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

Source abstract

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 $A\subset\overline A$, 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 $\overline D$ on $\overline A$. After possibly replacing $\overline A$ by another equivariant compactification depending only on $D$ and independent of $f$, for every $\varepsilon>0$, $$T_f(r,\overline D)\leq_{\mathrm{exc}}N_f^{[1]}(r,D)+\varepsilon T_f(r,\overline D).$$