{"paper":{"title":"A Symmetry Method for Key Vanishing Lemmas and Optimal $2$-Jet Thresholds for Hyperbolicity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Jianjun Liu, Lei Hou, Song-Yan Xie, Tao Cui","submitted_at":"2026-06-17T14:58:14Z","abstract_excerpt":"This manuscript reports on an ongoing project. We develop a new symmetry-based method for establishing vanishing results for negatively twisted invariant 2-jet differentials, both on generic surfaces in $\\mathbb{P}^3$ and on complements of generic plane curves. The approach improves upon the recent work of Hou--Huynh--Merker--Xie by using a two-term perturbation of Fermat-type equations that is invariant under an involution exchanging two coordinates. The induced linear action on the space of invariant 2-jet differentials, combined with an elementary eigenvector argument from the representatio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.19155","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.19155/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}