The moduli of flat PU(2,1) structures on Riemann surfaces

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Abstract

For a compact Riemann surface X of genus g > 1, Hom(π1(X), PU(p, q))/PU(p, q) is the moduli space of flat PU(p, q)-connections on X. There are two integer invariants, dP, dQ, associated with each σ ∈ Hom(π1(X), PU(p, q))/PU(p, q). These invariants are related to the Toledo invariant τ by τ = 2qdP-pdQ/p+q. This paper shows, via the theory of Higgs bundles, that if q = 1, then -2(g - 1) ≤ τ ≤ 2(g - 1). Moreover, Hom(π1(X), PU(2, 1))/PU(2, 1) has one connected component corresponding to each τ ∈ 2/3ℤ with -2(g - 1) ≤ τ ≤ 2(g - 1). Therefore the total number of connected components is 6(g - 1) + 1.

Original languageEnglish
Pages (from-to)231-256
Number of pages26
JournalPacific Journal of Mathematics
Volume195
Issue number1
DOIs
Publication statusPublished - 2000 Sept

All Science Journal Classification (ASJC) codes

  • General Mathematics

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