Analytic Ax-Schanuel Theorem for semi-abelian varieties and Nevanlinna theory

The purpose of this paper is to explore Nevanlinna theory of the entire curve $\exh_A f:=(\exp_Af,f):\C \to A \times \Lie(A)$ associated with an entire curve $f: \C \to \Lie(A)$, where $\exp_A:\Lie(A)\to A$ is an exponential map of a semi-abelian variety $A$. Firstly we give a Nevanlinna theoretic proof to the {\em analytic Ax-Schanuel Theorem} for semi-abelian varieties, which was proved by J. Ax 1972 in the case of formal power series (Ax-Schanuel Theorem). We assume some non-degeneracy condition for $f$ such that the elements of the vector-valued function $f(z)-f(0) \in \Lie(A)\iso \C^n$ are $\Q$-linearly independent in the case of $A=(\C^*)^n$. Then by making use of the Log Bloch-Ochiai Theorem and a key estimate which we show, we prove that $\td_\C\, \exh_A f \geq n+ 1$. Our next aim is to establish a {\em 2nd Main Theorem} for $\exh_A f$ and its $k$-jet lifts with truncated counting functions at level one.