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We prove that the group of $LPGL(2,\\mathbb{C})$ invariant holomorphic line bundles on $L\\mathbb{P}_1$ is isomorphic to an infinite dimensional Lie group. Further, we prove that the space of holomorphic sections of these bundles is finite dimensional, and compute the dimension for a generic bundle."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/0210017","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CV","submitted_at":"2002-10-02T01:16:17Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"3624e4c7edb2b3ea26ab388e85e19188db56117516a4e0821741f8e2a2afcf02","abstract_canon_sha256":"86161d18606e605ffd1df9491597948e8a181fe5ff1b72e30c2c679de8d82577"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:03:07.928790Z","signature_b64":"UaIAtuZXT5QnZEcObf4vZNjc3QQN39QfZ775FYP8Wzm5CVniEJmBrNyEBpmfiiEaDn4Zqz+jHIbq21YjfJSZBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"98df4ee19fc5d921337d3bad0410d7a93773ea78769bc7ca3e6b8c8cebc7a834","last_reissued_at":"2026-07-05T04:03:07.928313Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:03:07.928313Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Holomorphic line bundles on the loop space of the Riemann sphere","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.CV","authors_text":"Ning Zhang","submitted_at":"2002-10-02T01:16:17Z","abstract_excerpt":"The loop space $L\\mathbb{P}_1$ of the Riemann sphere consisting of all $C^k$ or Sobolev $W^{k,p}$ maps from the circle $S^1$ to $\\mathbb{P}_1$ is an infinite dimensional complex manifold. 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