×

Demailly-Semple jets of order 4 and 5 in dimension 2. (Jets de Demailly-Semple d’ordres 4 et 5 en dimension 2.) (French) Zbl 1161.13002

Summary: In dimension 2 and for jets of order 4, there are 9 Demailly-Semple invariant polynomials generated by bracketing the invariants for jets of order \(\leqslant \) 3 [cf. J.-P. Demailly, in: Algebraic geometry. Santa Cruz 1995: AMS Proc. Symp. Pure Math. 62(pt.2), 285–360 (1997; Zbl 0919.32014); E. Rousseau, Hyperbolicité des variétés complexes, arXiv:0709.3882]. They share 9 fundamental syzygies and an Euler characteristic computation shows that every entire holomorphic curve into a generic smooth complex projective algebraic surface \(X\subset P_3 (\mathbb C)\) satisfies global algebraic differential equations provided deg \(X\geqslant 9\).
For jets of order 5, the algebraic structure explodes in complexity. Bracketing gives 36 invariants. Removing redundancies leaves 24 fundamental invariants, 11 of which, denoted \(f_1^\prime, \Lambda ^3 , \Lambda ^5_1 , \Lambda ^7_{1,1} , \Lambda ^9_{1,1,1}, M^8 , M^{10}_1 , K^{12}_{1,1} , N^{12}, H^{14}_1\) and \(F^{16}_{1,1}\) and called bi-invariants, are meaningful for Euler characteristic estimates. Unexpectedly, 5 more appear: \(X^{18} , X^{19} , X^{21} , X^{23}\) and \(X^{25} \). The paper contains an algorithm generating all such (complicated) bi-invariants.

MSC:

13A50 Actions of groups on commutative rings; invariant theory
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
14J70 Hypersurfaces and algebraic geometry
14M12 Determinantal varieties
13D02 Syzygies, resolutions, complexes and commutative rings
14M15 Grassmannians, Schubert varieties, flag manifolds

Citations:

Zbl 0919.32014
PDFBibTeX XMLCite
Full Text: arXiv Link