Extrinsic projective curves X^1 in P^2(C): harmony with intrinsic cohomology

TL;DR

This paper explicitly constructs holomorphic jet differentials on smooth complex algebraic curves in P^2, linking their extrinsic projective geometry with intrinsic cohomological properties, and providing a detailed description of sections of the Green-Griffiths bundle.

Contribution

It introduces a new explicit family of holomorphic jet differentials for smooth projective curves in P^2, revealing their nonlinear features and connecting extrinsic and intrinsic geometric aspects.

Findings

Explicit formulas for jet differentials in terms of polynomial R and derivatives

Complete description of holomorphic sections of Green-Griffiths bundle

New insights into the nonlinear structure of jet differentials

Abstract

On a geometrically smooth complex algebraic curve X^1 in P^2(C), represented in complex affine coordinates (x,y) as the zero-locus R(x,y) = 0 of some polynomial R of degree d >= k+3, an explicit family of generating independent holomorphic jet differentials J_R^1, ..., J_R^k expressed in terms of R and its partial derivatives is exhibited with its new precious nonlinearity features as a complete explicitation of all holomorphic sections of the Green-Griffiths bundle of m-homogeneous polynomialized order k jets of local holomorphic maps from a complex disc into X^1.