This paper deals with holomorphic self-maps of the complex projective plane and the algebraic relations among the eigenvalues of the derivatives at the fixed points. These eigenvalues are constrained by certain index theorems such as the holomorphic Lefschetz fixed-point theorem. A simple dimensional argument suggests there must exist even more algebraic relations that the ones currently known. In this work we analyze the case of quadratic self-maps having an invariant line and obtain all such relations. We also prove that a generic quadratic self-map with an invariant line is completely determined, up to linear equivalence, by the collection of these eigenvalues. Under the natural correspondence between quadratic rational maps of \({\mathbb {P}}^2\) and quadratic homogeneous vector fields on \({\mathbb {C}}^3\), the algebraic relations among multipliers translate to algebraic relations among the Kowalevski exponents of a vector field. As an application of our results, we describe the sets of integers that appear as the Kowalevski exponents of a class of quadratic homogeneous vector fields on \({\mathbb {C}}^3\) having exclusively single-valued solutions.

中文翻译：

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具有不变线的射影平面二次自映射的不动点乘数

本文研究了复投影平面的全纯自映射以及不动点导数特征值之间的代数关系。这些特征值受到某些指数定理的约束，例如全纯的Lefschetz不动点定理。一个简单的维度论证表明必须存在比当前已知的更多的代数关系。在这项工作中，我们分析了具有不变线的二次自映射的情况，并获得了所有这样的关系。我们还证明，通过收集这些特征值，可以完全确定线性不变的通用二次自映射，直到线性对等为止。在\（{\ mathbb {P}} ^ 2 \）的二次有理图和二次齐次矢量场 之间的自然对应关系下 \（{{mathbb {C}} ^ 3 \），乘法器之间的代数关系转化为矢量场的Kowalevski指数之间的代数关系。作为结果的应用，我们描述了整数集，这些整数集以\（{\ mathbb {C}} ^ 3 \）上具有唯一单值解的一类二次齐次矢量场的Kowalevski指数形式出现 。