On the Multipliers at Fixed Points of Quadratic Self-Maps of the Projective Plane with an Invariant Line

Abstract

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.

Appendix A: A Relative Fixed-Point Formula

We give here an elementary proof of relation (6) following the lines of the proof of [31, Sect. 12, Thm. 12.4]. A more conceptual one, based on the Atiyah-Bott fixed-point theorem, appears in the unpublished notes [34].

Theorem 6

Let $f:{\mathbb {P}}^2\rightarrow {\mathbb {P}}^2$ be a rational map, $\ell \subset {\mathbb {P}}^2$ a line such that $f(\ell )\subset \ell$. Suppose that $f|_\ell$ has the fixed points $p_1$, ..., $p_{k}$ and that these are simple. Let $\lambda _i$ and $\mu _i$ be be the eigenvalues of Df at $p_i$, with $\lambda _i$ the eigenvalue tangent to $\ell$. Then,

$$\begin{aligned} \sum _{i=1}^{k} \frac{1-\mu _i}{1-\lambda _i}=1. \end{aligned}$$

Proof

In homogeneous coordinates $[z_1:z_2:z_3]$, let $\ell$ be given by $z_1=0$ and f be given by

$$\begin{aligned}{}[z_1:z_2:z_3]\mapsto [z_1{\widehat{P}}(z_1,z_2,z_3):{\widehat{Q}}(z_1,z_2,z_3):{\widehat{R}}(z_1,z_2,z_3)], \end{aligned}$$

for ${\widehat{Q}}$ and ${\widehat{R}}$ homogeneous polynomials of degree d and ${\widehat{P}}$ a homogeneous polynomial of degree $d-1$. Consider the affine chart $z_3\ne 0$ and suppose, without loss of generality, that it contains the $d+1$ fixed points of $f|_\ell$. In the coordinates [x : y : 1], f is given by

$$\begin{aligned} (x,y)\mapsto \left( \frac{xP(x,y)}{R(x,y)},\frac{Q(x,y)}{R(x,y)}\right) , \end{aligned}$$

where $P(x,y)={\widehat{P}}(x,y,1)$ and so on. Let $y_1$, ..., $y_{d+1}$ be the fixed points of f within $\ell$. Consider the rational one-form in $\ell$

$$\begin{aligned} \eta =\frac{1-\frac{P}{R}(0,y)}{y-\frac{Q}{R}(0,y)}dy. \end{aligned}$$

Since all the fixed points of $f|_\ell$ are within the chosen affine chart, $R(0,y)=cy^d+\cdots$ for some $c\ne 0$. This implies that $\lim _{y\rightarrow \infty } P(0,y)/R(0,y)=0$ and that $\lim _{y\rightarrow \infty } Q(0,y)/R(0,y)\in {\mathbb {C}}$. Hence, $\eta$ has a pole at $\infty$ with residue $-1$. The other poles of $\eta$ are given by the fixed points of $f|_\ell$. Let us calculate the corresponding residues. On the one hand, $y-Q(0,y)/R(0,y)$, which gives the restriction of f to $\ell$, has, for its Taylor development at $y_i$,

$$\begin{aligned} y_i+(y-y_i)-(y_i+f|_\ell '(y_i)(y-y_i)+\cdots )=(1-\lambda _i)(y-y_i)+\cdots . \end{aligned}$$

On the other, a straightforward computation shows that

$$\begin{aligned} \mu _i=\left. \frac{\partial }{\partial x} \left( x\frac{P}{R}\right) \right| _{(0,y_i)}=\frac{P(0,y_i)}{R(0,y_i)}. \end{aligned}$$

The Residue Theorem allows us to conclude.$\square$

Remark 3

It is worthwhile mentioning that there is no analogous result for other kinds of curves. Consider the following five-parameter family of quadratic maps of ${\mathbb {P}}^2$:

$$\begin{aligned}&[z_1:z_2:z_3]\mapsto [(a-1)^2a^2z_1^2+(a^2-c)a^2z_2^2-2(a-1)a^3z_1z_2+ca^2z_3z_1:\\&\quad (a^2-2ba+b-da)az_2^2-(a-1)(a-b)az_1z_2+ba^2z_2z_3+da^2z_3z_1: \\&\quad b^2z_3^2+([a-b]^2-ea^2)z_2^2+2b(a-b)z_2z_3+ea^2z_3z_1]. \end{aligned}$$

Every element f of this family preserves the conic Q given by $z_1z_3-z_2^2=0$. A generic one is holomorphic and has seven different fixed points. The three fixed points of f on Q are $p_0=[1:0:0]$, $p_1=[1:1:1]$ and $p_\infty =[0:0:1]$. A straightforward calculation shows that the map that to each f associates the set of values

$$\begin{aligned} (\mathrm {tr}(Df|_{p_0}),\mathrm {tr}(Df|_{p_1}),\mathrm {tr}(Df|_{p_\infty }),\det (Df|_{p_0}),\det (Df|_{p_1}),\det (Df|_{p_\infty })) \end{aligned}$$

has rank five at a generic point, this is, there is only one relation for the eigenvalues of the derivatives of f of the fixed points in Q. This relation is necessarily Fatou’s one (1) for $f|_Q$, and no other relation holds.