Multiple Kodaira Fiberings and Relative Irregularity

Definitions and mathematical statement

Let $X$ be a compact connected complex surface. Under the usual convention, a Kodaira fibration is a surjective holomorphic submersion $f\colon X\to B$ with connected fibers, where $B$ is a compact connected Riemann surface of genus at least $2$, the fibers are compact Riemann surfaces of genus at least $2$, and the family is nonisotrivial. Here nonisotrivial means that the smooth fibers are not all mutually biholomorphic.

1. Does there exist a compact connected complex surface $X$ admitting at least three distinct Kodaira fibrations \[ f_i\colon X\longrightarrow B_i,\qquad i=1,2,3? \] For a precise conventional reading, two fibrations $f_i\colon X\to B_i$ and $f_j\colon X\to B_j$ define the same fibering if there is a biholomorphism $\phi\colon B_i\to B_j$ with $f_j=\phi\circ f_i$; the question then requires three pairwise inequivalent fibrations. The supplied record does not specify its equivalence convention, including whether automorphisms of $X$ are also to be allowed. Thus this interpretation of distinctness requires source confirmation.

2. Let $X$ now be a compact connected Kähler surface and let $f\colon X\to B$ be a nonisotrivial surjective holomorphic map with connected fibers onto a compact connected Riemann surface. Let $F$ be a smooth general fiber and write \[ g=g(F),\qquad q(X)=\dim_{{\mathbb{C}}}H^1(X,\mathcal O_X),\qquad q_f=q(X)-g(B). \] Can one obtain improved upper bounds for $q_f$ in terms of $g$, or bounds that are asymptotically sharp as $g\to\infty$, especially when $f$ is a Kodaira fibration? In particular, seek functions $U(g)$ for which \[ q_f\le U\left(g(F)\right) \] holds for every fibration in the intended class, considering the general Kähler class and the Kodaira subclass separately.

   The general formulation above allows singular fibers, whereas a Kodaira fibration is smooth. The record does not explicitly specify whether singular fibers, multiple fibers, or additional minimality or base-genus hypotheses are intended in the general Kähler case. It also gives neither a comparison bound defining "improved" nor a precise asymptotic criterion defining "sharp." These qualifications remain unresolved; the source therefore poses a bounds research question rather than a specified numerical conjecture.

Short English statement

The problem asks whether one compact complex surface can carry at least three distinct Kodaira fiberings, and seeks stronger or asymptotically sharp bounds for the relative irregularity $q_f=q(X)-g(B)$ as the fiber genus grows. The second question concerns nonisotrivial fibered Kähler surfaces, with particular emphasis on Kodaira fibrations. The equivalence convention for fiberings, the full hypotheses of the general class, and the intended comparison and sharpness criteria require clarification.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, with catalog/problems.json as the supplementary registry; permanent ID $30003293$, title "Multiple Kodaira Fiberings and Relative Irregularity," source problem code catalogue entry. The supplied record gives proposed year $2016$ and names no individual proposer.
• Supplied workshop source: Surface Bundles (2017), Oberwolfach Reports, pp. 3149--3195, DOI 10.4171/owr/2016/56. https://doi.org/10.4171/owr/2016/56. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference, labeled "Llosa Isenrich and Py": https://arxiv.org/abs/2001.03694. No title or specific supporting result is provided in the record.
• Supplied literature reference, labeled "Oberwolfach Report 56 (2017)": https://doi.org/10.4171/owr/2016/56.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that later work gives constraints on multiple Kodaira fibrations and many relative-irregularity bounds, while existence of three or more Kodaira fibrations and asymptotically sharp bounds remain unresolved. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review. These are supplied editorial assertions, not independent verification of the mathematical formulation or current open status.