Polar Cremona Transformations and Monodromy of Polynomials

多元阿基米德zeta函数的极轨迹

Polar loci of multivariable archimedean zeta functions

本文确定了多元阿基米德zeta函数的极轨迹为单项支持轨迹，建立了重数与极阶的关系，推广了Barlet的结果，并应用于Bernstein-Sato型理想。

Oblique poles of $\int_X| {f}| ^{2λ}| {g}|^{2μ} \square$

Existence of oblique polar lines for the meromorphic extension of the current valued function $\int |f|^{2λ}|g|^{2μ}\square$ is given under the following hypotheses: $f$ and $g$ are holomorphic function germs in $\CC^{n+1}$ such that $g$ is non-singular, the germ $S:=\ens{\d f\wedge \d g =0}$ is one dimensional, and $g|_S$ is proper and finite. The main tools we use are interaction of strata for $f$ (see \cite{B:91}), monodromy of the local system $H^{n-1}(u)$ on $S$ for a given eigenvalue $\exp(-2iπu)$ of the monodromy of $f$, and the monodromy of the cover $g|_S$. Two non-trivial examples ar…

Singularities of variations of mixed Hodge structure

We prove that a variation of graded-polarizable mixed Hodge structure over a punctured disk with unipotent monodromy, has a limiting mixed Hodge structure at the puncture (i.e., it is admissible in the sense of [SZ]) which splits over $\R$, if and only if certain grading of the complexified weight filtration, depending smoothly on the Hodge filtration, extends across the puncture. In particular, the result exactly supplements Schmid's Theorem for pure structures, which holds for the graded variation, and gives a Hodge-theoretic condition for the relative monodromy weight filtration to exist.

The global derived period map

Abstract. We develop the global period map in the context of derived geometry, generalising Griffiths' classical period map as well as the infinitesimal derived period map. We begin by constructing the derived period domain which classifies Hodge filtrations and enhances the classical period domain. We analyze the monodromy action. Then we associate to any polarized smooth projective map of derived stacks a canonical morphism of derived analytic stacks from the base into the quotient of the derived period domain by monodromy. We conclude the paper by discussing a few examples and a derived Tor…

Topology of the Milnor fibrations of polar weighted homogeneous polynomials

In the present paper, we study deformations of polar weighted homogeneous polynomials which are also polar weighted homogeneous polynomials. We describe a round handle decomposition of the Milnor fibration of a deformation of a polar weighted homogeneous polynomial concretely and give the number of round handles by the number of positive and negative components of the links of singularities appearing before and after the deformation. We also give a formula of characteristic polynomials of these singularities by using the decomposition of the monodromy of the Milnor fibration induced by a round…

Homaloidal determinants

A form in a polynomial ring over a field is said to be homaloidal if its polar map is a Cremona map, i.e., if the rational map defined by the partial derivatives of the form has an inverse rational map. The object of this work is the search for homaloidal polynomials that are the determinants of sufficiently structured matrices. We focus on generic catatalecticants, with special emphasis on the Hankel matrix. An additional focus is on certain degenerations or specializations thereof. In addition to studying the homaloidal nature of these determinants, one establishes several results on the ide…

有理映射的奇点：基础与曲面

Singularities of rational maps: foundations and surfaces

本文建立有理映射奇点理论，引入相关不变量，详细研究曲面情形并关联多理论，还提出高维等方向的未来研究问题。

关于对数微分系统的单值映射

On the monodromy map for the logarithmic differential systems

本文研究对数微分系统的单值映射，证明在特定亏格和维数条件下该映射为浸入，并给出非浸入条件，推广了非奇异情形的结果。

Hodge型(1,2,2,1)周期映射的完备化

Completion of a period map of Hodge type (1,2,2,1)

本文通过Kato-Usui空间理论，借助镜像对称计算单值矩阵，完成了Hodge型(1,2,2,1)二维几何族周期映射的完备化，为第二个高维非厄米对称Mumford-Tate域的例子。

子表示的单值性与低次自守伽罗瓦表示的不可约性

Monodromy of subrepresentations and irreducibility of low degree automorphic Galois representations

本文研究了数域上代数簇的étale基本群表示系统，证明了在温和局部条件下A型不可约子表示的剩余不可约性，并证明了全实或CM域上次数不超过6的正则代数极化尖点自守表示对应的Galois表示的不可约性与剩余不可约性。

On the Kodaira-Spencer map of abelian schemes

Let $A$ be an abelian scheme over a smooth affine complex variety $S$, $\varOmega_A$ the $\sO_S$-module of $1$-forms of the first kind on $A$, $\sD_S\varOmega_A$ the $\sD_S$-module spanned by $\varOmega_A$ in the first algebraic De Rham cohomology module, and $θ_\partial: \varOmega_A \to \sD_S\varOmega_A/\varOmega_A$ the Kodaira-Spencer map attached to a tangent vector field $\partial$ on $S$. We compare the rank of $\sD_S\varOmega_A/\varOmega_A$ to the maximal rank of $θ_\partial$ when $\partial$ varies: we show that both ranks do not change when one passes to the "modular case", \ie when one…

Newton数、消失多面体与代数次数

Newton numbers, vanishing polytopes and algebraic degrees

本文分类了具有消失Newton数的便利Newton多面体为$B_k$-多面体，部分解决了Arnold单调性问题，并推广Newton数至$\ell$-与$e$-Newton数，后者计算代数次数并刻画对偶缺陷集。