Vanishing of principal value integrals on surfaces

余维数为三的通用消失子概形

Generic vanishing subschemes of codimension three

本文分类了维数至少为6的不可分解主极化阿贝尔簇中余维数为三的几何非退化GV子概形，证明其为雅可比簇中平移的$\pm W_{g-3}(C)$，并利用theta对偶性计算了相关上同调及刻画超椭圆情形。

整全阿廷动机 II：反常动机与阿廷消失定理

Integral Artin motives II: Perverse motives and Artin Vanishing Theorem

本文研究整系数阿廷平展动机范畴上的反常动机t-结构，构造了近似的反常同伦t-结构，在基概型维数≤2时得到了反常动机t-结构，证明其在维数4时不存在，且依赖于阿廷消失定理的类似结论。

Tube classes over elementary vanishing cycles

Let $X$ be a closed Riemann surface. When $X$ is embedded into a projective space, the first rational cohomology group can be concretely obtained from the monodromy in the family of its smooth hyperplane sections by C. Schnell's tube mapping. We generalize this result to the first integral homology group by relating the tube mapping with the topological Abel--Jacobi mapping. By making use of the mapping class group action, we prove that all tube classes constructed from the elementary vanishing cycles form a cofinite subgroup of the first integral homology group of $X$.

An integral version of Zariski decompositions on normal surfaces

We show that any pseudo-effective divisor on a normal surface decomposes uniquely into its "integral positive" part and "integral negative" part, which is an integral analog of Zariski decompositions. By using this decomposition, we give three applications: a vanishing theorem of divisors on surfaces (a generalization of Kawamata-Viehweg and Miyaoka vanishing theorems), Reider-type theorems of adjoint linear systems on surfaces (including a log version and a relative version of the original one) and extension theorems of morphisms defined on curves on surfaces (generalizations of Serrano and P…

Uniformity of stably integral points on principally polarized abelian surfaces

We prove, assuming that the conjecture of Lang and Vojta holds true, that there is a uniform bound on the number of stably integral points in the complement of the theta divisor on a principally polarized abelian surface defined over a number field. This gives a uniform version, in the spirit of a result of Caporaso-Harris-Mazur, of an unconditional theorem of Faltings. We utilize recent results of Alexeev and Nakamura on complete moduli for quasi-abelian varieties. We expect that a thorough understanding of current work of Alexeev should give a more general result for abelian varieties of an …

An Assympotic Vanishing Theorem for Generic Unions of Multiple Points

In this revised form, the proof of the principal lemma has been simplified and the main theorem has been extended to all characteristics for those varieties which are smooth in codimension one. This principal theorem essentially says the following : given an ample line bundle O(1) on a projective variety X and a fixed upper bound M on the multiplicities, there exists a lower bound D such that any generic union of multiple points of multiplicity at most M imposes independent conditions on the sections of O(d) for d>D. Here a multiple point is the closed subscheme defined by a power m of the ide…

Vanishing of Degree 3 Cohomological Invariants

For a complex algebraic variety $X$, we show that triviality of the sheaf cohomology group $H^0(X,\mathcal{H}^3)$ occurring on the second page of the Bloch-Ogus spectral sequence follows from a condition on the integral Chow group $CH^2X$ and the integral cohomology group $H^3(X, Z)$. In the case that $X$ is an appropriate approximation to the classifying stack $BG$ of a finite $p$-group $G$, this result states that the group $G$ has no degree three cohomological invariants. As a corollary we show that the nonabelian groups of order $p^3$ for odd prime $p$ have no degree three cohomological in…

Moments on Riemann surfaces and hyperelliptic Abelian integrals

In the present paper we solve the following different but interrelated problems: (a) the moment problem on Riemann surfaces, (b) the vanishing problem of polynomial Abelian integrals of dimension zero on the projective plane, (c) the vanishing problem of polynomial hyperelliptic Abelian integrals.

On the vanishing of local cohomology of the absolute integral closure in positive characteristic

The aim of this paper is to extend the main result of C. Huneke and G. Lyubeznik in [Adv. Math. 210 (2007), 498--504] to the class of rings that are images of Cohen-Macaulay local rings. Namely, let $R$ be a local Noetherian domain of positive characteristic that is an image of a Cohen-Macaulay local ring. We prove that all local cohomology of $R$ (below the dimension) maps to zero in a finite extension of the ring. As a direct consequence we obtain that the absolute integral closure of $R$ is a big Cohen-Macaulay algebra. Since every excellent local ring is an image of a Cohen-Macaulay local …

关于ELSV型公式及通过谱曲线形变得到的$\Omega$-积分之间的关系

On ELSV-type formulae and relations between $Ω$-integrals via deformations of spectral curves

本文通过结合拓扑递归、形变技术及多项式性质，推导出涉及$\Omega$-类的积分消失关系，并应用于三个谱曲线族，得到与亏格无关及依赖亏格的两类新关系。

主朗道行列式

Principal Landau Determinants

本研究为推进粒子物理计算，结合多面体几何等工具提出计算朗道奇点的新算法，定义费曼图的主朗道行列式，适配GKZ框架，完成多类图与标准模型过程的计算，实现为开源包PLD.jl。

关于积分渐近展开中的对数项

On the logarithmic terms in the asymptotic expansion of integrals

研究非退化牛顿函数定义的孤立奇点Milnor纤维上积分的渐近展开中的对数项，确定主导对数项的条件。

On the Topology of Foliations with a First Integral

The main objective of this article is to study the topology of the fibers of a generic rational function of the type $F^p/G^q$ in the projective space of dimension two. We will prove that the action of the monodromy group on a single Lefschetz vanishing cycle $δ$ generates the first homology group of a generic fiber of $F^p/G^q$. In particular, we will prove that for any two Lefschetz vanishing cycles $δ_0$ and $δ_1$ in a regular compact fiber of $F^p/G^q$, there exists a monodromy $h$ such that $h(δ_0)=\pm δ_1$.