Geodesic
The Action of a Plane Singular Holomorphic Flow on a Non-invariant Branch
Canadian journal of mathematics, Tome 72 (2020) no. 4, pp. 835-866

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DOI

We study the dynamics of a singular holomorphic vector field at $(\mathbb{C}^{2},0)$. Using the associated flow and its pullback to the blow-up manifold, we provide invariants relating the vector field, a non-invariant analytic branch of curve, and the deformation of this branch by the flow. This leads us to study the conjugacy classes of singular branches under the action of holomorphic flows. In particular, we show that there exists an analytic class that is not complete, meaning that there are two elements of the class that are not analytically conjugated by a local biholomorphism embedded in a one-parameter flow. Our techniques are new and offer an approach dual to the one used classically to study singularities of holomorphic vector fields.
DOI : 10.4153/S0008414X19000221
Mots-clés : holomorphic vector field, deformation, moduli, plane branch 
Ayuso, P. Fortuny; Ribón, J. The Action of a Plane Singular Holomorphic Flow on a Non-invariant Branch. Canadian journal of mathematics, Tome 72 (2020) no. 4, pp. 835-866. doi: 10.4153/S0008414X19000221
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     title = {The {Action} of a {Plane} {Singular} {Holomorphic} {Flow} on a {Non-invariant} {Branch}},
     journal = {Canadian journal of mathematics},
     pages = {835--866},
     year = {2020},
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