{"paper":{"title":"Degrees $d \\geqslant \\big( \\sqrt{n}\\, \\log\\, n\\big)^n$ and $d \\geqslant \\big( n\\, \\log\\, n\\big)^n$ in the Conjectures of Green-Griffiths and of Kobayashi","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.DG"],"primary_cat":"math.AG","authors_text":"Joel Merker (LM-Orsay), The-Anh Ta (LM-Orsay)","submitted_at":"2019-01-13T19:40:02Z","abstract_excerpt":"Once first answers in any dimension to the Green-Griffiths and Kobayashi conjectures for generic algebraic hypersurfaces $\\mathbb{X}^{n-1} \\subset \\mathbb{P}^n(\\mathbb{C})$ have been reached, the principal goal is to decrease (to improve) the degree bounds, knowing that the `celestial' horizon lies near $d \\geqslant 2n$.\nFor Green-Griffiths algebraic degeneracy of entire holomorphic curves, we obtain: \\[ d \\,\\geqslant\\, \\big(\\sqrt{n}\\,{\\sf log}\\,n\\big)^n, \\] and for Kobayashi-hyperbolicity (constancy of entire curves), we obtain: \\[ d \\,\\geqslant\\, \\big(n\\,{\\sf log}\\,n\\big)^n. \\] The latter im"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.04042","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":""},"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"}