Details:
Title  Application of computational invariant theory to Kobayashi hyperbolicity and to Green–Griffiths algebraic degeneracy  Author(s)  Joël Merker  Type  Article in Journal  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 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 biinvariants is generated by 16 mutually independent polynomials sharing 41 Gröbnerized syzygies. Nonconstant entire holomorphic curves valued in an algebraic 3fold (resp. 4fold) X^3 ⊂ P^4 ( C ) (resp. X^4 ⊂ 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–Poincaré 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 biinvariants is generated by 17 mutually independent polynomials sharing 105 Gröbnerized syzygies.  Keywords  Algebraic theory of invariants, Jet space, Nonreductive group action, Schur bundle, Euler characteristic, Gröbner bases, Algorithm, Commutative algebra, Finitely generated rings  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717110001033 
Language  English  Journal  Journal of Symbolic Computation  Volume  45  Number  10  Pages  986  1074  Year  2010  Edition  0  Translation 
No  Refereed 
No 
