  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.

Differential Geometry;Algebraic Geometry;Algebraic Topology;Geometric Topology