Gluing affine torus actions via divisorial fans

Generalizing the passage from a fan to a toric variety, we provide a combinatorial approach to construct arbitrary effective torus actions on normal, algebraic varieties. Based on the notion of a "proper polyhedral divisor" introduced in earlier work, we develop the concept of a "divisorial fan" and show that these objects encode the equivariant gluing of affine varieties with torus action. We characterize separateness and completeness of the resulting varieties in terms of divisorial fans, and we study examples like C*-surfaces and projectivizations of (nonsplit) vector bundles over toric varieties.