anabelian geometry

Noun

anabelian geometry (countable and uncountable, plural anabelian geometries)

1. A theory which describes the way in which the algebraic fundamental group G of an algebraic variety (or some related geometric object) V determines how V can be mapped into another geometric object W, under the assumption that G is very far from being abelian (commutative).
   • We shall say more about the noncommutative nature of anabelian geometries after we state the main theorem (Theorem 1.17 below) on these geometries.1999, Shinichi Mochizuki, Foundations of p-adic Teichmüller Theory, American Mathematical Society, page 63
   • The basic idea of Grothendieck's anabelian geometry is that under certain "anabelian hypotheses" the étale fundamental group of a scheme contains all the geometric and arithmetic information about the scheme in discussion, that is to say, the scheme is functorially encoded in its étale fundamental...
   • Among these we shall discuss applications to the inverse Galois problem, Belyi's theorem on covers of the projective line minus three points and some advanced results on ‘anabelian geometry’ of curves.2000, Florian Pop, “Alterations and Birational Anabelian Geometry”, in H. Hauser, J. Lipman, F. Oort, A. Quirós, editors, Resolution of Singularities, Birkhäuser, page 519