Quantum spaces associated to mixed polarizations and their limiting behavior on toric varieties
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914984120811520 |
|---|---|
| author | Wang, Dan |
| author_facet | Wang, Dan |
| contents | Let $(X, ω, J)$ be a toric variety of dimension $2n$ determined by a Delzant polytope $P$. As indicated in [40], $X$ admits a natural mixed polarization $\mathcal{P}_{k}$, induced by the action of a subtorus $T^{k}$. In this paper, we first establish the quantum space $\mathcal{H}_{k}$ for $\mathcal{P}_{k}$, identifying a basis parameterized by the integer lattice points of $P$. This confirms that the dimension of $\mathcal{H}_{k}$ aligns with those derived from Kähler and real polarizations. Secondly, we examine a one-parameter family of Kähler polarizations $\mathcal{P}_{k,t}$, defined via symplectic potentials, and demonstrate their convergence to $\mathcal{P}_{k}$. Thirdly, we verify that these polarizations $\mathcal{P}_{k,t}$ coincide with those induced by imaginary-time flow. Finally, we explore the relationship between the quantum space $\mathcal{H}_{k,0}$ and $\mathcal{H}_{k}$, establishing that ``$\lim_{t \rightarrow \infty} \mathcal{H}_{k,t} = \mathcal{H}_{k}$." |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_17130 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum spaces associated to mixed polarizations and their limiting behavior on toric varieties Wang, Dan Symplectic Geometry Mathematical Physics Differential Geometry Let $(X, ω, J)$ be a toric variety of dimension $2n$ determined by a Delzant polytope $P$. As indicated in [40], $X$ admits a natural mixed polarization $\mathcal{P}_{k}$, induced by the action of a subtorus $T^{k}$. In this paper, we first establish the quantum space $\mathcal{H}_{k}$ for $\mathcal{P}_{k}$, identifying a basis parameterized by the integer lattice points of $P$. This confirms that the dimension of $\mathcal{H}_{k}$ aligns with those derived from Kähler and real polarizations. Secondly, we examine a one-parameter family of Kähler polarizations $\mathcal{P}_{k,t}$, defined via symplectic potentials, and demonstrate their convergence to $\mathcal{P}_{k}$. Thirdly, we verify that these polarizations $\mathcal{P}_{k,t}$ coincide with those induced by imaginary-time flow. Finally, we explore the relationship between the quantum space $\mathcal{H}_{k,0}$ and $\mathcal{H}_{k}$, establishing that ``$\lim_{t \rightarrow \infty} \mathcal{H}_{k,t} = \mathcal{H}_{k}$." |
| title | Quantum spaces associated to mixed polarizations and their limiting behavior on toric varieties |
| topic | Symplectic Geometry Mathematical Physics Differential Geometry |
| url | https://arxiv.org/abs/2410.17130 |