Holomorphic sections of line bundles on the jet spaces of the Riemann sphere (Q2151143)

Let \(M\) be a complex manifold of positive dimension. The loops space of smooth functions from the torus (of real dimension one) to \(M\) was thoroughly studied for a few decades, from the point of view of complex analytic and complex algebraic geometry. Notable in bits respect are Laszló Lempert works, to cite only a few: \textit{L. Lempert} [Math. Proc. R. Ir. Acad. 104A, No. 1, 35--46 (2004; Zbl 1082.58004)], \textit{L. Lempert} and \textit{E. Szabó} [Asian J. Math. 11, No. 3, 485--496 (2007; Zbl 1136.14023)], \textit{L. Lempert} and \textit{N. Zhang} [Acta Math. 193, No. 2, 241--268 (2004; Zbl 1087.58005)]. The article under review is dedicated to Lempert's 70-th birthday and deals with the space of \(k\) jets, at a prescribed point of the torus, to \(M\). Specific bases of polynomial sections of this complex algebraic (jet space) manifold to holomorphic line bundles are exhibited.