Higgs bundles over elliptic curves - ICMAT

M(GL(n, C)) 0 of zero degree Higgs bundles we have the following morphism(det, tr) : M(GL(n, C)) 0 Pic 0 (X) × H 0 (X, O)(det, tr) : M(GL(n, C)) 0 Pic 0 (X) × H 0 (X, O)and we can consider the closed subvarieties[(E, Φ)] S [(det E, tr Φ)] S ,M det=Otr=0 (GL(n, C)) = (det, tr)−1 (O, 0) ⊂ M(GL(n, C)) 0andM (GL(n, C)) = (det, det=Otr=0 tr)−1 (O, 0) ⊂ M(GL(n, C)) 0 .Since M(GL(n, C)) 0 and M(GL(n, C)) 0 are coarse moduli spaces there exist two bijectionsα P : A (n,0) /∼ → M(GL(n, C)) 0 and α Q : A (n,0) /∼ → M(GL(n, C)) 0 satisfyingthat (3.6) and (3.7) are morphisms.If we have two Higgs bundles (E 1 , Φ 1 ) and (E 2 , Φ 2 ) with (E 1 , Φ 1 ) ∼ S (E 2 , Φ 2 ) then,certainly det E 1∼ = det E2 and tr Φ 1 = tr Φ 2 . This implies that  n / ∼ S injects intoA (n,0) /∼ S . We can take the restrictions of α P and α Q to Ân/∼ S which we denote by α P′and α Q ′ and we observe that (M (GL(n, C)), det=Otr=0 α′ P ) and (M (GL(n, C)), det=Otr=0 α′ Q ) satisfythe moduli conditions (3.6) and (3.7) and therefore they are respectively moduli spaces forthe functors Mod(Ân, P (n,0) , S) and Mod(Ân, Q (n,0) , S), i.e.andM(SL(n, C)) = (det, tr) −1 (O, 0) ⊂ M(GL(n, C)) 0 (3.13)M(SL(n, C)) = (det, tr) −1 (O, 0) ⊂ M(GL(n, C)) 0 . (3.14)After (3.13) we see that the existence of the coarse moduli space M(SL(n, C)) followsfrom [Ni] and [Si1].A PGL(n, C)-Higgs bundle over the elliptic curve X is pair (P(E), Φ) where E is avector bundle, P(E) is the projective bundle given by E and Φ ∈ H 0 (X, End E) has tr Φ =0. A PGL(n, C)-Higgs bundle has topological invariant ˜d ∈ Z n where ˜d = (d mod n).Remark 3.5.1. Over a curve, a projective bundle comes always from a vector bundle,although this fact is no longer true in higher dimension.A PGL(n, C)-Higgs bundle (P(E), Φ) is semistable, stable or polystable if (E, Φ) isrespectively a semistable, stable or polystable Higgs bundle. If gr(E, Φ) = (E ′ , Φ ′ ), wedefine the associated graded object of the semistable PGL(n, C)-Higgs bundle (P(E), Φ)as the pair (P(E ′ ), Φ ′ ).We denote by Ǎn, ˜dthe collection of semistable PGL(n, C)-Higgs bundles of degree˜d ∈ Z n . A family of PGL(n, C)-Higgs bundles E → X × T is a pair (P, Φ) where P is43