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Differential equations on hypersurfaces in \(\mathbb P^4\). (Equations différentielles sur les hypersurfaces de \(\mathbb P^4\).) (French) Zbl 1115.14009
Here the author proves that for all integers \(d \geq 97\) (resp. \(d \geq 92\)) and every smooth degree \(d\) hypersurface \(X \subset \mathbb {P}^4_{\mathbb {C}}\) every entire curve of \(X\) (resp. \(\mathbb {P}^4_{\mathbb {C}}\backslash X\)) satisfies an algebraic differential equation of degree \(3\). The main tool is the logarithmic version of Demailly vector bundle of differential jets [G. Dethloff and S. Lu, Osaka J. Math. 38, No. 1, 185–237 (2001; Zbl 0982.32022)]. Apart from this fundamental tool the main trick is to use the decomposition of the graded object obtained from the vector bundle of jets of order \(3\) into Schur’s irreducible representations.

MSC:
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14J70 Hypersurfaces and algebraic geometry
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
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