# Holomorphic vector fields with a non-degenerate isolated zero

Let $$v$$ be a holomorphic vector field defined in a neighbourhood of $$0$$ on $$\mathbb C^n$$ with an isolated zero at $$0$$. Let $$\sum_{i,j}{a_{ij}}z_i\frac{\partial}{\partial z_j}$$ be the linear term of $$v$$ and suppose that the matrix $$a_{ij}$$ is invertible and all its eigenvalues have modulus different from $$1$$. Is it true that for some holomorphic coordinates $$w_i$$ in a neighbourhood of $$0$$ we have $$v=\sum_{i,j}{a_{ij}}w_i\frac{\partial}{\partial w_j}$$?

If yes, where could I find such a statement? If not, what would be a counterexample? I am happy to assume that the eigenvalues of $$A$$ all have modulus less than $$1$$.

• If all have modulus less than 1, I know this from a paper of Kodaira about complex surfaces. Oct 21, 2020 at 15:16
• That's nice! What is the exact reference? You can leave it as an answer (and not a comment) if you wish Oct 21, 2020 at 15:41
• I think there can be resonances even in the case all eigenvalues are less than one in modulus. They can prevent linearization. Oct 23, 2020 at 13:08
• Arnaud, that's very interesting, if you can find an explicit example/reference, that would be great Oct 23, 2020 at 13:10

A resonance of a zero of a vector field is an equation $$\lambda_i=\sum_{j\ne i} m_j \lambda_j$$ among eigenvalues of the linearization of the vector field about the zero, where $$\{m_j\}$$ are integers, $$m_j\ge 0$$, and $$\sum_j m_j\ge 2$$.
• I am curious, in the case when all the eigenvalues of $A$ are smaller than $1$ in modulus, can it happen (say when there are resonants) that the mapping is not holomorphically conjugate to a linear one in a small neighbourhood of $0$? Oct 23, 2020 at 12:58