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A necessary condition is that the categorical quotients $Q_X$ and $Q_Y$ are biholomorphic and that the biholomorphism $\\phi$ sends the Luna strata of $Q_X$ isomorphically onto the corresponding Luna strata of $Q_Y$. Fix $\\phi$. We demonstrate two homotopy principles in this situation. The first result says that if there is a $G$-diffeomorphism $\\Phi\\colon X\\to Y$, i"},"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":"1503.00797","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-03-03T01:19:39Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"97a374635aa14f4b30ffe0375d529e21484afda4059a15c8e54099cefa3168f4","abstract_canon_sha256":"76444f78079e27d795b1a8db47b08f3034988c4ac3291541bc1434b6bfc81c6f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:16.573005Z","signature_b64":"bsKW1J0ddXBiBahGY2BPCMRLOgXPOO4ofMvLmkcQvHwZ2IB0KXuO9APaDj9jl92VwKlj7D1M28Ze35mC3iONDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9d88fa4312ca1905e22fd519b94a333eb079aa9d48a3aaccae32c77d6111cf78","last_reissued_at":"2026-05-18T00:45:16.572489Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:16.572489Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Homotopy principles for equivariant isomorphisms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.CV","authors_text":"Finnur Larusson, Frank Kutzschebauch, Gerald W. Schwarz","submitted_at":"2015-03-03T01:19:39Z","abstract_excerpt":"Let $G$ be a reductive complex Lie group acting holomorphically on Stein manifolds $X$ and $Y$. Let $p_X\\colon X\\to Q_X$ and $p_Y\\colon Y\\to Q_Y$ be the quotient mappings. When is there an equivariant biholomorphism of $X$ and $Y$? A necessary condition is that the categorical quotients $Q_X$ and $Q_Y$ are biholomorphic and that the biholomorphism $\\phi$ sends the Luna strata of $Q_X$ isomorphically onto the corresponding Luna strata of $Q_Y$. Fix $\\phi$. We demonstrate two homotopy principles in this situation. 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