{"paper":{"title":"Holomorphic family of strongly pseudoconvex domains in a K\\\"ahler manifold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.CV","authors_text":"Sungmin Yoo, Young-Jun Choi","submitted_at":"2019-08-16T04:39:16Z","abstract_excerpt":"Let $p:X\\rightarrow Y$ be a surjective holomorphic mapping between K\\\"ahler manifolds. Let $D$ be a bounded smooth domain in $X$ such that every generic fiber $D_y:=D\\cap p^{-1}(y)$ for $y\\in Y$ is a strongly pseudoconvex domain in $X_y:=p^{-1}(y)$, which admits the complete K\\\"ahler-Einstein metric. This family of K\\\"ahler-Einstein metrics induces a smooth $(1,1)$-form $\\rho$ on $D$. In this paper, we prove that $\\rho$ is positive-definite on $D$ if $D$ is strongly pseudoconvex. We also discuss the extensioin of $\\rho$ as a positive current across singular fibers."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.05842","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/1908.05842/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"}