{"paper":{"title":"Symmetries and stabilization for sheaves of vanishing cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.DG"],"primary_cat":"math.AG","authors_text":"Balazs Szendroi, Christopher Brav, Delphine Dupont, Dominic Joyce, Vittoria Bussi","submitted_at":"2012-11-14T10:07:28Z","abstract_excerpt":"Let $U$ be a smooth $\\mathbb C$-scheme, $f:U\\to\\mathbb A^1$ a regular function, and $X=$Crit$(f)$ the critical locus, as a $\\mathbb C$-subscheme of $U$. Then one can define the \"perverse sheaf of vanishing cycles\" $PV_{U,f}$, a perverse sheaf on $X$.\n  This paper proves four main results:\n  (a) Suppose $\\Phi:U\\to U$ is an isomorphism with $f\\circ\\Phi=f$ and $\\Phi\\vert_X=$id$_X$. Then $\\Phi$ induces an isomorphism $\\Phi_*:PV_{U,f}\\to PV_{U,f}$. We show that $\\Phi_*$ is multiplication by det$(d\\Phi\\vert_X)=1$ or $-1$.\n  (b) $PV_{U,f}$ depends up to canonical isomorphism only on $X^{(3)},f^{(3)}$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3259","kind":"arxiv","version":4},"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"}