DCDS
On the integrability of holomorphic vector fields
Leonardo Câmara Bruno Scárdua
We determine topological and algebraic conditions for a germ of holomorphic foliation $\mathcal{F}_X$ induced by a generic vector field $X$ on $(\mathbb{C}^{3},0)$ to have a holomorphic first integral, i.e., a germ of holomorphic map $F$ : $(\mathbb{C}^{3},0)\rightarrow(\mathbb{C}^{2},0)$ such that the leaves of $\mathcal{F}_X$ are contained in the level curves of $F$.
keywords: maps tangent to the identity. holonomy groups First integrals
DCDS
Holomorphic foliations transverse to manifolds with corners
Toshikazu Ito Bruno Scárdua
We study the geometrical and dynamical properties of a holomorphic vector field on a complex surface, assumed to be transverse to the boundary of a domain which is a non-smooth manifold with boundary and corners. We obtain hyperbolicity and prove a compact leaf result. For a pseudoconvex domain with boundary diffeomorphic to the boundary of a bidisc in $\mathbb C^2$ the foliation is pull-back of a liner hyperbolic foliation. If moreover the diffeomorphism is transversely holomorphic then we have linearization.
keywords: Holomorphic foliation manifold with corners hyperbolic holonomy.

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