Moduli spaces of surfaces and monodromy invariants

TL;DR

This survey reviews recent findings on abc-surfaces, focusing on monodromy invariants of Lefschetz fibrations, and explores complex deformation, symplectomorphism, and diffeomorphism relationships through explicit examples.

Contribution

It consolidates recent results on monodromy invariants of abc-surfaces and discusses their implications for complex and symplectic geometry.

Findings

Monodromy invariants distinguish different deformation types

Explicit examples illustrate relationships between complex and symplectic structures

New remarks extend previous results on abc-surfaces

Abstract

This paper is a survey of the authors' recent results on "abc-surfaces" and the monodromy of their natural Lefschetz fibrations and projections to P^1 x P^1, see (arXiv:0910.2142). The results being surveyed explore various fundamental questions about complex deformation vs. symplectomorphism vs. diffeomorphism from the perspective of one explicit family of examples; the paper also contains some remarks which were not in earlier papers.