…
In this paper we obtain an ordinary differential equation ${\sf H}$ from a Picard-Fuchs equation associated with a nowhere vanishing holomorphic $n$-form. We work on a moduli space ${\sf T }$ constructed from a Calabi-Yau $n$-fold $W$ together with a basis of the middle complex de Rham cohomology of $W$. We verify the existence of a unique vector field ${\sf H}$ on ${\sf T }$ such that its composition with the Gauss-Manin connection satisfies certain properties. The ordinary differential equation given by ${\sf H}$ is a generalization of differential equations introduced by Darboux, Halphen an…
$L^p$ Hodge theory for bounded subanalytic manifolds
本文研究有界子解析流形上满足 Dirichlet 或 Neumann 条件的 $L^p$ 微分形式的上同调，证明了 $L^p$ Hodge 分解、de Rham 定理及 Lefschetz 对偶，并通过稠密性结果给出边界条件的显式刻画。
For a germ of a variety $\mathcal{V}, 0 \subset \mathbb C^N, 0$, a singularity $\mathcal{V}_0$ of type $\mathcal{V}$, is given by a germ $f_0 : \mathbb C^n, 0 \to \mathbb C^N, 0$ which is transverse to $\mathcal{V}$ in an appropriate sense so that $\mathcal{V}_0 = f_0^{-1}(\mathcal{V})$. For these singularities, we introduce "characteristic cohomology" to capture the contribution of the topology of $\mathcal{V}$ to that of $\mathcal{V}_0$, for the Milnor fiber (for $\mathcal{V}, 0$ a hypersurface), and complement and link of $\mathcal{V}_0$ (in the general case). The characteristic cohomology …
We introduce the notion of a prism, which may be regarded as a "deperfection" of the notion of a perfectoid ring. Using prisms, we attach a ringed site -- the prismatic site -- to a $p$-adic formal scheme. The resulting cohomology theory specializes to (and often refines) most known integral $p$-adic cohomology theories. As applications, we prove an improved version of the almost purity theorem allowing ramification along arbitrary closed subsets (without using adic spaces), give a co-ordinate free description of $q$-de Rham cohomology as conjectured by the second author, and settle a vanishin…
The purpose of this paper is to establish an injectivity theorem generalized to pseudo-effective line bundles with transcendental (non-algebraic) singular hermitian metrics and multiplier ideal sheaves. As an application, we obtain a Nadel type vanishing theorem. For the proof, we study the asymptotic behavior of the harmonic forms with respect to a family of regularized metrics, and give a method to obtain L2-estimates of solutions of the dbar-equation by using the de Rham-Weil isomorphism between the dbar-cohomology and the check{C}ech cohomology.
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