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Topological invariants of generalized Kummer manifolds

Let L ⊆ C be a lattice and consider the compact complex torus XL = C/L. Let Γ 6 Aut(L) ∩ SL2m C be a (finite) group of automorphisms of L whose action on C is semifree (i.e., free away from 0). X L consists of finitely many isolated fixed points, so the orbifold XL/Γ has finitely many isolated singularities. We can resolve these singularities by ‘blowing up’ each of the singular points. In fact, we can do this once and for all by blowing up the fixed point set X L and then passing to a quotient manifold. Locally, at each p ∈ X L choose a holomorphic chart z = (z1, . . . , z2m) : U −→ U ′ ⊆ C with z(p) = 0. Then define Ũ = {(u, [w]) ∈ U × CP2m−1 : zi(u)wj = zj(u)wi}. This can be extended to define a submanifold of XL × CP2m−1.