Moduli spaces of surfaces and monodromy invariants

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This paper is a survey of the authors' recent results on "abc-surfaces" and the monodromy of their natural Lefschetz fibrations and projections to P¹ x P¹, 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.