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Demailly-Semple jets in dimension 3. (Étude des jets de Demailly-Semple en dimension 3.) (French) Zbl 1092.58003
The author presents an algebraic characterization of Demailly-Semple jets in dimension \(3\) by using the invariant theory of non reductive groups. In dimension \(3\), the algebra of differential operators is generated by the first order jets \(f^\prime_i\), the expressions \(w_{ij}=f^\prime_if^{\prime \prime}_j-f^\prime_jf^{\prime \prime}_i\) involving jets of first and second order, some expressions \(w^k_{ij}\) involving jets of first, second and third order and a determinant \(W\) defined by the jets of first, second and third order of the components \(f_1,f_2,f_3\). The degree is \(7\). The main geometric applications consist in the characterization of the graded bundle of jets of order \(3\) on a three-dimensional complex manifold and in the computation of the Euler-Poincaré characteristic of a smooth hypersurface of degree \(d\) in \(\mathbb{P}^4\).

MSC:
58A20 Jets in global analysis
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
13A50 Actions of groups on commutative rings; invariant theory
06B05 Structure theory of lattices
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