Four dimensional almost complex torus manifolds
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866908334997635072 |
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| author | Jang, Donghoon |
| author_facet | Jang, Donghoon |
| contents | In dimension 4, we extend the correspondence between compact nonsingular toric varieties and regular fans to a correspondence between almost complex torus manifolds and families of multi-fans in a geometric way, where an (almost) complex torus manifold is a $2n$-dimensional compact connected (almost) complex manifold equipped with an effective action of a real $n$-dimensional torus $T^n$ that has fixed points.
Let $M$ be a 4-dimensional almost complex torus manifold. To $M$, we associate two equivalent combinatorial objects, a family $Δ$ of multi-fans and a graph $Γ$, which encode the data on the fixed point set. We find a necessary and sufficient condition for each of $Δ$ and $Γ$.
Moreover, we provide a minimal model and operations for each of $Δ$ and $Γ$. We introduce operations on a multi-fan and a graph that correspond to blow up and down of a manifold, and show that we can blow up and down $M$ to a minimal manifold $M'$ whose weights at the fixed points are unit vectors in $\mathbb{Z}^2$, $Δ$ to a family of minimal multi-fans that has unit vectors only, and $Γ$ to a minimal graph whose edges all have unit vectors as labels.
As an application, if $M$ is complex, $Δ$ is a fan and determines $M$, $Γ$ encodes the equivariant cohomology of $M$, and $M'$ is $\mathbb{CP}^1 \times \mathbb{CP}^1$. This implies that any two 4-dimensional complex torus manifolds are obtained from each other by equivariant blow up and down. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11024 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Four dimensional almost complex torus manifolds Jang, Donghoon Differential Geometry Algebraic Geometry Algebraic Topology Geometric Topology In dimension 4, we extend the correspondence between compact nonsingular toric varieties and regular fans to a correspondence between almost complex torus manifolds and families of multi-fans in a geometric way, where an (almost) complex torus manifold is a $2n$-dimensional compact connected (almost) complex manifold equipped with an effective action of a real $n$-dimensional torus $T^n$ that has fixed points. Let $M$ be a 4-dimensional almost complex torus manifold. To $M$, we associate two equivalent combinatorial objects, a family $Δ$ of multi-fans and a graph $Γ$, which encode the data on the fixed point set. We find a necessary and sufficient condition for each of $Δ$ and $Γ$. Moreover, we provide a minimal model and operations for each of $Δ$ and $Γ$. We introduce operations on a multi-fan and a graph that correspond to blow up and down of a manifold, and show that we can blow up and down $M$ to a minimal manifold $M'$ whose weights at the fixed points are unit vectors in $\mathbb{Z}^2$, $Δ$ to a family of minimal multi-fans that has unit vectors only, and $Γ$ to a minimal graph whose edges all have unit vectors as labels. As an application, if $M$ is complex, $Δ$ is a fan and determines $M$, $Γ$ encodes the equivariant cohomology of $M$, and $M'$ is $\mathbb{CP}^1 \times \mathbb{CP}^1$. This implies that any two 4-dimensional complex torus manifolds are obtained from each other by equivariant blow up and down. |
| title | Four dimensional almost complex torus manifolds |
| topic | Differential Geometry Algebraic Geometry Algebraic Topology Geometric Topology |
| url | https://arxiv.org/abs/2310.11024 |