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Achiral Lefschetz fibrations and the moduli space of curves

Sardor Yakupov

TL;DR

The paper extends Smith's classifying-map framework for Lefschetz fibrations to the achiral setting by employing a fibered normalized Ricci flow to produce smooth classifying maps into the Deligne–Mumford compactification $\overline{\mathcal{M}}_g$ for genus $g\ge 4$. It proves the existence and uniqueness (up to isotopy) of such classifying maps and shows they can be made smooth across singular fibers, with a refined construction yielding a map into the automorphism-free locus under mild assumptions. Additionally, it generalizes the Smith signature formula to achiral fibrations by leveraging the Meyer signature cocycle and a cohomological decomposition, providing an elementary proof that avoids a direct index-theoretic route. Overall, the work links 4-manifold fibrations with moduli-space geometry via the fibered Ricci-flow approach and yields computable signature data from the classifying map.

Abstract

Symplectic Lefschetz fibrations can be described via classifying maps with values in the Deligne-Mumford compactification of the moduli space of curves, by means of constructions relying on symplectic geometry. In this note we prove the existence of classifying maps for achiral Lefschetz fibrations using Riemannian geometry. We further extend the Smith signature formula to the achiral case, providing a more elementary proof to this statement.

Achiral Lefschetz fibrations and the moduli space of curves

TL;DR

The paper extends Smith's classifying-map framework for Lefschetz fibrations to the achiral setting by employing a fibered normalized Ricci flow to produce smooth classifying maps into the Deligne–Mumford compactification for genus . It proves the existence and uniqueness (up to isotopy) of such classifying maps and shows they can be made smooth across singular fibers, with a refined construction yielding a map into the automorphism-free locus under mild assumptions. Additionally, it generalizes the Smith signature formula to achiral fibrations by leveraging the Meyer signature cocycle and a cohomological decomposition, providing an elementary proof that avoids a direct index-theoretic route. Overall, the work links 4-manifold fibrations with moduli-space geometry via the fibered Ricci-flow approach and yields computable signature data from the classifying map.

Abstract

Symplectic Lefschetz fibrations can be described via classifying maps with values in the Deligne-Mumford compactification of the moduli space of curves, by means of constructions relying on symplectic geometry. In this note we prove the existence of classifying maps for achiral Lefschetz fibrations using Riemannian geometry. We further extend the Smith signature formula to the achiral case, providing a more elementary proof to this statement.
Paper Structure (7 sections, 22 theorems, 56 equations)

This paper contains 7 sections, 22 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Achiral Lefschetz fibrations
  4. Moduli space of complex curves
  5. Ricci flow for surfaces
  6. Construction of the classifying map
  7. Signature of achiral Lefschetz fibrations

Key Result

Theorem A

Let $f:X\to S$ be an achiral Lefschetz fibration of fiber genus $g\geq 4$. Then there exists a classifying map $\phi_f: S\to \mathop{\mathrm{\overline{\mathcal{M}}}}\nolimits_g$, which is unique up to isotopy preserving the geometric intersection number between $S$ and $\mathop{\mathrm{\partial \mat

Theorems & Definitions (45)

  • Theorem A: see Theorem \ref{['thm:main1']}
  • Theorem B: see Theorem \ref{['NFRF']}
  • Theorem C: see Theorem \ref{['thm:main2']}
  • Definition 2.1
  • Remark
  • Definition 2.2
  • Remark
  • Theorem 2.1: Gompf
  • Theorem 2.2: Donaldson
  • Definition 2.3
  • ...and 35 more