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完整後設資料紀錄
DC 欄位 | 值 | 語言 |
---|---|---|
dc.contributor.advisor | 陳榮凱(Jung-Kai Chen) | |
dc.contributor.author | Chi-Kang Chang | en |
dc.contributor.author | 張繼剛 | zh_TW |
dc.date.accessioned | 2021-06-17T03:41:06Z | - |
dc.date.available | 2018-02-23 | |
dc.date.copyright | 2018-02-23 | |
dc.date.issued | 2018 | |
dc.date.submitted | 2018-02-07 | |
dc.identifier.citation | [Ba01] Lucian B˘ adescu, Algebraic surfaces, Universitext, Springer-Verlag, New York, 2001, Translated from the 1981 Romanian original by Vladimir Ma¸sek and revised by the author. MR 1805816
[Bea96] Arnaud Beauville, Complex algebraic surfaces, second ed., London Mathematical Society Student Texts, vol. 34, Cambridge University Press, Cambridge, 1996, Translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid. MR 1406314 [HL10] Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, second ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010. MR 2665168 [Kir85] Frances Clare Kirwan, Partial desingularisations of quotients of nonsingular varieties and their Betti numbers, Ann. of Math. (2) 122 (1985), no. 1, 41~85. MR 799252 [KLS06] D. Kaledin, M. Lehn, and Ch. Sorger, Singular symplectic moduli spaces, Invent. Math. 164 (2006), no. 3, 591~614. MR 2221132 [LS06] Manfred Lehn and Christoph Sorger, La singularit´e de O’Grady, J. Algebraic Geom. 15 (2006), no. 4, 753~770. MR 2237269 [Muk84] Shigeru Mukai, Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math. 77 (1984), no. 1, 101~116. MR 75113365 [O’G96] Kieran G. O’Grady, Relations among Donaldson polynomials of certain algebraic surfaces. I, Forum Math. 8 (1996), no. 1, 1~61. MR1366533 [Rap08] Antonio Rapagnetta, On the Beauville form of the known irreduciblesymplectic varieties, Math. Ann. 340 (2008), no. 1, 77~95. MR2349768 [Yos99] Kota Yoshioka, Irreducibility of moduli spaces of vector bundles on k3 surfaces, DEPARTMENT OF MATHEMATICS, SUNY AT STONY BROOK, 1999. [Yos01] , Moduli spaces of stable sheaves on abelian surfaces, Math. Ann. 321 (2001), no. 4, 817~884. MR 1872531 | |
dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/70056 | - |
dc.description.abstract | 本篇文章主要內容為探討與整理 K.G. O'Grady 在 1998 年的論文
“Desingularized moduli spaces of sheaves on a K3” 。 作者於該篇文章中建構以及探討一複 K3 曲面上之第一陳類為 0,而第二陳類為 c 的二階 Gieseker-丸山半穩定層所形成的模空間開始, 當中 c 為一不小於四之偶數。該一模空間的建構是由一 Quot 概形之幾何不變量商所形成,並且其上存在有奇異點。 而後第二步為根據F.Kirwan在1985年提出的方法對於已建構好的模空間上之部分的嚴格半穩定點以拉開的方式來進行奇點解消。 而重點是當 c=4 時,可以利用森重文 所提出的方法,來對由此奇點解消所建構之平滑流形上之一除子做出壓縮而形成一 Hyperkähler 流形,且由此 HK 流形到原先之模空間的自然雙有理映射為一態射,故此一 HK 流形為原先模空間的一個 symplectic 奇點解消,且其在雙有理等價和 deformation 等價的意義下皆不等價於目前熟知的兩種 HK 流形:對應到點的 Hilbert 概形與 Kummer 多樣體。 關鍵字:層之模空間,半穩定層,幾何不變量理論, symplectic 奇點解消,Hyperkähler 多樣體。 | zh_TW |
dc.description.abstract | Abstract
The aim of this article is to study Kieran G. O’Grady’s paper 'Desingularized moduli spaces of sheaves on a K3' in 1998, where the author constructs the moduli space of rank two torsion-free semistable sheaves on a non-singular K3 surface with c1 = 0 and c2 = c a even number not less then 4. This moduli space is denoted by Mc, which is a G.I.T. quotient from the Quot-scheme and is singular. By using Kirwan’s method of successive blow ups of the strictly semistable loci with reductive stabilizer, one can obtain a desingularization Mcc of Mc. What’s surprising is that when c = 4, there is a Mori extremal divisorial contraction of Mc4 so that the outcome is a hyperk¨ahler manifold Mf4. Moreover, the natural map from Mf4 to M4 is a morphism and hence a simplectic desingularization of M4. The hyperk¨ahler manifold Mf4 is not birational/deformation equivalence to another two typical constructions of HK manifolds: the Hilbert schemes of points and Kummer varieties. Key words: moduli space of sheaves, semistable sheaves, geometric invariant theory, symplectic resolution, hyperk¨ahler variety. | en |
dc.description.provenance | Made available in DSpace on 2021-06-17T03:41:06Z (GMT). No. of bitstreams: 1 ntu-107-R04221023-1.pdf: 942063 bytes, checksum: 34a596d21415578cbe514fbf7629a729 (MD5) Previous issue date: 2018 | en |
dc.description.tableofcontents | Contents
1 Introduction 3 2 K3 surfaces 4 3 Semistable sheaves and the construction of Mc 6 4 Kirwan’s desingularization of Mc 8 5 Luna’s ´etale slice 16 6 Normal cones and deformations of sheaves 19 7 The normal cone of Σ0Q 22 8 The normal cone of Ω0Q 28 9 Description of Ωss R 34 10 Description of Σss R 36 11 Analysis of Kirwan’s desingularization 47 12 The two-form on the moduli space 55 13 A symplectic desingularization of M4 56 14 Discussion 62 | |
dc.language.iso | en | |
dc.title | O'Grady的十維HK射影流形的建構 | zh_TW |
dc.title | Desingularized moduli spaces of torsion-free semistable sheaves on a K3 surface | en |
dc.type | Thesis | |
dc.date.schoolyear | 106-1 | |
dc.description.degree | 碩士 | |
dc.contributor.oralexamcommittee | 賴青瑞(Ching-Jui Lai),莊武諺(Wu-Yen Chuang) | |
dc.subject.keyword | 層之模空間,半穩定層,幾何不變量理論,symplectic奇點解消,Hyperkahler多樣體, | zh_TW |
dc.subject.keyword | moduli space of sheaves,semistable sheaves,geometric invariant theory,symplectic resolution,Hyperkahler variety, | en |
dc.relation.page | 66 | |
dc.identifier.doi | 10.6342/NTU201800370 | |
dc.rights.note | 有償授權 | |
dc.date.accepted | 2018-02-08 | |
dc.contributor.author-college | 理學院 | zh_TW |
dc.contributor.author-dept | 數學研究所 | zh_TW |
顯示於系所單位: | 數學系 |
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