Application of computational invariant theory to Kobayashi hyperbolicity and to Green-Griffiths algebraic degeneracy
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 detail. 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 Grobnerized syzygies. Nonconstant entire holomorphic curves valued in an algebraic 3-fold (resp. 4-fold) X(3) subset of P(4)(C) (resp. X(4) subset of P(5)(C)) of degree d satisfy global differential equations as soon as d >= 72 (resp. d >= 259). 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 Grobnerized syzygies. (C) 2010 Elsevier Ltd. All rights reserved.
algorithm schur bundle algebraic theory of invariants jet space nonreductive group action euler characteristic grobner bases commutative algebra finitely generated rings