Vanishing of leafwise Dolbeault cohomology for complex foliations with closed Stein leaves

Determine whether, for a complex foliation (M, F) in which every leaf is a Stein manifold and is closed in M, the leafwise Dolbeault cohomology groups H_F^{0,q}(M) vanish for all q ≥ 1.

Background

In Section 5, the authors reformulate the Morse-Novikov-Treves complex for Levi flat structures and specialize to complex foliations, leading to a leafwise Morse-Novikov-Dolbeault complex. They prove vanishing results under additional convexity assumptions on an exhaustion function (Corollary \ref{vlmnc2}).

They then relate their result to a classical problem in foliation theory: whether leafwise Dolbeault cohomology vanishes when the leaves are Stein and closed in the ambient manifold. This question, attributed to El Kacimi Alaoui, remains a benchmark for understanding global leafwise cohomology under strong local holomorphic convexity along leaves.

The quoted passage explicitly frames the question as an open one and states it precisely, positioning the paper’s results as related but under stronger geometric hypotheses.