Global topology of the Hitchin system - GEOM

Proper Hitchin map
(E, φ) ∈ T ∗N ❀ E is stable; to projectivize χ we need to
allow E to become unstable.
A Higgs bundle is a pair (E, φ) where E is a vector bundle on
C and φ ∈ H0 (C, End(E) ⊗ K) is a Higgs field.
a Higgs bundle (E, φ) is (semi-)stable if for every φ-invariant
proper subbundle E we have µ(F ) (≤)
< µ(E)
Md the moduli space of (semi-)stable Higgs bundles, a
non-singular quasi-projective and symplectic variety,
containing T ∗N ⊂ Md as an open dense subvariety
extend χ : Md → A in the obvious way
Theorem (Hitchin 1987, Nitsure 1991, Faltings 1993)
χ is a proper algebraically completely integrable Hamiltonian
system. Its generic fibres are abelian varieties.
as codim(Md \ T ∗N d ) ≥ 2 ⇒ C[Md ] ∼ = C[T ∗N d ] ⇒ thus
by the Theorem
A ∼= Spec(C[Md ]) ∼ = Spec(C[T ∗N d ]) 15 / 39