2023/10/17 by Donghoon Jang, Jang, Donghoon
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Commutative Algebra and Its Applications #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)
paper · pdf · doi:10.48550/arxiv.2310.11024
openalex publication_date 2023/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Tn 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 ℤ2, 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 \mathbbCP1 × \mathbbCP1. This implies that any two 4-dimensional complex torus manifolds are obtained from each other by equivariant blow up and down.