令X 為個帶有凱勒形式(Kähler form ω) 以及全純四形式( holomorphic four- form Ω )的四維緊致卡拉比丘空間(Calabi-Yau manifolds) 。在一些假設條件下，通過研究Donaldson- Thomas方程所決定的模空間，我們定義了四維Donaldson-Thomas不變量。我們也對四維局部卡拉比丘空間(local Calabi-Yau four-folds) 定義了四維Donaldson-Thomas 不變量，並且將之聯繫到三維Fano空間的Donaldson- Thomas 不變量。在一些情況下，我們還研究了DT/GW不變量對應。最后，我們在模空間光滑時計算了一些四維Donaldson- Thomas不變量。 / Let X be a complex four-dimensional compact Calabi-Yau manifold equipped with a Kahler form ω and a holomorphic four-form Ω. Under certain assumptions, we de ne Donaldson-Thomas type deformation invariants by studying the moduli space of the solutions of Donaldson-Thomas equations on the given Calabi-Yau manifold. We also study sheaves counting on local Calabi-Yau four-folds. We relate the sheaves countings over X = KY with the Donaldson- Thomas invariants for the associated compact three-fold Y . In some specialcases, we prove the DT/GW correspondence for X. Finally, we compute the Donaldson-Thomas invariants of certain Calabi-Yau four-folds when the moduli spaces are smooth. / Detailed summary in vernacular field only. / Cao, Yalong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 100-105). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- The *4 operator --- p.18 / Chapter 2.1 --- The *4 operator for bundles --- p.18 / Chapter 2.2 --- The *4 operator for general coherent sheaves --- p.20 / Chapter 3 --- Local Kuranishi structure of DT₄ moduli spaces --- p.22 / Chapter 4 --- Compactification of DT₄ moduli spaces --- p.34 / Chapter 4.1 --- Stable bundles compactification of DT₄ moduli spaces --- p.34 / Chapter 4.2 --- Attempted general compactification of DT₄ moduli spaces --- p.36 / Chapter 5 --- Virtual cycle construction --- p.39 / Chapter 5.1 --- Virtual cycle construction for DT₄ moduli spaces --- p.40 / Chapter 5.2 --- Virtual cycle construction for generalized DT₄ moduli spaces --- p.48 / Chapter 6 --- DT4 invariants for compactly supported sheaves on local CY₄ --- p.52 / Chapter 6.1 --- The case of X = KY --- p.52 / Chapter 6.2 --- The case of X = T*S --- p.57 / Chapter 7 --- DT₄ invariants on toric CY₄ via localization --- p.66 / Chapter 8 --- Computational examples --- p.70 / Chapter 8.1 --- DT₄=GW correspondence in some special cases --- p.71 / Chapter 8.1.1 --- The case of Hol(X) = SU(4) --- p.72 / Chapter 8.1.2 --- The case of Hol(X) = Sp(2) --- p.77 / Chapter 8.2 --- Some remarks on cosection localizations for hyper-kähler four-folds --- p.79 / Chapter 8.3 --- Li-Qin's examples --- p.80 / Chapter 8.4 --- Moduli space of ideal sheaves of one point --- p.83 / Chapter 9 --- Appendix --- p.85 / Chapter 9.1 --- Local Kuranishi models of Mc° --- p.85 / Chapter 9.2 --- Some remarks on the orientability of the determinant line bundles on the (generalized) DT₄ moduli spaces --- p.87 / Chapter 9.3 --- Seidel-Thomas twists --- p.90 / Chapter 9.4 --- A quiver representation of Mc --- p.92
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_328768 |
| Date | January 2013 |
| Contributors | Cao, Yalong., Chinese University of Hong Kong Graduate School. Division of Mathematics. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, bibliography |
| Format | electronic resource, electronic resource, remote, 1 online resource (105 leaves) |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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