"Elliptic-Parabolic-Hyperbolic trichotomy in mathematics"의 두 판 사이의 차이

수학노트
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imported>Pythagoras0
imported>Pythagoras0
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==introduction==
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* spherical-Euclidean-hyperbolic
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* finite-affine-indefinite
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* {{수학노트|url=리만_사상_정리_Riemann_mapping_theorem_and_the_uniformization_theorem}}
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* {{수학노트|url=이차곡선(원뿔곡선)}}
  
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==algebraic geometry==
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* Let $X$ be a smooth complex projective variety. There are three main types of varieties.
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* Not every variety is of one of these three types, but minimal model theory relates every variety to one of these extreme types
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# Fano. This means that $−K_X$ is ample. (We recall the definition of ampleness in section 2.)
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# Calabi-Yau. We define this to mean that $K_X$ is numerically trivial.
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# ample canonical bundle. This means that $K_X$ is ample; it implies that $X$ is of general type.”
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* Here, for $X$ of complex dimension $n$, the canonical bundle $K_X$ is the line bundle $\Omega^n_X$ of $n$-forms.
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* We write $−K_X$ for the dual line bundle $K^∗_X$, the determinant of the tangent bundle.
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==memo==
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* http://mathoverflow.net/questions/120612/trichotomies-in-mathematics
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==articles==
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* Terragni, T. “Data about Hyperbolic Coxeter Systems.” arXiv:1503.08764 [math], March 30, 2015. http://arxiv.org/abs/1503.08764.
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* Rastegar, Arash. ‘EPH-Classifications in Geometry, Algebra, Analysis and Arithmetic’. arXiv:1503.07859 [math], 26 March 2015. http://arxiv.org/abs/1503.07859.
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* Totaro, Burt. ‘Algebraic Surfaces and Hyperbolic Geometry’. ArXiv E-Prints 1008 (1 August 2010): 3825.
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[[분류:abstract concepts]]
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[[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
  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


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.