{"paper":{"title":"Toric degenerations and symplectic geometry of smooth projective varieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.SG","authors_text":"Kiumars Kaveh","submitted_at":"2015-08-03T05:33:00Z","abstract_excerpt":"Let $X$ be an $n$-dimensional smooth complex projective variety embedded in $\\mathbb{C}\\mathbb{P}^{N}$. We construct a smooth family $\\mathcal{X}$ over $\\mathbb{C}$ with an embedding in $\\mathbb{C}\\mathbb{P}^{N} \\times \\mathbb{C}$ whose generic fiber is $X$ and the special fiber is the torus $(\\mathbb{C}^*)^n$ sitting in $\\mathbb{C}\\mathbb{P}^{N}$ via a monomial embedding. We use this to show that if $\\omega$ is an integral K\\\"ahler form on $X$ then for any $\\epsilon > 0$ there is an open subset $U_\\epsilon \\subset X$ such that $vol(X \\setminus U_\\epsilon) < \\epsilon$ and $U_\\epsilon$ is sympl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.00316","kind":"arxiv","version":3},"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"}