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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 林學庸 | zh_TW |
| dc.contributor.advisor | Hsueh-Yung Lin | en |
| dc.contributor.author | 張志煥 | zh_TW |
| dc.contributor.author | Chih-Huan Chang | en |
| dc.date.accessioned | 2025-07-11T16:10:23Z | - |
| dc.date.available | 2025-07-12 | - |
| dc.date.copyright | 2025-07-11 | - |
| dc.date.issued | 2025 | - |
| dc.date.submitted | 2025-06-30 | - |
| dc.identifier.citation | [AHLH18] Jarod Alper, Daniel Halpern-Leistner, and Jochen Heinloth. Existence of moduli spaces for algebraic stacks. Inventiones mathematicae, 234:949 –1038, 2018.
[AHR19] Jarod Alper, Jack Hall, and David Rydh. The étale local structure of algebraic stacks. arXiv preprint, 2019. [Alp13] Jarod Alper. Good moduli spaces for artin stacks. Annales de l’Institut Fourier, 63(6):2349–2402, 2013. [Alp25] Jarod Alper. Lecture notes on the theory of algebraic stacks and moduli. https://sites.math.washington.edu/~jarod/moduli.pdf, 2025. Lecture notes, University of Washington. Accessed May 2025. [AZ01] M. Artin and J. J. Zhang. Abstract Hilbert schemes. Algebras and Representation Theory, 4(4):305–394, 2001. [BBBBJ15] Oren Ben-Bassat, Christopher Brav, Vittoria Bussi, and Dominic Joyce. A ’darboux theorem’ for shifted symplectic structures on derived artin stacks, with applications. Geometry & Topology, 19(3):1287–1371, 2015. [BBD+15] Christopher Brav, Vittoria Bussi, Delphine Dupont, Dominic Joyce, and 45 Balázs Szendrői. Symmetries and stabilization for sheaves of vanishing cycles. Journal of Singularities, 11:85–151, 2015. [BBJ19] Christopher Brav, Vittoria Bussi, and Dominic Joyce. A darboux theorem for derived schemes with shifted symplectic structure. Journal of the American Mathematical Society, 32(2):399–443, 2019. [BBS13] Kai Behrend, Jim Bryan, and Balázs Szendrői. Motivic degree zero donaldson–thomas invariants. Inventiones Mathematicae, 192(1):111–160, 2013. [Beh09] Kai Behrend. Donaldson–thomas type invariants via microlocal geometry. Annals of Mathematics, 170(3):1307–1338, 2009. [BJM19] Vittoria Bussi, Dominic Joyce, and Sven Meinhardt. On motivic vanishing cycles of critical loci. Journal of Algebraic Geometry, 28(3):405–438, 2019. [BNK25] Chenjing Bu, Andrés Ibáñez Núñez, and Tasuki Kinjo. Intrinsic donaldson-thomas theory. ii. stability measures and invariants. arXiv preprint, 2025. [Bri10] Tom Bridgeland. An introduction to motivic hall algebras. arXiv preprint arXiv:1002.4372, 2010. [Bri11] Tom Bridgeland. Hall algebras and curve-counting invariants. Journal of the American Mathematical Society, 24(4):969–998, 2011. [Bu24] Chenjing Bu. A motivic integral identity for (-1)-shifted symplectic stacks. arXiv preprint, 2024. [Bu25] Chenjing Bu. Orthosymplectic donaldson–thomas theory. arXiv preprint arXiv:2503.20667, 2025. [DM15] Ben Davison and Sven Meinhardt. Donaldson–thomas theory for categories of homological dimension one with potential. arXiv preprint arXiv:1512.08898, 2015. [GZLMH06] S.M. Gusein-Zade, I. Luengo, and A. Melle-Hernández. Power structure over the grothendieck ring of varieties and generating series of hilbert schemes of points. Michigan Mathematical Journal, 54(2):353–359, 2006. [HL14] Daniel Halpern-Leistner. On the structure of instability in moduli theory. arXiv preprint arXiv:1411.0627, 2014. [Joy06] Dominic Joyce. Configurations in abelian categories. i. basic properties and moduli stacks. Advances in Mathematics, 203(1):194–255, 2006. [Joy07a] Dominic Joyce. Configurations in abelian categories. ii. ringel–hall algebras. Advances in Mathematics, 210(2):635–706, 2007. [Joy07b] Dominic Joyce. Motivic invariants of artin stacks and ’stack functions’. The Quarterly Journal of Mathematics, 58(3):345–392, 2007. [Joy15] Dominic Joyce. A classical model for derived critical loci. Journal of Differential Geometry, 101(2):289–367, 2015. [JS12] Dominic Joyce and Yinan Song. A theory of generalized Donaldson–Thomas invariants, volume 217 of Memoirs of the American Mathematical Society. American Mathematical Society, 2012. [JU21] Dominic Joyce and Markus Upmeier. Orientation data for moduli spaces of coherent sheaves over calabi–yau 3-folds. Advances in Mathematics, 381:107627, 2021. [KPS24] Tasuki Kinjo, Hyeonjun Park, and Pavel Safronov. Cohomological hall algebras for 3-calabi–yau categories. arXiv preprint arXiv:2406.12838, 2024. [KS08] Maxim Kontsevich and Yan Soibelman. Stability structures, motivic donaldson–thomas invariants and cluster transformations. arXiv preprint, 2008. [Lie06] Max Lieblich. Moduli of complexes on a proper morphism. Journal of Algebraic Geometry, 15:175–206, 2006. [Lur09] Jacob Lurie. Higher Topos Theory, volume 170 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2009. [MNOP06a] D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande. Gromov–witten theory and donaldson–thomas theory. i. Compositio Mathematica, 142(5):1263–1285, 2006. [MNOP06b] D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande. Gromov–witten theory and donaldson–thomas theory. ii. Compositio Mathematica, 142(5):1286–1304, 2006. [Par23] John Pardon. Universally counting curves in calabi–yau threefolds. arXiv preprint, 2023. [Par24] Hyeonjun Park. Introduction to shifted symplectic structures. Lecture notes, available at https://drive.google.com/file/d/1pHUn0OqyMIPOwq47ShzwgYM7Zr-SwjH6/view, 2024. [PT09] R. Pandharipande and R. P. Thomas. Curve counting via stable pairs in the derived category. Inventiones mathematicae, 178(2):407–447, 2009. [PTVV13] Tony Pantev, Bertrand Toën, Michel Vaquié, and Gabriele Vezzosi. Shifted symplectic structures. Publications mathématiques de l’IHÉS, 117:271-328, 2013. [STV15] Timo Schürg, Bertrand Toën, and Gabriele Vezzosi. Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes. Journal für die reine und angewandte Mathematik, 2015(702):1–40, 2015. [Tod10] Yukinobu Toda. Curve counting theories via stable objects i. dt/pt correspondence. Journal of the American Mathematical Society, 23(4):1119–1157, 2010. [Tod20] Yukinobu Toda. Hall algebras in the derived category and higher rank DT invariants. Algebraic Geometry, 7(3):240–262, 2020. [TV08] Bertrand Toën and Gabriele Vezzosi. Homotopical Algebraic Geometry II: Geometric Stacks and Applications, volume 193 of Memoirs of the American Mathematical Society. American Mathematical Society, 2008. | - |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/97684 | - |
| dc.description.abstract | 本論文證明了在假設半穩定物件的模疊具有好模空間的情況下，卡拉比丘三維複流形上 Donaldson–Thomas／Pandharipande–Thomas（DT/PT）對應的動機版本。我們採用Toda所構造的 t-結構心臟與弱穩定條件，在此架構下定義動機 DT 與 PT 不變量。本文的主要結果為一組動機等式，揭示兩類不變量之間的關係。證明方法延續 Bridgeland 與 Toda 的穿牆架構，並依賴 卜辰璟近期證明的動機積分恆等式。本研究為數值 DT/PT 對應提供了動機上的提升。 | zh_TW |
| dc.description.abstract | This thesis proves a motivic version of the Donaldson–Thomas/Pandharipande–Thomas (DT/PT) correspondence on Calabi–Yau threefolds under the assumption on the existence of some good moduli spaces. Working with the heart constructed and a weak stability condition constructed by Toda, we define motivic DT and PT invariants via vanishing cycles. The main result establishes a motivic identity between these invariants, following Bridgeland and Toda's wall-crossing approach. The argument relies on the motivic integral identity recently proved by Bu. This provides a refinement of the numerical DT/PT correspondence and supports further development of categorified curve-counting theories. | en |
| dc.description.provenance | Submitted by admin ntu (admin@lib.ntu.edu.tw) on 2025-07-11T16:10:23Z No. of bitstreams: 0 | en |
| dc.description.provenance | Made available in DSpace on 2025-07-11T16:10:23Z (GMT). No. of bitstreams: 0 | en |
| dc.description.tableofcontents | Acknowledgements iii
摘要 v Abstract vii Contents ix Chapter 1 Introduction 1 1.1 Curve counting theories on Calabi–Yau threefolds . . . . . . . . . . 1 1.2 Categorification of Donaldson–Thomas Theory . . . . . . . . . . . . 2 Chapter 2 Stability conditions and moduli spaces 5 2.1 Weak stability conditions on triangulated categories . . . . . . . . . . 5 2.1.1 Constructions of weak stability conditions . . . . . . . . . . . . . . 6 2.2 Weak stability condition on DX . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Moduli stacks of semistable objects . . . . . . . . . . . . . . . . . . 8 2.4 The Good moduli space . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4.1 AHLH’s Existence Theorem . . . . . . . . . . . . . . . . . . . . . 11 2.4.2 Good moduli space for Semistable objects on the wall . . . . . . . . 12 2.4.3 Some properties for semistable objects . . . . . . . . . . . . . . . . 13 Chapter 3 Shifted symplectic structure and orientation 17 3.1 Shifted symplectic structure . . . . . . . . . . . . . . . . . . . . . . 17 3.1.1 Shifted symplectic structure . . . . . . . . . . . . . . . . . . . . . . 18 3.1.2 d-critical structures . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Results on the moduli stack of objects on a Calabi Yau threefold . . . 21 Chapter 4 Motivic invariants and Integral identity 23 4.1 Rings of motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.1.1 Rings of Motives for stacks . . . . . . . . . . . . . . . . . . . . . . 23 4.1.2 Motives with monodromic action . . . . . . . . . . . . . . . . . . . 25 4.2 Motivic Behrend Functions . . . . . . . . . . . . . . . . . . . . . . . 26 4.3 Graded and Filtered objects . . . . . . . . . . . . . . . . . . . . . . 28 4.3.1 Morphisms of stacks of graded and filtered objects . . . . . . . . . 29 4.3.2 Derived version . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.4 Motivic Integral Identity . . . . . . . . . . . . . . . . . . . . . . . . 31 Chapter 5 Motivic Donaldson Thomas invariants 33 5.1 Donaldson Thomas type invariants . . . . . . . . . . . . . . . . . . . 33 5.2 Motivic Hall Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.3 Motivic Integration Map . . . . . . . . . . . . . . . . . . . . . . . . 35 5.4 DT/PT correspondence . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.4.1 The invariants N mot n . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 References . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 | - |
| dc.language.iso | en | - |
| dc.subject | 動機積分恆等式 | zh_TW |
| dc.subject | 卡拉比丘流形 | zh_TW |
| dc.subject | 好模空間 | zh_TW |
| dc.subject | 動機 Donaldson–Thomas 不變量 | zh_TW |
| dc.subject | Motivic integral identity | en |
| dc.subject | Motivic Donaldson-Thomas invariants | en |
| dc.subject | Good moduli space | en |
| dc.subject | Calabi-Yau threefolds | en |
| dc.title | 動機化 DT/PT 對應 | zh_TW |
| dc.title | Motivic DT/PT Correspondence | en |
| dc.type | Thesis | - |
| dc.date.schoolyear | 113-2 | - |
| dc.description.degree | 碩士 | - |
| dc.contributor.oralexamcommittee | 李元斌;莊武諺 | zh_TW |
| dc.contributor.oralexamcommittee | Yuan-Pin Lee;Wu-Yen Chuang | en |
| dc.subject.keyword | 動機 Donaldson–Thomas 不變量,好模空間,卡拉比丘流形,動機積分恆等式, | zh_TW |
| dc.subject.keyword | Motivic Donaldson-Thomas invariants,Good moduli space,Calabi-Yau threefolds,Motivic integral identity, | en |
| dc.relation.page | 49 | - |
| dc.identifier.doi | 10.6342/NTU202501351 | - |
| dc.rights.note | 同意授權(全球公開) | - |
| dc.date.accepted | 2025-07-01 | - |
| dc.contributor.author-college | 理學院 | - |
| dc.contributor.author-dept | 數學系 | - |
| dc.date.embargo-lift | 2025-07-12 | - |
| 顯示於系所單位： | 數學系 | |
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