Higgs bundles over elliptic curves - ICMAT

Chapter 4Moduli spaces of Higgs bundles4.1 Some results for elliptic curvesLet (A, a 0 ) be a commutative abelian group. If A is a quasiprojective variety and themultiplication map µ : A × A → A is a morphism, we can define the following morphismα A,h : Sym h A A(4.1)[a 1 , . . . , a h ] Sh ∑ hi=1 a i.Thanks to the group structure of A, the symmetric group S h acts on the (h−1)-th Cartesianproduct of A. Consider the following morphism between cartesian produtcs of Av A,h : A× h−1 . . . ×A A× . h . . ×A(4.2)(a 1 , . . . , a h−1 ) (a 1 , . . . , a h−1 , − ∑ h−1i=1 a i),and the projection on the first h − 1 factorsq A,h : A× . h . . ×A A× h−1 . . . ×A(a 1 , . . . , a h−1 , a h ) (a 1 , . . . , a h−1 ).The action of σ ∈ S h on the (h − 1)-tuple (a 1 , . . . , a h−1 ) ∈ A× h−1 . . . ×A is defined byNote that the morphismσ · (a 1 , . . . , a h−1 ) = q A,h (σ · (v A,h (a 1 , . . . , a h−1 ))). (4.3) u A,h : (A× h−1 . . . ×A) / S hker α A,h(4.4)[a 1 , . . . , a h−1 ] Sh [a 1 , . . . , a h−1 , − ∑ h−1i=1 a i] Sh50