The Moduli of Flat PU(p,p)-Structures with Large Toledo Invariants

Details The Moduli of Flat PU(p,p)-Structures with Large Toledo Invariants The Moduli of Flat PU(p,p)-Structures with Large Toledo Invariants

2000-09-22

arxiv.org/abs/math/0009203v2

Mathematics

Algebraic Geometry

16 pages

Scientific paper

For a compact Riemann surface $X$ of genus $g > 1$, $\Hom(\pi_1(X), PU(p,q))/PU(p,q)$ is the moduli space of flat $PU(p,q)$-connections on $X$. There are two invariants, the Chern class $c$ and the Toledo invariant $\tau$ associated with each element in the moduli. The Toledo invariant is bounded in the range $-2min(p,q)(g-1) \le \tau \le 2min(p,q)(g-1)$. This paper shows that the component, associated with a fixed $\tau > 2(max(p,q)-1)(g-1)$ (resp. $\tau < -2(max(p,q)-1)(g-1)$) and a fixed Chern class $c$, is connected (The restriction on $\tau$ implies $p=q$).

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