SHERP

Toric Hirzebruch-Riemann-Roch
토릭 힐체브루흐-리만-로흐

DC Field Value Language
dc.contributor.advisorAtanas Iliev-
dc.contributor.authorLee, Jae Hwang-
dc.date.accessioned2018-12-03T01:56:26Z-
dc.date.available2018-12-03T01:56:26Z-
dc.date.issued2018-
dc.identifier.other000000153123-
dc.identifier.urihttp://hdl.handle.net/10371/144242-
dc.description학위논문 (석사)-- 서울대학교 대학원 : 자연과학대학 수리과학부, 2018. 8. Atanas Iliev.-
dc.description.abstract이 논문에서 우리의 주 목적은 매끄러운 완전 토릭 다양체에 대한 힐체부르흐-리만-로흐

정리이다. 이것을 위해 우리는 아핀 토릭 다양체와 그것을 어떻게 만드는지부터 출발한다.

또한, 사영 혹은 추상 토릭 다양체들이, 그것들의 팬과 궤도 대응을 포함하여 설명될 것이다.

다음으로, 우리는 간단하게 디바이저, 선 다발, 그리고 토릭 다양체에 대한 코호몰로지를 탐

험한다. 마지막으로, 우리는 등변 버전의 힐체브루흐-리만-로흐 정리를 작업한 뒤, 브리온의

등식을 이용하고 비등변 극한을 취하여 힐체브루흐-리만-로흐 정리를 증명한다.
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dc.description.abstractIn this thesis, our main goal is the Hirzebruch-Riemann-Roch theorem for smooth

complete toric varieties. To do this, we start from affine toric varieties and how to construct

them. Also, projective and abstract toric varieties will be introduced, including

their fans and the orbit-correspondence. Next, we briefly explore divisors, line bundles,

and cohomology for toric varieties. Finally, after working the equivariant version of the

Hirzebruch-Riemann-Roch, using Brion’s equality and taking the nonequivariant limit,

we prove the Hirzebruch-Riemann-Roch theorem.
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dc.description.tableofcontents1 Introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 History of toric varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Affine Toric varieties 8

2.1 Algebraic Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Affine Toric Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Cones and Affine Toric Varieties . . . . . . . . . . . . . . . . . . . . . . . 15

3 Projective Toric Varieties 20

3.1 Projective Toric Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Polytopes and Projective Toric Varieties . . . . . . . . . . . . . . . . . . . 23

4 The Toric variety of a Fan 26

4.1 Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 The Orbit-Cone Correspondence . . . . . . . . . . . . . . . . . . . . . . . 28

5 Divisors and Line Bundles on Toric Varieties 30

5.1 Divisors on Toric Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2 The Sheaf of a Torus-Invariant Divisor . . . . . . . . . . . . . . . . . . . . 34

5.3 Line bundles on Toric Varieties . . . . . . . . . . . . . . . . . . . . . . . . 35

6 Cohomology and Topology of Toric Varieties 36

6.1 Decomposing Cohomology of Toric Varieties . . . . . . . . . . . . . . . . . 36

6.2 Vanishing Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.3 The Cohomology Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7 Toric Hirzebruch-Riemann-Roch 43

7.1 Chern Characters and Todd Classes . . . . . . . . . . . . . . . . . . . . . 43

7.2 Brions Equalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7.3 Toric Equivariant Riemannn-Roch . . . . . . . . . . . . . . . . . . . . . . 50

7.4 Applications and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 59

A Homological algebra 63

A.1 Gysin maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

A.2 Spectral Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

A.3 Serre Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

The bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

국문초록 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
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dc.formatapplication/pdf-
dc.format.mediumapplication/pdf-
dc.language.isoen-
dc.publisher서울대학교 대학원-
dc.subject.ddc510-
dc.titleToric Hirzebruch-Riemann-Roch-
dc.title.alternative토릭 힐체브루흐-리만-로흐-
dc.typeThesis-
dc.contributor.AlternativeAuthor이재황-
dc.description.degreeMaster-
dc.contributor.affiliation자연과학대학 수리과학부-
dc.date.awarded2018. 8-
Appears in Collections:
College of Natural Sciences (자연과학대학)Dept. of Mathematical Sciences (수리과학부)Theses (Master's Degree_수리과학부)
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