"Elliptic-Parabolic-Hyperbolic trichotomy in mathematics"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
imported>Pythagoras0 |
imported>Pythagoras0 |
||
1번째 줄: | 1번째 줄: | ||
+ | ==introduction== | ||
+ | * spherical-Euclidean-hyperbolic | ||
+ | * finite-affine-indefinite | ||
+ | * {{수학노트|url=리만_사상_정리_Riemann_mapping_theorem_and_the_uniformization_theorem}} | ||
+ | * {{수학노트|url=이차곡선(원뿔곡선)}} | ||
+ | |||
+ | ==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== | ||
+ | * http://mathoverflow.net/questions/120612/trichotomies-in-mathematics | ||
+ | |||
+ | |||
+ | ==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. | ||
+ | |||
+ | |||
+ | |||
+ | [[분류:abstract concepts]] | ||
+ | [[분류:migrate]] |
2020년 11월 13일 (금) 07:53 판
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.