Elliptic-Parabolic-Hyperbolic trichotomy in mathematics
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introduction
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
- Fano. This means that \(−K_X\) is ample. (We recall the definition of ampleness in section 2.)
- Calabi-Yau. We define this to mean that \(K_X\) is numerically trivial.
- 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
articles
- Terragni, T. “Data about Hyperbolic Coxeter Systems.” arXiv:1503.08764 [math], March 30, 2015. http://arxiv.org/abs/1503.08764.
- 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.