Invariant Jet differentials and Asymptotic Serre duality

• Published in: arXiv.org 2024-11
• Main Author: Rahmati, Mohammad Reza
• Format: Article
• Language: English
• Subjects: Asymptotic properties; Bundles; Bundling; Invariants
• ISSN: 2331-8422

We generalize the main result of Demailly \cite{D2} for the bundles \(E_{k,m}^{GG}(V^*)\) of jet differentials of order \(k\) and weighted degree \(m\) to the bundles \(E_{k,m}(V^*)\) of the invariant jet differentials of order \(k\) and weighted degree \(m\). Namely, Theorem 0.5 from \cite{D2} and Theorem 9.3 from \cite{D1} provide a lower bound \(\frac{c^k}{k}m^{n+kr-1}\) on the number of the linearly independent holomorphic global sections of \(E_{k,m}^{GG} V^* \bigotimes \mathcal{O}(-m \delta A)\) for some ample divisor \(A\). The group \(G_k\) of local reparametrizations of \((\mathbb{C},0)\) acts on the \(k\)-jets by orbits of dimension \(k\), so that there is an automatic lower bound \(\frac{c^k}{k} m^{n+kr-1}\) on the number of the linearly independent holomorphic global sections of \(E_{k,m}V^* \bigotimes \mathcal{O}(-m \delta A)\). We formulate and prove the existence of an asymptotic duality along the fibers of the Green-Griffiths jet bundles over projective manifolds. We also prove a Serre duality for asymptotic sections of jet bundles. An application is also given for partial application to the Green-Griffiths conjecture.