Nonrational polytopes and fans in toric geometry

Abstract

In this survey, we first examine the notion of nonrational polytope and nonrational fan in the context of toric geometry. We then discuss and interrelate some recent developments in the subject.

Toric Quasifolds

“…We would like to point out that most of the alternative nonrational toric viewpoints are founded on variations on the theme of the fundamental triple (2), beginning, first and foremost, with the quasilattice Q. In joint work with Battaglia [6], we describe in detail how many of these different perspectives connect with each other and with ours; a dictionary is provided, in the hope that it will offer clarity and facilitate future interaction in the field.…”

“…1The starting point in the algebraic and complex category is actually, more generally, a fan instead of a polytope, but the basic idea that follows applies verbatim (see [6]).…”