Four dimensional almost complex torus manifolds

Abstract

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 $\Delta$ of multi-fans and a graph $\Gamma$, which encode the data on the fixed point set. We find a necessary and sufficient condition for each of $\Delta$ and $\Gamma$. Moreover, we provide a minimal model and operations for each of $\Delta$ and $\Gamma$. 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 $\mathbb{Z}^2$, $\Delta$ to a family of minimal multi-fans that has unit vectors only, and $\Gamma$ to a minimal graph whose edges all have unit vectors as labels. As an application, if $M$ is complex, $\Delta$ is a fan and determines $M$, $\Gamma$ 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.