Cohomological $\chi $ -dependence of ring structure for the moduli of one-dimensional sheaves on $\mathbb {P}^2$
We prove that the cohomology rings of the moduli space $M_{d,\chi }$ of one-dimensional sheaves on the projective plane are not isomorphic for general different choices of the Euler characteristics. This stands in contrast to the $\chi $ -independence of the Betti numbers of these modul...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2024-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424000318/type/journal_article |
| _version_ | 1827300289513783296 |
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| author | Woonam Lim Miguel Moreira Weite Pi |
| author_facet | Woonam Lim Miguel Moreira Weite Pi |
| author_sort | Woonam Lim |
| collection | DOAJ |
| description | We prove that the cohomology rings of the moduli space
$M_{d,\chi }$
of one-dimensional sheaves on the projective plane are not isomorphic for general different choices of the Euler characteristics. This stands in contrast to the
$\chi $
-independence of the Betti numbers of these moduli spaces. As a corollary, we deduce that
$M_{d,\chi }$
are topologically different unless they are related by obvious symmetries, strengthening a previous result of Woolf distinguishing them as algebraic varieties. |
| first_indexed | 2024-04-24T15:53:03Z |
| format | Article |
| id | doaj.art-d299760bef184bdb85157b22de53181b |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-24T15:53:03Z |
| publishDate | 2024-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-d299760bef184bdb85157b22de53181b2024-04-01T10:52:44ZengCambridge University PressForum of Mathematics, Sigma2050-50942024-01-011210.1017/fms.2024.31Cohomological $\chi $ -dependence of ring structure for the moduli of one-dimensional sheaves on $\mathbb {P}^2$Woonam Lim0https://orcid.org/0000-0001-9488-0384Miguel Moreira1https://orcid.org/0000-0002-9896-3002Weite Pi2https://orcid.org/0009-0002-4151-1547Utrecht University – Department of Mathematics, Budapestlaan 6, 3584 CD Utrecht, The Netherlands; E-mail:Massachusetts Institute of Technology – Department of Mathematics, 77 Massachusetts Avenue, Cambridge MA, 02139, USA; E-mail:Yale University – Department of Mathematics, 219 Prospect St, New Haven, CT 06511, USAWe prove that the cohomology rings of the moduli space $M_{d,\chi }$ of one-dimensional sheaves on the projective plane are not isomorphic for general different choices of the Euler characteristics. This stands in contrast to the $\chi $ -independence of the Betti numbers of these moduli spaces. As a corollary, we deduce that $M_{d,\chi }$ are topologically different unless they are related by obvious symmetries, strengthening a previous result of Woolf distinguishing them as algebraic varieties.https://www.cambridge.org/core/product/identifier/S2050509424000318/type/journal_article14D2014C15 |
| spellingShingle | Woonam Lim Miguel Moreira Weite Pi Cohomological $\chi $ -dependence of ring structure for the moduli of one-dimensional sheaves on $\mathbb {P}^2$ Forum of Mathematics, Sigma 14D20 14C15 |
| title | Cohomological $\chi $ -dependence of ring structure for the moduli of one-dimensional sheaves on $\mathbb {P}^2$ |
| title_full | Cohomological $\chi $ -dependence of ring structure for the moduli of one-dimensional sheaves on $\mathbb {P}^2$ |
| title_fullStr | Cohomological $\chi $ -dependence of ring structure for the moduli of one-dimensional sheaves on $\mathbb {P}^2$ |
| title_full_unstemmed | Cohomological $\chi $ -dependence of ring structure for the moduli of one-dimensional sheaves on $\mathbb {P}^2$ |
| title_short | Cohomological $\chi $ -dependence of ring structure for the moduli of one-dimensional sheaves on $\mathbb {P}^2$ |
| title_sort | cohomological chi dependence of ring structure for the moduli of one dimensional sheaves on mathbb p 2 |
| topic | 14D20 14C15 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424000318/type/journal_article |
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