{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-the-chern-numbers-of-the-generalised","title":"On the Chern numbers of the generalised Kummer varieties","arxiv_id":"math/0204197","date":"2002-04-15","proceeding":null,"authors":["Marc Arnold Nieper-Wisskirchen"],"abstract":"Let $A^{[[n]]}$ denote the $2(n - 1)$-dimensional generalised Kummer variety constructed from the abelian surface $A$. Further, let $X$ be an arbitrary smooth projective surface with $\\int_X c_1(X)^2 \\neq 0$, and $X^{[k]}$ the Hilbert scheme of zero-dimensional subschemes of $X$ of length $k$. We give a formula which expresses the value of any complex genus on $A^{[[n]]}$ in terms of Chern numbers of the varieties $X^{[k]}$. It is shown by Ellingsrud and Stroemme how to use Bott's residue formula to effectively calculate the Chern numbers of the Hilbert schemes $(\\IP^2)^{[k]}$ of points on the projective plane. Since $\\int_{\\IP^2} c_1(\\IP^2)^2 = 9 \\neq 0$ we can use these numbers and our formula to calculate the Chern numbers of the generalised Kummer varieties. A table with all Chern numbers of the generalised Kummer varieties $A^{[[n]]}$ for $n \\leq 8$ is included.","url_abs":"https://arxiv.org/abs/math/0204197v2","url_pdf":"https://arxiv.org/pdf/math/0204197v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"on-the-chern-numbers-of-the-generalised","repo_url":"https://github.com/8d1h/bott","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}