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As a corollary, we show that over a surface, the formula is true if and only if the Milnor number equals the Tjurina number at each singularity of $D$. We also show the Rimann-Roch type of formula is true if the Jacobian scheme of $D$ is nonsingular or a complete intersection."},"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":"1201.6110","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-01-30T06:20:27Z","cross_cats_sorted":[],"title_canon_sha256":"e111ba846f471f84907c42d75423233b763ef7459ba1bcb28baa539c28a95b25","abstract_canon_sha256":"54ea891ad8d1ddc406a194b0a6861577658458063f13b398e135cf84052fda77"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:36.768953Z","signature_b64":"tAyGBAxttyu40EQZxpTWXH2al1TtvrptMJbL52s64yVPPJZTjjk9QL0F6UR5rHtECTxWq2Wntn/tJKt8gcq2AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8c43d0bc4d0a057fd7c9beb0f9287588e13c075fb286ce74166b45a61429498d","last_reissued_at":"2026-05-18T00:32:36.768206Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:36.768206Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Chern Classes of Logarithmic Vector Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Xia Liao","submitted_at":"2012-01-30T06:20:27Z","abstract_excerpt":"Let $X$ be a nonsingular complex variety and $D$ a reduced effective divisor in $X$. 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