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Complex) manifold this way there is still a lot of talking about how to compare diﬀerent atlases, what are equivalence classes of atlases, what are maximal atlases and so on. With our deﬁnition we know what it means that a map h : X −→ Y between two C ∞ (resp. complex) manifolds is a C ∞ (resp. holomorphic) map. Such a map should be continuous and then we use the atlases to formulate what else should be true, namely that the maps induced by the local charts should be C ∞ (resp. holomorphic). But we see that there is a diﬀerent way of formulating that h is C ∞ (resp.

Of 2πik course f (z) = ζz + a1 z 2 . . where ζ = e m and (k,m) = 1. We can ﬁnd a smaller open set D ⊂ U such that f (D) = D. This deﬁnes a holomorphic action of the cyclic group G =< f ν > of order n on D and we can form the quotient under this action. This is the space D/G = B. Let π : D −→ B the projection map. We deﬁne a sheaf OB on B : For any open set V ⊂ B we deﬁne OB (V ) as the ring of holomorphic functions on the inverse image π −1 (V ) ⊂ D which are invariant under the action of the cyclic group G.

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