Entire holomorphic curves into ${\mathbb{P}}^n(\mathbb{C})$ intersecting $n+1$ general hypersurfaces

Abstract

Let $\{D_i{\}}_{i=1}^{n+1}$ be $n+1$ hypersurfaces in ${\mathbb{P}}^n(\mathbb{C})$ with total degrees ${\sum }_{i=1}^{n+1}°D_i⩾n+2$, in general position and satisfying a generic geometric condition: every $n$ hypersurfaces intersect only at smooth points and the intersection is transversal. Then, for every algebraically nondegenerate entire holomorphic curve $f:\mathbb{C}\to {\mathbb{P}}^n(\mathbb{C})$, we show a Second Main Theorem: $$\sum _{i=1}^{n+1}{\delta }_f(D_i)<n+1$$ in terms of defect inequality in Nevanlinna theory. This is the first result in the literature on Second Main Theorem for $n+1$ general hypersurfaces in ${\mathbb{P}}^n(\mathbb{C})$ with optimal total degrees.

arXiv comment20 pages, 6 figures, comments are welcome