Elliptic-Parabolic-Hyperbolic trichotomy in mathematics

introduction

• spherical-Euclidean-hyperbolic
• finite-affine-indefinite

algebraic geometry

• Let $X$ be a smooth complex projective variety. There are three main types of varieties.
• Not every variety is of one of these three types, but minimal model theory relates every variety to one of these extreme types

1. Fano. This means that $−K_X$ is ample. (We recall the definition of ampleness in section 2.)
2. Calabi-Yau. We define this to mean that $K_X$ is numerically trivial.
3. ample canonical bundle. This means that $K_X$ is ample; it implies that $X$ is of general type.”

• Here, for $X$ of complex dimension $n$, the canonical bundle $K_X$ is the line bundle $\Omega^n_X$ of $n$-forms.
• We write $−K_X$ for the dual line bundle $K^∗_X$, the determinant of the tangent bundle.

memo

• http://mathoverflow.net/questions/120612/trichotomies-in-mathematics

articles

• Rastegar, Arash. ‘EPH-Classifications in Geometry, Algebra, Analysis and Arithmetic’. arXiv:1503.07859 [math], 26 March 2015. http://arxiv.org/abs/1503.07859.
• Totaro, Burt. ‘Algebraic Surfaces and Hyperbolic Geometry’. ArXiv E-Prints 1008 (1 August 2010): 3825.