{"paper":{"title":"An obstruction to the existence of constant scalar curvature K\\\"ahler metrics","license":"","headline":"","cross_cats":["math.AG"],"primary_cat":"math.DG","authors_text":"J. Ross, R. P. Thomas","submitted_at":"2004-12-29T20:15:28Z","abstract_excerpt":"We prove that polarised manifolds that admit a constant scalar curvature K\\\"ahler (cscK) metric satisfy a condition we call slope semistability. That is, we define the slope $\\mu$ for a projective manifold and for each of its subschemes, and show that if $X$ is cscK then $\\mu(Z)\\le\\mu(X)$ for all subschemes $Z$.\n  This gives many examples of manifolds with K\\\"ahler classes which do not admit cscK metrics, such as del Pezzo surfaces and projective bundles. If $\\PP(E)\\to B$ is a projective bundle which admits a cscK metric in a rational K\\\"ahler class with sufficiently small fibres, then $E$ is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0412518","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0412518/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"}