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Such varieties are called fake products of projective lines or fake $(\\mathbb P^1)^n$. These are higher dimensional analogs of fake quadrics. In this paper we show that the number of fake $(\\mathbb P^1)^n$ is finite (independently of $n$), we give examples of fake $(\\mathbb P^1)^4$ and show that for $n>4$ there are no fake $(\\"},"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":"1411.3384","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-11-12T22:39:18Z","cross_cats_sorted":[],"title_canon_sha256":"403c8fcb69bfccb6da05bec1ccded4de2b197e9f4340d676da11f58989d5958d","abstract_canon_sha256":"c80558de5cb27246754208251366160861537844dd6a743ad2fdbdfe4190a302"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:37:43.916376Z","signature_b64":"p9Cx43+7XARnmZfmq3ymHCoJffVWRXHSoQc5fHrI8Ubo/yd8oMTGh+/KjoYBYYSH/e8U2Uc7GeNTbNE5GkttBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ea60e4974c62c727a7bea7486fe229905cbdc3ceb87408e1fe17784f3dda1d9b","last_reissued_at":"2026-05-18T02:37:43.915950Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:37:43.915950Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Varieties of general type with the same Betti numbers as $\\mathbb P^1\\times \\mathbb P^1\\times\\ldots\\times \\mathbb P^1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Amir D\\v{z}ambi\\'c","submitted_at":"2014-11-12T22:39:18Z","abstract_excerpt":"We study quotients $\\Gamma\\backslash \\mathbb H^n$ of the $n$-fold product of the upper half plane $\\mathbb H$ by irreducible and torsion-free lattices $\\Gamma < PSL_2(\\mathbb R)^n$ with the same Betti numbers as the $n$-fold product $(\\mathbb P^1)^n$ of projective lines. 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