Global topology of the Hitchin system - GEOM

Character varieties
the GLn-character variety:
Md B := {(Ai, Bi)i=1..g ∈ GL2g n | [A1, B1] . . . [Ag , Bg ] = ζd n In}//PGLn
non-singular, affine
the SLn-character variety:
ˇM d B := {(Ai, Bi)i=1..g ∈ SL2g n | [A1, B1] . . . [Ag , Bg ] = ζd n In}//PGLn
non-singular, affine
for PGLn note that (C × ) 2g acts on Md B and
Γ ∼ = (Zn) 2g ⊂ (C × ) 2g acts on ˇM d
M
ˆd
B := M
ˇd
B /Γ ∼ = Md B /(C× ) 2g is an affine orbifold
Theorem (Non-Abelian Hodge Theorem; Simpson, Corlette)
ˆˇM d Dol
diff RH
∼= ˆˇMDR
∼= ˆˇMB
RH is complex analytic ∼ =; so SYZ satisfied by ˇM d B and ˆM d B
Conjecture (Hausel-Villegas, 2004)
(d, n) = (e, n) = 1 E( ˇM d ˆB d
B ; u, v) = Est ( ˆM e B ; u, v) 30 / 39