Atoms meet symbols
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915904731742208 |
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| author | Cavenaghi, Leonardo F. Katzarkov, Ludmil Kontsevich, Maxim |
| author_facet | Cavenaghi, Leonardo F. Katzarkov, Ludmil Kontsevich, Maxim |
| contents | This paper introduces a novel framework for constructing invariants in $G$-equivariant birational geometry by unifying two recent approaches: the theory of atoms recently developed by Katzarkov, Kontsevich, Pantev, and Yu, and the theory of modular symbols due to Kontsevich, Tschinkel, and Pestun.
We initiate the theory of Chen-Ruan atoms. Assuming the blowup formula for the quantum Chen-Ruan cohomology, we outline how to extend the theory of atoms to global quotient orbifolds and present some striking applications.
In addition, we develop a separate class of purely geometric invariants for $\mathbb{Z}/2$- and $\mathbb{Z}/3$-actions on surfaces and threefolds.
We provide many examples of non-$G$-linearizable $G$-actions on projective varieties treated with these new techniques. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_15831 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Atoms meet symbols Cavenaghi, Leonardo F. Katzarkov, Ludmil Kontsevich, Maxim Algebraic Geometry Differential Geometry Symplectic Geometry This paper introduces a novel framework for constructing invariants in $G$-equivariant birational geometry by unifying two recent approaches: the theory of atoms recently developed by Katzarkov, Kontsevich, Pantev, and Yu, and the theory of modular symbols due to Kontsevich, Tschinkel, and Pestun. We initiate the theory of Chen-Ruan atoms. Assuming the blowup formula for the quantum Chen-Ruan cohomology, we outline how to extend the theory of atoms to global quotient orbifolds and present some striking applications. In addition, we develop a separate class of purely geometric invariants for $\mathbb{Z}/2$- and $\mathbb{Z}/3$-actions on surfaces and threefolds. We provide many examples of non-$G$-linearizable $G$-actions on projective varieties treated with these new techniques. |
| title | Atoms meet symbols |
| topic | Algebraic Geometry Differential Geometry Symplectic Geometry |
| url | https://arxiv.org/abs/2509.15831 |