A Second Main Theorem for Moving Hypersurface Targets

TL;DR

This paper extends the Second Main Theorem to slowly moving hypersurfaces in projective space, providing a more general defect relation and an effective truncation estimate, advancing value distribution theory.

Contribution

It generalizes Shiffman's conjecture to moving hypersurfaces and introduces an effective truncation level in the Second Main Theorem.

Findings

Proved a generalized defect relation for moving hypersurfaces.

Established an effective estimate for the truncation level.

Extended previous results to a broader class of hypersurfaces.

Abstract

In 1979, B. Shiffman conjectured that if f is an algebraically nondegenerate holomorphic map of C into P^n and D_1,...,D_q are hypersurfaces in P^n in general position, then the sum of the defects is at most n+1. This conjecture was proved by M. Ru in 2004. In this paper, the Shiffman conjecture is proved more generally in the case of slowly moving hypersurfaces in (weakly) general position. Moreover, we introduce a truncation in the corresponding Second Main Theorem, with an effective estimate on the truncation level, thus generalizing a result of An-Phuong.