Problem 13 — (Kernel foliation).

Definitions and mathematical statement

For a zero-order partition $\alpha'=(\alpha'_1,\ldots,\alpha'_n)$, the abelian-differential stratum $\mathcal H(\alpha')$ consists of translation surfaces $(X,\omega)$ with the specified zeros of $\omega$, with $\sum_i\alpha'_i=2g-2$. In local period coordinates, periods lie in $H^1(X,Z;{\mathbb{C}})$, where $Z$ is the zero set. The kernel, or absolute-period, foliation fixes the image in $H^1(X;{\mathbb{C}})$ while relative periods vary. Its local relative directions have complex dimension $n-1$ in the usual labeled-zero setting.

Problem 13 (kernel foliation). For the space $N$ constructed in the omitted preceding discussion, ask:

1. Is $N$ a complex-analytic orbifold? Is it a complex-algebraic orbifold?
2. Under what conditions does \[ \dim_{{\mathbb{C}}}N=\dim_{{\mathbb{C}}}O+n-m \] hold?
3. Under what conditions does $N$ coincide with the entire connected component of its enveloping stratum $\mathcal H(\alpha'_1,\ldots,\alpha'_n)$?

The extracted record does not define $N$, $O$, or $m$, or state whether closures or kernel-leaf saturations are used. Although $n$ indexes the zeros of the target stratum, the role of $m$ cannot be certified. These missing constructions are indispensable: the displayed dimension formula is a proposed relation for the source's objects, not a general identity for arbitrary subsets or for every kernel-foliation leaf. In particular, the local relative dimension $n-1$ does not by itself justify the formula with $n-m$.

The source cites McMullen [McM3] for classification of $\mathrm{GL}_2({\mathbb{R}})$ orbit closures in $\mathcal H(1,1)$ and $\mathcal H(2)$, using homologous saddle connections. A saddle connection is a straight flat segment between zeros with no zero in its interior. It describes cutting a surface in $\mathcal H(1,1)$ along a pair joining the distinct zeros to obtain a two-torus description and use Ratner's theorem. This genus-two methodological context does not define the missing $N$ and $O$ or settle the three stated questions.

Short English statement

For the source's kernel-foliation construction, determine when its resulting space is analytic or algebraic, when the stated dimension formula holds, and when it fills a connected stratum component. The construction of the spaces and the parameter $m$ is missing.

Sources

• Local UnsolvedMath registry, permanent ID 11000220, source code catalogue entry; no proposer or proposal year supplied.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics, Problem 13, PDF page 261; https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. The registry records PDF-font-structure extraction and access on 2026-07-29; the inventory distinguishes 329 labeled headings by source page.
• McMullen [McM3] and Ratner's theorem are named in the source, without complete bibliography entries or exact theorem hypotheses in this record.