Classification of totally real elliptic Lefschetz fibrations via necklace diagrams

TL;DR

This paper introduces a combinatorial classification method for totally real elliptic Lefschetz fibrations with real sections using necklace diagrams, linking geometric structures to algebraic matrix decompositions.

Contribution

It develops a novel combinatorial approach to classify these fibrations via necklace diagrams and extends the classification to those without real sections.

Findings

Necklace diagrams correspond to isomorphism classes of fibrations.

An explicit algorithm for matrix decompositions yields classification.

Refined diagrams uniquely determine fibrations without real sections.

Abstract

We show that totally real elliptic Lefschetz fibrations that admit a real section are classified by their "real loci" which is nothing but an $S^1$-valued Morse function on the real part of the total space. We assign to each such real locus a certain combinatorial object that we call a \emph{necklace diagram}. On the one hand, each necklace diagram corresponds to an isomorphism class of a totally real elliptic Lefschetz fibration that admits a real section, and on the other hand, it refers to a decomposition of the identity into a product of certain matrices in $PSL(2,\Z)$. Using an algorithm to find such decompositions, we obtain an explicit list of necklace diagrams associated with certain classes of totally real elliptic Lefschetz fibrations. Moreover, we introduce refinements of necklace diagrams and show that refined necklace diagrams determine uniquely the isomorphism classes of the totally real elliptic Lefschetz fibrations which may not have a real section. By means of necklace diagrams we observe some interesting phenomena underlying special feature of real fibrations.