EQUIVALENCE OF NEIGHBORHOODS OF EMBEDDED COMPACT COMPLEX MANIFOLDS AND HIGHER CODIMENSION FOLIATIONS

Xianghong Gong (1) , Laurent Stolovitch (2)

1 Department of Mathematics [Madison]
2 LJAD - Laboratoire Jean Alexandre Dieudonné

We consider an embedded $n$-dimensional compact complex manifold in $n+d$ dimensional complex manifolds. We are interested in the holomorphic classification of neighborhoods as part of Grauert's formal principle program. We will give conditions ensuring that a neighborhood of $C_n$ in $M_{n+d}$ is biholomorphic to a neighborhood of the zero section of its normal bundle. This extends Arnold's result about neighborhoods of a complex torus in a surface. We also prove the existence of a holomorphic foliation in $M_{n+d }$ having $C_n$ as a compact leaf, extending Ueda's theory to the high codimension case. Both problems appear as a kind linearization problem involving small divisors condition arising from solutions to their cohomological equations.

Domaines: Systèmes dynamiques [math.DS]; Variables complexes [math.CV]

Xianghong Gong, Laurent Stolovitch. EQUIVALENCE OF NEIGHBORHOODS OF EMBEDDED COMPACT COMPLEX MANIFOLDS AND HIGHER CODIMENSION FOLIATIONS. Arnold Mathematical Journal, 2022, 8, pp.61-145. ⟨10.1007/s40598-021-00192-w⟩. ⟨hal-02895577⟩