Lefschetz fibrations on adjoint orbits - PMA

Topological <strong>Lefschetz</strong> fibration - definition
Let X be a compact complex variety of dimension n together with
a surjective map f : X → P 1 . We say that f is a topological
<strong>Lefschetz</strong> fibration (TLF) if:
1. The differential df is surjective outside a finite set of points
{Q1, . . . , Qµ} ⊂ X .
2. For any p ∈ S 2 \ {f (Q1), . . . , f (Qµ)} the fibre f −1 (p) is a
smooth complex variety of dimension n.
3. The images qi := f (Qi) are distinct for different Qi.
4. Let Xi be the fibre f containing Qi. Then for each i there
exist small discs qi ∈ Ui and Qi ⊂ UQi ⊂ X such that
f : UQi → Ui is a holomorphic Morse function and in
coordinates (x0, . . . , xn) ∈ UQi e z ∈ Ui,
z = f (x0, . . . , xn) = x 2 0 + · · · + x 2 n.
5. f |X − {Xi} is locally trivial.
Elizabeth Gasparim <strong>Lefschetz</strong> <strong>fibrations</strong> on adjoint orbits