Global SYZ mirror symmetry and homological mirror symmetry for principally polarized abelian varieties
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909554364645376 |
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| author | Azam, Haniya Cannizzo, Catherine Lee, Heather Liu, Chiu-Chu Melissa |
| author_facet | Azam, Haniya Cannizzo, Catherine Lee, Heather Liu, Chiu-Chu Melissa |
| contents | For any positive integer $g$, we introduce the moduli space $\mathcal{A}^F_g =[\mathcal{H}_g/P_g(\mathbb{Z})]$ parametrizing $g$-dimensional principally polarized abelian varieties $V_τ$ together with a Strominger-Yau-Zalsow (SYZ) fibration, where $τ\in \mathcal{H}_g$ is the genus-$g$ Seigel upper half space and $P_g(\mathbb{Z}) \subset \mathrm{Sp}(2g,\mathbb{Z})$ is the integral Siegel parabolic subgroup. We study global SYZ mirror symmetry over the global moduli $\mathcal{H}_g$ and $\mathcal{A}^F_g$, relating the B-model on $V_τ$ and the A-model on its mirror, a compact $2g$-dimensional torus $\mathbb{T}^{2g}$ equipped with a complexified symplectic form.
For each $V_τ$, we establish a homological mirror symmetry (HMS) result at the cohomological level over $\mathbb{C}$. This implies core HMS at the cohomological level over $\mathbb{C}$ and a graded $\mathbb{C}$-algebra isomorphism known as Seidel's mirror map. We study global HMS where Floer cohomology groups $HF^*(\hat{\ell}, \hat{\ell}')$ form coherent sheaves over a complex manifold parametrizing triples $(τ, \hat{\ell}, \hat{\ell}')$ where $τ\in \mathcal{H}_g$ defines a complexified symplectic form $ω_τ$ on $\mathbb{T}^{2g}$ and $\hat{\ell}$, $\hat{\ell} '$ are affine Lagrangian branes in $(\mathbb{T}^{2g}, ω_τ)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20948 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global SYZ mirror symmetry and homological mirror symmetry for principally polarized abelian varieties Azam, Haniya Cannizzo, Catherine Lee, Heather Liu, Chiu-Chu Melissa Symplectic Geometry Algebraic Geometry 53D37 (primary) 14J33 (secondary) For any positive integer $g$, we introduce the moduli space $\mathcal{A}^F_g =[\mathcal{H}_g/P_g(\mathbb{Z})]$ parametrizing $g$-dimensional principally polarized abelian varieties $V_τ$ together with a Strominger-Yau-Zalsow (SYZ) fibration, where $τ\in \mathcal{H}_g$ is the genus-$g$ Seigel upper half space and $P_g(\mathbb{Z}) \subset \mathrm{Sp}(2g,\mathbb{Z})$ is the integral Siegel parabolic subgroup. We study global SYZ mirror symmetry over the global moduli $\mathcal{H}_g$ and $\mathcal{A}^F_g$, relating the B-model on $V_τ$ and the A-model on its mirror, a compact $2g$-dimensional torus $\mathbb{T}^{2g}$ equipped with a complexified symplectic form. For each $V_τ$, we establish a homological mirror symmetry (HMS) result at the cohomological level over $\mathbb{C}$. This implies core HMS at the cohomological level over $\mathbb{C}$ and a graded $\mathbb{C}$-algebra isomorphism known as Seidel's mirror map. We study global HMS where Floer cohomology groups $HF^*(\hat{\ell}, \hat{\ell}')$ form coherent sheaves over a complex manifold parametrizing triples $(τ, \hat{\ell}, \hat{\ell}')$ where $τ\in \mathcal{H}_g$ defines a complexified symplectic form $ω_τ$ on $\mathbb{T}^{2g}$ and $\hat{\ell}$, $\hat{\ell} '$ are affine Lagrangian branes in $(\mathbb{T}^{2g}, ω_τ)$. |
| title | Global SYZ mirror symmetry and homological mirror symmetry for principally polarized abelian varieties |
| topic | Symplectic Geometry Algebraic Geometry 53D37 (primary) 14J33 (secondary) |
| url | https://arxiv.org/abs/2503.20948 |