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Title: | O'Grady的十維HK射影流形的建構 Desingularized moduli spaces of torsion-free semistable sheaves on a K3 surface |
Authors: | Chi-Kang Chang 張繼剛 |
Advisor: | 陳榮凱(Jung-Kai Chen) |
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, moduli space of sheaves,semistable sheaves,geometric invariant theory,symplectic resolution,Hyperkahler variety, |
Publication Year : | 2018 |
Degree: | 碩士 |
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。 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. |
URI: | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/70056 |
DOI: | 10.6342/NTU201800370 |
Fulltext Rights: | 有償授權 |
Appears in Collections: | 數學系 |
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