Kirby Problem 2.20

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $S_g^b$ be a compact oriented genus-$g$ surface with $b$ boundary components, and $\operatorname{Mod}(S_g^b)$ its orientation-preserving mapping class group fixing boundary pointwise. Write $t_c$ for the right-handed Dehn twist about a simple closed curve $c$ and $[a,b]=aba^{-1}b^{-1}$. Monodromy of a genus-$g$ Lefschetz fibration over a genus-$h$ surface is encoded by a relation \[t_{c_1}\cdots t_{c_\ell}[a_1,b_1]\cdots[a_h,b_h]=1\] in $\operatorname{Mod}(S_g)$, with positive twists about essential vanishing cycles at the critical values. Boundary-marked lifts can have the product of boundary twists on the right; for a pencil these boundary components encode its base points and the base genus is $h=0$. A surface bundle is the case $\ell=0$. A Lefschetz fibration has isolated critical points locally modeled on $(z_1,z_2)\mapsto z_1z_2$; a pencil is defined away from finitely many base points with local ratio model $z_1/z_2$. Holomorphicity here requires integrable complex structures on the total space and base, making the fibration or bundle holomorphic and the pencil meromorphic with these base points. An almost-complex structure alone does not suffice.

Statement recorded in the supplied source.

Determine properties of the monodromy factorization that detect whether the represented Lefschetz pencil, Lefschetz fibration, or surface bundle over a surface admits such a holomorphic realization. In particular, characterize the extra restrictions satisfied by monodromy factorizations of holomorphic fibrations, for all three classes of maps. [S2]

Short English statement

Find conditions on monodromy factorizations that distinguish integrable holomorphic pencils, fibrations and surface bundles from their more general topological or symplectic counterparts.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2768 (KP-2.20). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, eds., K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295, American Mathematical Society (2026), Problem 2.20 and its remarks; Chapters 1 and 2 for the stated conventions. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.