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Connections and the Second Main Theorem for Holomorphic Curves

By means of $C^\infty$-connections we will prove a general second main theorem and some special ones for holomorphic curves. The method gives a geometric proof of H. Cartan's second main theorem in 1933. By applying the same method, we will prove some second main theorems in the case of the product space $(\pone)^2$ of the Riemann sphere.