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Given $q\\in\\set{0,1,\\ldots,n-1}$, let $\\Box^{(q)}_{b,k}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$ forms with values in $L^k$. For $\\lambda\\geq0$, let $\\Pi^{(q)}_{k,\\leq\\lambda}:=E((-\\infty,\\lambda])$, where $E$ denotes the spectral measure of $\\Box^{(q)}_{b,k}$. In this work, we prove that $\\Pi^{(q)}_{k,\\leq k^{-N_0}}F^*_k$, $F_k\\Pi^{(q)}_{k,\\leq k^{-N_0}}F^*_k$, $N_0\\geq1$, admit asymptotic ex"},"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":"1401.6647","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2014-01-26T13:33:37Z","cross_cats_sorted":["math.AP","math.DG"],"title_canon_sha256":"2aa7fb1c417a69e39ee95994171a4eadfb0c88f4bbb034c8cced2d616b27a037","abstract_canon_sha256":"6c9f0455ba231b212fc366876f63ab97354993391607af338f73533d16510d27"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:57.198033Z","signature_b64":"lqCsC20DvS1BHKDCu1wcKO5XBvdfknMYXS9kl74KB20HMn4nPn2OBNFg78ATCy+Bwbdr7z6VJSNUHXBfB6EuDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b65bb867f93ea2124cf2a8ff7bc4617ad7fb77ef6eb6e89ebe618441cdd38cf","last_reissued_at":"2026-05-18T00:39:57.197336Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:57.197336Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Szeg\\\"o kernel asymptotics for high power of CR line bundles and Kodaira embedding theorems on CR manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.DG"],"primary_cat":"math.CV","authors_text":"Chin-Yu Hsiao","submitted_at":"2014-01-26T13:33:37Z","abstract_excerpt":"Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n\\geqslant2$, and let $L^k$ be the $k$-th tensor power of a CR complex line bundle $L$ over $X$. 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