{"paper":{"title":"The Calabi-Yau equation for $T^2$-bundles over $\\mathbb{T}^2$: the non-Lagrangian case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Anna Fino, Ernesto Buzano, Luigi Vezzoni","submitted_at":"2012-01-13T14:21:04Z","abstract_excerpt":"In the spirit of [10,2], we study the Calabi-Yau equation on $T^2$-bundles over $\\mathbb{T}^2$ endowed with an invariant non-Lagrangian almost-K\\\"ahler structure showing that for $T^2$-invariant initial data it reduces to a Monge-Amp\\`ere equation having a unique solution. In this way we prove that for every total space $M^4$ of an orientable $T^2$-bundle over $\\mathbb{T}^2$ endowed with an invariant almost-K\\\"ahler structure the Calabi-Yau problem has a solution for every normalized $T^2$-invariant volume form."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.2846","kind":"arxiv","version":2},"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"}