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 ${\text{exh}}_{A}f:=({\exp }_{A}f,f):\text{C}\to A\times \text{Lie}(A)$ associated with an entire curve $f:\text{C}\to \text{Lie}(A)$, where ${\exp }_A:\text{Lie}(A)\to A$ 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 $f(z)-f(0)\in \text{Lie}(A)\text{iso}{\text{C}}^n$ are $\text{Q}$-linearly independent in the case of $A=({\text{C}}^{\ast }{)}^n$. 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 ${\text{exh}}_{A}f$ and its $k$-jet lifts with truncated counting functions at level one.