Parameter Constraints and Real Structures in Quadratic Semicomplete Vector Fields on C3

It is a remarkable fact that among the known examples of quadratic semicomplete vector fields on C3, it is always possible to find linear coordinates where the corresponding vector field has all—or “almost all†—coefficients in the real numbers. Indeed, the coefficients are very often integral. The space of quadratic vector fields on C3, up to linear equivalence, is a complex 9-dimensional family. The main result of this work establishes that the degree of freedom in determining the coefficients of a semicomplete vector field (under very mild generic assumptions) is at most 3. In other words, there are 3 parameters from which all remaining parameters are determined. Moreover, if these 3 parameters are real, then so is the vector field. We start by considering a generic quadratic vector field Z on Cn that is homogeneous and is not a multiple of the radial vector field. The first step in our work will be to construct a canonical form for the induced vector field X on CPn−1. This canonical form will be invariant under the action of a specific group of symmetries. When n=3, we then push further our approach by studying the singularities not lying on the exceptional divisor but at the hyperplane at infinity Δ≅CP2. In this setting, the dynamics of the foliation turn out to be quite simple while the singularities tend to be degenerated. The advantage is that we can deal with degenerated singularities with the technique of successive blowups. This leads to simple expressions for the eigenvalues directly in terms of the coefficients of X.