A defect relation for holomorphic curves intersecting hypersurfaces

Abstract

In 1933, H. Cartan proved a defect relation Σ q j =1 δ f ( H j ) ≤ n + 1 for a linearly nondegenerate holomorphic curve f : [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] and hyperplanes H j , 1 ≤ j ≤ q , in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] in general position. This paper extends it to holomorphic curves intersecting hypersurfaces. In 1979, B. Shiffman conjectured that if f : [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] is an algebraically non-degenerate holomorphic map, and D 1 , . . . , D q are hypersurfaces in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /] in general position, then Σ q j =1 δ f ( D j ) ≤ n + 1. This paper proves this conjecture.