Applications of computational invariant theory to Kobayashi hyperbolicity and to Green-Griffiths algebraic degeneracy

TL;DR

This paper develops algorithms to generate invariants of jet differentials under non-reductive group actions, advancing understanding of Kobayashi hyperbolicity and algebraic degeneracy in complex geometry.

Contribution

It provides a complete algorithm for generating invariants of jet differentials in arbitrary dimensions and orders, with detailed case studies for specific jet orders and dimensions.

Findings

Generated invariants for jets of order 4 in dimension 4 with 2835 polynomials.

Established differential equations for entire holomorphic curves in algebraic 3-folds and 4-folds.

Derived an asymptotic formula for the Euler-Poincare characteristic of Schur bundles.

Abstract

A major unsolved problem (according to Demailly 1997) towards the Kobayashi hyperbolicity conjecture in optimal degree is to understand jet differentials of germs of holomorphic discs that are invariant under any reparametrization of the source. The underlying group action is not reductive, but we provide a complete algorithm to generate all invariants, in arbitrary dimension n and for jets of arbitrary order k. Two main new situations are studied in great details. For jets of order 4 in dimension 4, we establish that the algebra of Demailly-Semple invariants is generated by 2835 polynomials, while the algebra of bi-invariants is generated by 16 mutually independent polynomials sharing 41 groebnerized syzygies. Nonconstant entire holomorphic curves valued in an algebraic 3-fold (resp. 4-fold) X^3 in P^4 (C) (resp. X^4 in P^5(C)) of degree d satisfy global differential equations as soon as d > 71 (resp. d > 258). A useful asymptotic formula for the Euler-Poincare characteristic of Schur bundles in terms of Giambelli's determinants is derived. For jets of order 5 in dimension 2, we establish that the algebra of Demailly-Semple invariants is generated by 56 polynomials, while the algebra of bi-invariants is generated by 17 mutually independent polynomials sharing 105 groebnerized syzygies.