Moscow Mathematical Journal

Volume 21, Issue 2, April–June 2021 pp. 365–382.

Spectra of Quadratic Vector Fields on $\mathbb{C}^2$: the Missing Relation

Authors: Yury Kudryashov (1) and Valente Ramírez (2)
Author institution:(1) University of Toronto Mississauga, 3359 Mississauga Road, Mississauga, ON, L5L 1C6
(2) University of Twente, Faculty of Electrical Engineering, Mathematics and Computer Science, Zilverling, P.O. Box 217, 7500 AE Enschede, The Netherlands

Summary:

Consider a quadratic vector field on $\mathbb{C}^2$ having an invariant line at infinity and isolated, non-degenerate singularities only. We define the extended spectra of singularities to be the collection of the spectra of the linearization matrices of each singular point over the affine part, together with all the characteristic numbers (i.e., Camacho–Sad indices) at infinity. This collection consists of 11 complex numbers, and is invariant under affine equivalence of vector fields.

In this paper we describe all polynomial relations among these numbers. There are 5 independent polynomial relations; four of them follow from the Euler–Jacobi, the Baum–Bott, and the Camacho–Sad index theorems, and are well-known. The fifth relation was, until now, completely unknown. We provide an explicit formula for the missing 5th relation, discuss it's meaning and prove that it cannot be formulated as an index theorem.

2020 Math. Subj. Class. 37F75, 32M25, 32S65.

Keywords: Quadratic vector fields, spectra of singularities, holomorphic foliations, index theorems.

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