Symplectic resolution of orbifolds with homogeneous isotropy

Abstract

We construct the symplectic resolution of a symplectic orbifold whose isotropy locus consists of disjoint submanifolds with homogeneous isotropy, that is, all its points have the same isotropy groups.

Gompf connected sum for orbifolds and K-contact Smale-Barden manifolds

“…For every chart as in Definition 5(i), we have on Ṽi = Ũi × R m a scalar product a(x) = a ij (x)y i y j on R m , depending on x ∈ Ũi , which is Υ iinvariant. Using partitions of unity as in [12,Proposition 6], we can see that there is a metric on any orbivector bundle.…”

“…The function (x) can be taken continuous, and if X is compact, then we can take 0 = min{ (x)| x ∈ X} > 0. If (Z, ω) is a symplectic orbifold, then at every point x ∈ Z there are orbifold Darboux charts [12,Proposition 10], that is an orbifold chart (U, Ũ , Γ, ϕ) such that Γ < U(n) and ω = dx i ∧ dy i , in these coordinates (x 1 , y 1 , . .…”

“…Given a symplectic orbifold (Z, ω), an orbifold almost complex J is compatible if the orbifold tensor g defined by g(−, −) = ω(−, J(−)), is an orbifold Riemannian metric. They always exist [12,Proposition 8].…”

“…Proof. We follow the argument in [12,Proposition 8]. First, we construct a compatible almost complex structure J 0 on X.…”

“…This gives a chart Ṽ × Ũ on which ω | Ṽ ×{0} = ω 0 is the standard form. It can be extended to a Darboux chart on some Ṽ × Ṽ ⊂ Ṽ × Ũ by using the argument in the proof of [12,Proposition 10]. Noting that (ω − ω 0 ) | Ṽ = 0, the form µ ∈ Ω 1 ( Ṽ × Ũ ) so that ω − ω 0 = dµ, can be arranged to be zero over Ṽ .…”