Which 4-manifolds are toric varieties? (Q1323392)

We present a topological definition of (compact) toric varieties due to R. MacPherson. We use it to classify nonsingular 4-dimensional toric varieties, first in the topological category by explicitly calculating the intersection form and applying Freedman's classification theorem, and then in the smooth category using a handlebody decomposition and Kirby calculus. We conclude in both cases that the different toric varieties are the complex projective plane \(\mathbb{C} P^ 2\), the product of spheres \(S^ 2 \times S^ 2\) and the connected sum of \(\mathbb{C} P^ 2\) with a finite number of \(-\mathbb{C} P^ 2\). These results could also be deduced from a theorem of Oda which uses techniques from algebraic geometry and combinatorial properties of regular fans in \(\mathbb{R}^ 2\).