Versal deformation of transversely holomorphic flows on the boundary of strongly convex domains of $\mathbb C^n$

TL;DR

The paper studies versal deformation of transversely holomorphic foliations on the boundary of a bounded strongly convex domain in $\mathbb C^n$, given by restricting the orbit foliation of a holomorphic vector field $\xi$ to the boundary. By reducing to the unit ball model using Brunella and Ito’s results and employing Haefliger’s deformation theory, it frames deformations via the Kodaira–Spencer map on the resonant vector-field space $\mathcal g_\lambda$. The main result constructs a versal deformation parameterized by a small subset $S$ of $\mathcal g_\lambda$ complementary to the span of $\Phi_*\xi$ and $L_{\Phi_*\xi}(\mathcal g_\lambda)$, with $\mathcal F_0^S=(\mathcal F_0(\xi+\Phi^*s))_{s\in S}$ capturing all germ deformations. This extends Haefliger’s sphere-based framework to general strongly convex domains with smooth boundary and removes restrictive singularity assumptions on $\xi$, yielding a robust tool for boundary foliations in several complex variables.

Abstract

In this article, we give a versal deformation for any transversely holomorphic foliation $\mathcal{F}_0$ given by the intersection of the orbits of a holomorphic vector field $ξ$ defined on a neighborhood of the closure of a bounded strongly convex open domain $Ω\subset\mathbb C^n$ ($n\geq2$) with smooth boundary, with its boundary $\partial Ω$. That is, any germ of deformation of $\mathcal{F}_0$ is also obtained by intersecting the orbits of a deformation of $ξ$ with the boundary of $Ω$.

Theorems & Definitions (62)

• Theorem A
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• Corollary 2.5.1
• Proposition 2.6
• proof
• Lemma 2.7