A Symmetry Method for Key Vanishing Lemmas and Optimal $2$-Jet Thresholds for Hyperbolicity

Abstract

This manuscript reports on an ongoing project. The finite full-column-rank computations underlying the Key Vanishing Lemmas isolated below are currently undergoing exact computer-assisted verification. Conditional on successful completion of that verification, the arguments presented here would establish the two positivity endpoints of Demailly's invariant two-jet Riemann--Roch calculation: every very general surface in $\mathbb{P}^3$ of degree $15$ would be Kobayashi hyperbolic, and the complement of every general smooth plane curve of degree $11$ would be Kobayashi hyperbolic. At the logarithmic endpoint the same conditional argument would also give a Second Main Theorem with coefficient $1422$; the preceding work treated all degrees $d\geqslant12$. The argument separates an infinite geometric range from a finite algebraic obstruction.An $\mathcal{O}(3)$ slanted-vector-field package combines a quadratic acceleration with a classical matrix lift. Their incompatible actions on the invariant fibre ring, together with the common-twist, prime-ideal, and relative-family interfaces, exclude the dependence alternative for every weight $m\geqslant3$. This improves the differentiation range from $t\geqslant8$ to $t\geqslant4$ and leaves, beyond the twist-one ranges, only five compact and thirteen logarithmic systems. The residual systems are reduced by symmetry. A two-term Fermat perturbation carries an involution, so a hypothetical negatively twisted two-jet differential has an invariant or anti-invariant component. Imposing this eigencondition before elimination adds exact linear constraints without adding unknowns. The required full-column-rank statements are isolated in the Key Vanishing Lemmas as finite computational targets. This paper records their finite reduction and proof-to-certificate interface. Their exact computer-assisted verification is in progress.