TY - JOUR
T1 - The moduli of flat PU(2,1) structures on Riemann surfaces
AU - Xia, Eugene Z.
N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
PY - 2000/9
Y1 - 2000/9
N2 - 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.
AB - 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.
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U2 - 10.2140/pjm.2000.195.231
DO - 10.2140/pjm.2000.195.231
M3 - Article
AN - SCOPUS:0001718031
SN - 0030-8730
VL - 195
SP - 231
EP - 256
JO - Pacific Journal of Mathematics
JF - Pacific Journal of Mathematics
IS - 1
ER -