Lagrangian Intersections, Symplectic Reduction and Kirwan Surjectivity
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912899402825728 |
|---|---|
| author | Leung, Naichung Conan Xie, Ying Yau, Yu Tung |
| author_facet | Leung, Naichung Conan Xie, Ying Yau, Yu Tung |
| contents | Given a smooth holomorphic symplectic variety $X$ with a Hamiltonian $G$-action, $G$-invariant Lagrangians $C's$ induce Lagrangians in the symplectic quotient $X// G$. Given clean intersections $B=C_1\cap C_2$ whose conormal sequence splits, we show that $$C_1/G\times_{X// G} C_2/G\cong T^{\vee}[-1](B/G).$$ When $det(N_{B/C_2})$ is torsion, we have $Ext^{\bullet}_{X// G}(\mathcal{O}_{C_1/G}, \mathcal{O}_{C_2/G})\cong H^{\bullet}_G(B, det(N_{B/C_2})_δ)$ provided that the Hodge-to-de Rham degeneracy holds. Furthermore, we have a generalized version of Kirwan surjectivity $Ext^{\bullet}_{X// G}(\mathcal{O}_{C_1/G}, \mathcal{O}_{C_2/G})\twoheadrightarrow Ext^{\bullet}_{X^{ss}// G}(\mathcal{O}_{C_1^{ss}/G}, \mathcal{O}_{C_2^{ss}/G})$ if $B$ is proper. When $C_1=C_2$, this is the Kirwan surjectivity, which is now interpreted as the symmetry commutes with reduction problem in 3d B-model. We also obtain similar results for $K_{C_1/G}^{1/2}$ and $K_{C_2/G}^{1/2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_11718 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lagrangian Intersections, Symplectic Reduction and Kirwan Surjectivity Leung, Naichung Conan Xie, Ying Yau, Yu Tung Algebraic Geometry Symplectic Geometry 14F08, 18G15, 53D12, 53D20 Given a smooth holomorphic symplectic variety $X$ with a Hamiltonian $G$-action, $G$-invariant Lagrangians $C's$ induce Lagrangians in the symplectic quotient $X// G$. Given clean intersections $B=C_1\cap C_2$ whose conormal sequence splits, we show that $$C_1/G\times_{X// G} C_2/G\cong T^{\vee}[-1](B/G).$$ When $det(N_{B/C_2})$ is torsion, we have $Ext^{\bullet}_{X// G}(\mathcal{O}_{C_1/G}, \mathcal{O}_{C_2/G})\cong H^{\bullet}_G(B, det(N_{B/C_2})_δ)$ provided that the Hodge-to-de Rham degeneracy holds. Furthermore, we have a generalized version of Kirwan surjectivity $Ext^{\bullet}_{X// G}(\mathcal{O}_{C_1/G}, \mathcal{O}_{C_2/G})\twoheadrightarrow Ext^{\bullet}_{X^{ss}// G}(\mathcal{O}_{C_1^{ss}/G}, \mathcal{O}_{C_2^{ss}/G})$ if $B$ is proper. When $C_1=C_2$, this is the Kirwan surjectivity, which is now interpreted as the symmetry commutes with reduction problem in 3d B-model. We also obtain similar results for $K_{C_1/G}^{1/2}$ and $K_{C_2/G}^{1/2}$. |
| title | Lagrangian Intersections, Symplectic Reduction and Kirwan Surjectivity |
| topic | Algebraic Geometry Symplectic Geometry 14F08, 18G15, 53D12, 53D20 |
| url | https://arxiv.org/abs/2602.11718 |