Differential graded categories in holomorphic symplectic geometry
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911574609887232 |
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| author | Mladenov, Borislav |
| author_facet | Mladenov, Borislav |
| contents | Let $(\mathrm{X},σ)$ be a holomorphic symplectic manifold. We study the differential graded category of canonical Lagrangian $\mathrm{D}$-branes $\mathcal{D}_\mathrm{Lag}(\mathrm{X},σ)$ along with its deformation quantisation, spanned by quantised orientations, $\mathcal{DQ}(\mathrm{X},σ)$, and the virtual de Rham category $\mathcal{DR}^{\mathrm{vir}}(\mathrm{X},σ)$. We prove the formality of these dg categories when localised at a countable collection of orientable compact Kähler Lagrangian submanifolds with pairwise clean intersections. Along the way, we define Kaledin classes of minimal $\mathrm{A}_\infty$-categories and show that they are the obstructions to formality. In addition, we obtain a formality criterion for flat weakly proper Calabi-Yau dg categories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06630 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Differential graded categories in holomorphic symplectic geometry Mladenov, Borislav Algebraic Geometry Quantum Algebra Representation Theory 14A22, 14F08, 14J42, 53D55 Let $(\mathrm{X},σ)$ be a holomorphic symplectic manifold. We study the differential graded category of canonical Lagrangian $\mathrm{D}$-branes $\mathcal{D}_\mathrm{Lag}(\mathrm{X},σ)$ along with its deformation quantisation, spanned by quantised orientations, $\mathcal{DQ}(\mathrm{X},σ)$, and the virtual de Rham category $\mathcal{DR}^{\mathrm{vir}}(\mathrm{X},σ)$. We prove the formality of these dg categories when localised at a countable collection of orientable compact Kähler Lagrangian submanifolds with pairwise clean intersections. Along the way, we define Kaledin classes of minimal $\mathrm{A}_\infty$-categories and show that they are the obstructions to formality. In addition, we obtain a formality criterion for flat weakly proper Calabi-Yau dg categories. |
| title | Differential graded categories in holomorphic symplectic geometry |
| topic | Algebraic Geometry Quantum Algebra Representation Theory 14A22, 14F08, 14J42, 53D55 |
| url | https://arxiv.org/abs/2604.06630 |