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

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 $\epsilon >0$, \[T_f(r,\overline D)\leq_{\mathrm{exc}}N_f^{[1]}(r,D)+\varepsilon T_f(r,\overline D).\]