Rationality and symmetry of stable pairs generating series of Fano 3-folds
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913012033519616 |
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| author | Karpov, Ivan Moreira, Miguel |
| author_facet | Karpov, Ivan Moreira, Miguel |
| contents | The generating series of descendent invariants of stable pairs on 3-folds is conjectured to be rational and to satisfy a $q\leftrightarrow q^{-1}$ symmetry. We prove this conjecture for Fano 3-folds. We utilize the same path of stability conditions that Toda used in his proof of the Calabi--Yau version of the conjecture, relating stable pairs and $L$ invariants, and work of the two authors that allows an extension of Joyce's descendent wall-crossing formula to non-standard hearts of $D^b(X)$. We use Ehrhart theory to deal with the combinatorics coming out of the wall-crossing formula. Furthermore, we specialize the wall-crossing formula to primary insertions and prove a strong rationality result predicted by the Pandharipande--Thomas/Gopakumar--Vafa correspondence. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_06023 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rationality and symmetry of stable pairs generating series of Fano 3-folds Karpov, Ivan Moreira, Miguel Algebraic Geometry Combinatorics 14N35, 05E45 The generating series of descendent invariants of stable pairs on 3-folds is conjectured to be rational and to satisfy a $q\leftrightarrow q^{-1}$ symmetry. We prove this conjecture for Fano 3-folds. We utilize the same path of stability conditions that Toda used in his proof of the Calabi--Yau version of the conjecture, relating stable pairs and $L$ invariants, and work of the two authors that allows an extension of Joyce's descendent wall-crossing formula to non-standard hearts of $D^b(X)$. We use Ehrhart theory to deal with the combinatorics coming out of the wall-crossing formula. Furthermore, we specialize the wall-crossing formula to primary insertions and prove a strong rationality result predicted by the Pandharipande--Thomas/Gopakumar--Vafa correspondence. |
| title | Rationality and symmetry of stable pairs generating series of Fano 3-folds |
| topic | Algebraic Geometry Combinatorics 14N35, 05E45 |
| url | https://arxiv.org/abs/2604.06023 |