Invariant Jet differentials and Asymptotic Serre duality

We generalize the main result of Demailly \cite{D2} for the bundles $E_{k,m}^{GG}(V^{\ast })$ of jet differentials of order $k$ and weighted degree $m$ to the bundles $E_{k,m}(V^{\ast })$ 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)$forsomeampledivisor$A$\mathrm{.}Thegroup$G_k$oflocalreparametrizationsof$($\mathbb{C}$,0)$actsonthe$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.