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TitleApplication of computational invariant theory to Kobayashi hyperbolicity and to Green–Griffiths algebraic degeneracy
Author(s) Joël Merker
TypeArticle in Journal
AbstractA 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 Gröbnerized syzygies. Nonconstant entire holomorphic curves valued in an algebraic 3-fold (resp. 4-fold) 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 bi-invariants is generated by 17 mutually independent polynomials sharing 105 Gröbnerized syzygies.
KeywordsAlgebraic theory of invariants, Jet space, Nonreductive group action, Schur bundle, Euler characteristic, Gröbner bases, Algorithm, Commutative algebra, Finitely generated rings
ISSN0747-7171
URL http://www.sciencedirect.com/science/article/pii/S0747717110001033
LanguageEnglish
JournalJournal of Symbolic Computation
Volume45
Number10
Pages986 - 1074
Year2010
Edition0
Translation No
Refereed No
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