Positive Sasakian Structures on 5-manifolds

A quasi–regular Sasakian structure on a manifold L is equivalent to writing L as the unit circle subbundle of a holomorphic Seifert C-bundle over a complex algebraic orbifold (X,∆ = ∑ (1− 1 mi )Di). The Sasakian structure is called positive if the orbifold first Chern class c1(X) − ∆ = −(KX + ∆) is positive. We are especially interested in the case when the Riemannian metric part of the Sasakian structure is Einstein. By a result of [Kob63, BG00], this happens iff (1) −(KX +∆) is positive, (2) the first Chern class of the Seifert bundle c1(L/X) is a rational multiple of −(KX +∆), and (3) there is an orbifold Kähler–Einstein metric on (X,∆). If the first two conditions hold, we say that f : L → (X,∆) is a pre-SE (or pre– Sasakian–Einstein) Seifert bundle. Note that if H2(L,Q) = 0 then H2(S,Q) ∼= Q and so the second condition is automatic. While it is not true that all pre-SE Seifert bundles carry a Sasakian–Einstein structure, all known topological obstructions to the existence of a Sasakian–Einstein structure in dimension 5 are consequences of the pre-SE condition. A 2–dimensional orbifold (S,∆) such that−(KS+∆) is positive is also called a log Del Pezzo surface. For any such, S is a rational surface with quotient singularities. By [Kol05, 2.4], Seifert C-bundles over (S, ∑ (1 − 1 mi )Di) are uniquely classified by a homology class B ∈ H2(S,Z) and integers 0 < bi < mi with (bi,mi) = 1. We denote the corresponding Seifert C-bundle (resp. S-bundle) by