Bounded Leaf Volumes and Analytic Quotients of Singular Holomorphic Foliations

Abstract

Let $\mathcal{F}$ be a singular holomorphic foliation of dimension $k$ on a compact Kähler manifold $M$ of dimension $n$. Suppose that the volumes of its regular leaves are uniformly bounded, $\dim(Sing\mathcal{F})<k$, and the union of the leaf closures meeting $Sing\mathcal{F}$ has finite $2k$-dimensional Hausdorff measure. We prove that the quotient by chains of intersecting leaf closures is a normal compact complex analytic space of dimension $n-k$. The quotient map is proper and holomorphic, with connected fibers. The proof uses Bishop's theorems, Remmert and Stein extension, Barlet's cycle space, and Stein factorization. Away from the images of separatrices, the quotient has local charts given by finite holonomy groups. Examples show that these images need not coincide with the singular set of the quotient, and that bounded leaf volumes alone do not imply Hausdorffness. In codimension one, Malgrange's theorem makes the separatrix measure assumption automatic when $codim(Sing\mathcal{F})\geq 3$.