Holomorphic Maps from ${\mathbb{C}}^p$ into Semi-Abelian Varieties

We prove the Second Main Theorem with truncation level one for holomorphic maps with Zariski-dense image intersecting a reduced effective Cartier divisor: Let $A$ be a semi-abelian variety with an equivariant compactification $A\subset \bar{A}$, and let $f:{\mathbb{C}}^p\to A$ be a holomorphic map with Zariski-dense image. Let $D$ be a reduced effective divisor on $A$ extending to a divisor $\bar{D}$ on $\bar{A}$. After possibly replacing $\bar{A}$ by another equivariant compactification depending only on $D$ and independent of $f$, for every $\epsilon >0$, $$T_f(r,\bar{D}){\le }_{\mathrm{e}\mathrm{x}\mathrm{c}}{N}_f^{[1]}(r,D)+\epsilon T_f(r,\bar{D})\mathrm{.}$$