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We show that the quantization of the corresponding toric manifold $X_{P}$ in toric K\\\"ahler polarizations and in the toric real polarization are related by analytic continuation of Hamiltonian flows evaluated at time $t = \\sqrt{-1} s$. We relate the quantization of $X_{P}$ in two different toric K\\\"ahler polarizations by taking the time-$\\sqrt{-1} s$ Hamiltonian \"flow\" of strongly convex functions on the moment polytope $P$. By taking $s$ to infinity, we obtain the quantization of $X_{P}$ in the (singular) real toric polarization.\n  Recall that $X_{P}$ has an ope"},"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":"1411.2793","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-11-11T13:09:17Z","cross_cats_sorted":[],"title_canon_sha256":"555f037917fdc1b6e8e40dfe42315953eb16ea5f4e51f30458be6696e9777c40","abstract_canon_sha256":"3aba901cff7da53d185376d5ecc3aa4bf4da3a89f1305738a200fb9d2107035f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:37:53.029782Z","signature_b64":"0Q9PVHNoGY/cBmhmObxoTeXlHPBCf/VB/JzJ+SlRa18EW5ZM9QHZwXD17ETN7fubRAd+APEWapMCIlP7aeZhAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"028330891c2db6034f487037b8266460c5ca664df9a2ecf8f6d89d8b3d1e5bd9","last_reissued_at":"2026-05-18T02:37:53.028381Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:37:53.028381Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Complex symplectomorphisms and pseudo-K\\\"ahler islands in the quantization of toric manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jo\\~ao P. 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