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The Poincaré Dual of a Geodesic Algebraic Curve in a Quotient of the 2-Ball

We shall consider an irreducible, non-singular, totally geodesic holomorphic curve N in a compact quotient M = Γ\D of the unit ball D = {(z, w):|z|2 + |w|2 < 1} in C 2 with the Kahler structure provided by the Bergman metric. The main result of this paper is an explicit construction of the harmonic form of type (1,1) which is dual to N. Our construction is as follows. Let p:D → Γ\D be the universal covering map. Choose a component D 1 in the inverse image of N under p. The choice of D 1 corresponds to choosing an embedding of the fundamental group of N into Γ. We denote the image by Γ 1. Let π : D → D 1 be the fiber bundle obtained by exponentiating the normal bundle of D 1 in D. Let μ be the volume form of D 1.