Ruochuan Liu
This paper concerns arithmetic families of $φ$-modules over reduced affinoid spaces. For such a family, we first prove that the slope polygons is lower semicontinuous around any rigid point. If the slope polygons are locally constant around a rigid point, we further prove that around this point, the family has a global slope filtration after base change to some extended Robba ring.
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Many slope filtrations occur in algebraic geometry, asymptotic analysis, ramification theory, p-adic theories, geometry of numbers... These functorial filtrations, which are indexed by rational (or sometimes real) numbers, have a lot of common properties. We propose a unified abstract treatment of slope filtrations, and survey how new ties between different domains have been woven by dint of deep correspondences between different concrete slope filtrations.
In this book we construct fundamental filtered complexes in filtered derived categories. These are generalizations of a filtered complex constructed by Mokrane. As an application of these constructions, we develop theory of limits of weight filtrations and limits of slope filtrations on the infinitesimal cohomologies of proper smooth schemes over the fraction field of a complete discrete valuation ring of mixed characteristics with perfect residue field.
Slope filtrations of log $p$-divisible groups
本文研究了 log p-可除群的斜过滤，推广了 Oort-Zink 关于 p-可除群的同源性结果。
Weight filtrations via slopes
本文在André的斜率形式化框架下，提出了一个抽象框架来解释混合结构为何具有共同特征，特别是其权重过滤。
The notion of Harder-Narasimhan filtration was firstly introduced by Harder and Narasimhan in the setting of vector bundles on a non-singular projective curve. Curiously, analogous constructions have been discovered in other branches of mathematics which motivate categorical constructions of Harder-Narasimhan filtration. In this article, we introduce a categorical construction of Harder-Narasimhan filtration via slope function method which does not need the additive condition in degree function. We also give a method to prove the existence and uniqueness theorem of the Harder-Narasimhan filtra…
We apply the theory of families of (phi,Gamma)-modules to trianguline families as defined by Chenevier. This yields a new definition of Kisin's finite slope subspace as well as higher dimensional analogues. Especially we show that these finite slope spaces contain eigenvarieties for unitary groups as closed subspaces. This implies that the representations arising from overconvergent p-adic automorphic forms on certain unitary groups are trianguline when restricted to the local Galois group.
In this article we are interested in morphisms without slope for mixed Hodge modules. We first show the commutativity of iterated nearby cycles and vanishing cycles applied to a mixed Hodge module in the case of a morphism without slope. Then we define the notion "strictly without slope" for a mixed Hodge module and we show the preservation of this condition under the direct image by a proper morphism. As an application we prove the compatibility of the Hodge filtration and Kashiwara-Malgrange filtrations for some pure Hodge modules with support an hypersurface with quasi-ordinary singularitie…
A p-divisible group over a base scheme in characteristic p in general does not admit a slope filtration. Let X be a p-divisible group with constant Newton polygon over a normal noetherian scheme S; we prove that there exists an isogeny from X to Y such that Y admits a slope filtration. In case S is regular this was proved by N. Katz for dim(S) = 1 and by T. Zink for dim(S) > 0. We give an example of a p-divisible group over a non-normal base which does not admit an isogeny to a p-divisible group with a slope filtration.
We give an expression for the Łojasiewicz exponent of a wide class of n-tuples of ideals $(I_1,..., I_n)$ in $Ø_n$ using the information given by a fixed Newton filtration. In order to obtain this expression we consider a reformulation of Łojasiewicz exponents in terms of Rees mixed multiplicities. As a consequence, we obtain a wide class of semi-weighted homogeneous functions $(\mathbb{C}^n,0)\to (\mathbb{C},0)$ for which the Łojasiewicz of its gradient map $\nabla f$ attains the maximum possible value.
Twisted Triple Product $p$-adic $L$-function for Finite Slope Families of Hilbert Modular Forms
本文构造了Hilbert模簇上相对de Rham上同调类的过收敛族，证明其Gauss-Manin联络的$p$-adic迭代收敛性，并据此构造了Hilbert模形式有限斜率族的$p$-adic扭曲三重积$L$-函数。
A $p$-adic Gross-Zagier formula for twisted triple product $p$-adic $L$-functions attached to finite slope families
本文推广了关于扭曲三重积 $p$-进 $L$-函数的 $p$-进 Gross-Zagier 公式，将其扩展至 Hilbert 模形式的有限斜率族，并允许素数 $p$ 在实二次域中为惰性。
Harder-Narasimhan filtration of torsion F-gauges
本文在正特征完美域上建立了挠 $F$-规范丛的 Harder-Narasimhan 理论，完全分类了其稳定对象，并纠正了 Ekedahl 的分类，同时将其提升为 Bridgeland 稳定性条件。
We prove that the perverse Leray filtration for the Hitchin morphism is locally constant in families, thus providing some evidence towards the validity of the $P=W$ conjecture due to de Cataldo, Hausel and Migliorini in non Abelian Hodge theory.
Characteristic cycles and the Gevrey filtration of irregularity in hypergeometric families
该研究证明了A-超几何系统特征标圈重数的上半连续性，证实了Schulze与Walther的猜想，还构造了特定半群环揭示Gevrey跳跃轨迹的性质。
We define a pro-$p$ Abelian sheaf on a modular curve of a fixed level $N \geq 5$ divisible by a prime number $p \neq 2$. Every $p$-adic representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ associated to an eigenform is obtained as a quotient of its étale cohomology. For any compact $\mathbb{Z}_p[[1 + N \mathbb{Z}_p]]$-algebra $Λ_1$ satisfying certain suitable conditions, we construct a representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ over $Λ_1$ associated to a $Λ_1$-adic cuspidal eigenform of finite slope as a scalar extension of a quotient of the étale cohomology.
In this paper, we investigate the general notion of the slope for families of curves $f: X \to Y$. The main result is an answer to the above question when $\dim Y = 2$, and we prove a lower bound for this new slope in this case over fields of any characteristic. Both the notion and the slope inequality are compatible with the theory for $\dim Y = 0, 1$ in a very natural way, and this gives a strong evidence that the slope for an $n$-fold fibration of curves $f: X \to Y$ may be $K_{X/Y}^n / \mathrm{ch}_{n-1}(f_* ω_{X/Y})$. Rather than the usual stability methods, the whole proof of the slope in…
We establish sharp bounds for the slopes of curves in $\bar{M}_g$ that sweep the locus of trigonal curves, proving Stankova-Frenkel's conjectured bound of $7+6/g$ for even $g$ and obtaining the bound $7+20/(3g+1)$ for odd $g$. For even $g$, we find an explicit expression of the so-called Maroni divisor in the Picard group of the space of admissible triple covers. For odd $g$, we describe the analogous extremal effective divisor and give a similar explicit expression.
On families of elliptic curves $E_{p,q}:y^2=x^3-pqx$ that intersect the same line $L_{a,b}:y=\frac{a}{b}x$ of rational slope
本文研究了椭圆曲线族 $E_{p,q}$ 与有理斜率直线 $L_{a,b}$ 相交的充分条件。
Let $S\to C$ be a smooth projective surface with numerically trivial canonical bundle fibered onto a curve. We prove the multiplicativity of the perverse filtration with respect to the cup product on $H^*(S^{[n]},\mathbb{Q})$ for the natural morphism $S^{[n]}\to C^{(n)}$. We also prove the multiplicativity for five families of Hitchin systems obtained in a similar way and compute the perverse numbers of the Hitchin moduli spaces. We show that for small values of $n$ the perverse numbers match the predictions of the numerical version of the de Cataldo-Hausel-Migliorini $P=W$ conjecture and of t…
Slope inequalities for KSB-stable and K-stable families
本文证明经典斜率不等式的高维推广，并借助 Harder–Narasimhan 滤过及 Castelnuovo、Noether 不等式，将其用于 KSB/K 稳定对族与模空间丰富锥研究。
We introduce several families of filtrations on the space of vector bundles over a smooth projective variety. These filtrations are defined using the large k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted by the kth power of an ample line bundle. The filtrations measure the failure of the bundle to admit a holomorphic structure. We study compatibility under the Chern isomorphism of these filtrations with the Hodge filtration on cohomology.
Let p be an odd prime and g an integer greater or equal to 2. We prove that a finite slope Siegel cuspidal eigenform of genus g can be p-adically deformed over the g-dimensional weight space. The proof of this result relies on the construction of a family of sheaves of locally analytic overconvergent modular forms.
New slope inequalities for families of complete intersections
本文证明完全交纤维化中 $\mathcal{O}_X(1)$ 的 $f$-正性，给出整体完全交情形下幂次与相对典范层 $f$-正性的数值充要条件，并导出 Chow 不稳定条件。
Semistability and asymptotics of geometric flows
本文证明在斜率半稳定全纯向量丛上，Hermitian-Yang-Mills流的渐近行为由Haiden-Katzarkov-Kontsevich-Pandit迭代滤过代数几何地决定，并推广到Calabi流和矩映射流，将无限维流简化为有限维流。
Codimension of jumping loci
本文研究光滑射影簇上向量丛限制到曲线族时Harder-Narasimhan过滤的变化，确定了斜率跳跃轨迹余维数线性依赖跳跃量的几何条件，并应用于研究射影平面秩2丛及曲线模空间奇点轨迹。
Consider a smooth, projective family of canonically polarized varieties over a smooth, quasi-projective base manifold Y, all defined over the complex numbers. It has been conjectured that the family is necessarily isotrivial if Y is special in the sense of Campana. We prove the conjecture when Y is a surface or threefold. The proof uses sheaves of symmetric differentials associated to fractional boundary divisors on log canonical spaces, as introduced by Campana in his theory of Orbifoldes Geometriques. We discuss a weak variant of the Harder-Narasimhan Filtration and prove a version of the Bo…
For a family of principal bundles with a reductive structure group on a family of curves in characteristic zero, it is known that the Harder Narasimhan type of its restriction to each fiber varies semicontinuously over the parameter scheme of the family. This defines a stratification of the parameter scheme by locally closed subsets, known as the Harder-Narasimhan stratification. In this note, we show how to endow each Harder-Narasimhan stratum with the structure of a locally closed subscheme of the parameter scheme, which enjoys the universal property that under any base change the pullback f…
For any family of principal bundles with a reductive structure group G on a family X/S of smooth projective varieties in characteristic zero, it is known that the parameter scheme S has a set theoretic stratification by locally closed subsets which correspond to the Harder-Narasimhan types of the restriction of the principal bundle to the various fibers of X/S. We show that each of these subsets has in fact the structure of a locally closed subscheme of the parameter scheme S, with the following universal property: Under any base change, the pullback family admits a relative Harder-Narasimhan …
Duality for asymptotic invariants of graded families
本文研究了渐进族的对偶性，通过交换子加性和超加性序列并反转其渐近增长常数，揭示了其在代数几何中的应用，包括Macaulay-Matlis对偶性和jet分离序列的对偶关系。
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