Unipotent Jacobian Matrices and Univalent Maps

A variant of Neron models over curves

We study a variant of the Neron models over curves which is recently found by the second named author in a more general situation using the theory of Hodge modules. We show that its identity component is a certain open subset of an iterated blow-up along smooth centers of the Zucker extension of the family of intermediate Jacobians and that the total space is a complex Lie group over the base curve and is Hausdorff as a topological space. In the unipotent monodromy case, the image of the map to the Clemens extension coincides with the Neron model defined by Green, Griffiths and Kerr. In the ca…

The Jacobian map, the Jacobian group and the group of automorphisms of the Grassmann algebra

There are nontrivial dualities and parallels between polynomial algebras and the Grassmann algebras. This paper is an attempt to look at the Grassmann algebras at the angle of the Jacobian conjecture for polynomial algebras (which is the question/conjecture about the $ $ {\em Jacobian set} -- the set of all algebra endomorphisms of a polynomial algebra with the Jacobian 1 -- the Jacobian conjecture claims that the Jacobian set is a {\em group}). In this paper, we study in detail the Jacobian set for the Grassmann algebra which turns out to be a {\em group} -- the {\em Jacobian group} $Σ$ -- a …

\(\mathbb{C}^5\)中雅可比映射全局单射性的一个反例及其解析根

A Counterexample to the Global Injectivity of a Jacobian Mapping in \mathbb{C}^5 and Its Analytical Roots

研究由特定6次齐次多项式诱导的\(\mathbb{C}^5\)到\(\mathbb{C}^5\)的多项式映射\(G\)，证明其雅可比矩阵幂幺但映射非全局单射，构造出\(G(a)=G(b)=\vec{0}\)的点对，还对梯度场零集结构分类得到37个解析复解。