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研究生:王賜聖
研究生(外文):Sz-Sheng Wang
論文名稱:三維卡拉比-丘空間奇異點及模空間連結性研究
論文名稱(外文):The Connectedness Problem of Calabi--Yau Moduli Spaces
指導教授:王金龍王金龍引用關係
口試委員:林惠雯余正道齊震宇莊武諺
口試日期:2015-06-03
學位類別:博士
校院名稱:國立臺灣大學
系所名稱:數學研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2015
畢業學年度:103
語文別:英文
論文頁數:62
中文關鍵詞:卡拉比-丘錐過渡變換
外文關鍵詞:Calabi-Yau threefoldconifold transitionsmall contractiondeterminantal contractionstandard web
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本文探討在雙有理映射及形變理論的操作下,給出判別三維卡拉比-丘簇的奇異點是否為節點(即米爾諾數等於一)的條件。同時也對於P.S. Green和T. Hübsch教授的結果:在乘積射影空間裡的三維完全交集卡拉比—丘流形皆可由錐過渡變換連接,提供一個詳細的證明。

We develop criteria for a Calabi--Yau 3-fold to be a conifold, i.e. to admit only ODPs as singularities, in the context of extremal transitions. There are birational contraction and smoothing involved in the process, and we give such a criterion in each aspect.

More precisely, given a small projective resolution pi : widehat{X} rightarrow X of Calabi--Yau 3-fold X, we show that (1) If the fiber over a singular point P in X is irreducible then P is a cA_1 singular point, and an ODP if and only if there is a normal surface which is smooth in a neighborhood of the fiber. (2) If the natural closed immersion Def(widehat{X}) hookrightarrow Def(X) is an isomorphism then X has only ODPs as singularities.

There are topological constraints associated to a smoothing widetilde{X} of X. It is well known that $e(widehat{X}) - e(widetilde{X}) = 2 | Sing(X) | if and only if X is a conifold. Based on this and a Bertini-type theorem for degeneracy loci of vector bundle morphisms, we supply a detailed proof of the result by P.S.~Green and T.~Hübsch that all complete intersection Calabi--Yau 3-folds in product of projective spaces are connected through projective conifold transitions (known as the standard web).

口試委員會審定書 . . . . . . . . . . . . . . . . . . . . . .I
中文摘要. . . . . . . . . . . . . . . . . . . . . . . . . .II
Abstract. . . . . . . . . . . . . . . . . . . . . . . . . III
Acknowledgements. . . . . . . . . . . . . . . . . . . . . IV
1 Introduction 2
2 Preliminaries 8
2.1 Singularities . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Small Birational Morphisms . . . . . . . . . . . . . . 11
2.3 Normal Gorenstein Surfaces . . . . . . . . . . . . . 17
2.4 Augmented Base Locus . . . . . . . . . . . . . . . . .22
2.5 Bertini-type theorems . . . . . . . . . . . . . . . . 23
2.6 Mixed Hodge Structures on Varieties with Normal Crossings 25
3 Deformation Theory of Calabi–Yau threefolds 30
3.1 Unobstructedness Theorem . . . . . . . . . . . . . . .30
3.2 Smoothings . . . . . . . . . . . . . . . . . . . . . .31
4 Decompositions of Small Transitions 33
4.1 Criteria for small reduced fibers . . . . . . . . . . 33
4.2 Decomposition Process of Small Transition . . . . . . 37
5 A Connectedness Theorem of Moduli spaces 42
5.1 Configurations and Parameter Spaces . . . . . . . . . 42
5.2 Determinantal Contractions . . . . . . . . . . . . . .51
5.3 Connecting the CICY Web . . . . . . . . . . . . . . . 54

1. M. Andreatta, J.A. Wisniewski; On contractions of smooth varieties, J. Alg. Geom. 7 (1998), 253–312.

2. J. Brevik, S. Nollet; Noether–Lefschetz theorem with base locus, Int. Math. Res. Not. 6 (2011), 1220–1244.

3. P. Candelas, A.M. Dale, C.A. Lütken, R. Schimmrigk; Complete intersection Calabi–Yau manifolds, Nucl. Phys. B298 (1988), 493–525.

4. K.A. Chandler, A. Howard, A.J. Sommese; Reducible hyperplane sections I, J. Math. Soc. Japan 51 (1999), 887–910.

5. H. Clemens, J. Kollár, S. Mori; Higher dimensional complex geometry, Astárisque, 1988.

6. O. Debarre; Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001.

7. A. Dimca; Singularities and Topology of Hypersurfaces, Universitext, Springer-Verlag, New York, 1992.

8. D. Eisenbud; Commutative algebra, with a view toward algebraic geometry, Graduate Texts in Math., no.150, Springer-Verlag, New York, 1995.

9. D. Eisenbud, J. Harris; The Geometry of Schemes, Graduate Texts in Math., no.197, Springer-Verlag, New York, 2002. 59

10. R. Friedeman; Simultaneous Resolution of Threefold Double Points, Math. Ann. 274 (1986), 671–689.

11. R.Friedeman; On threefolds with trivial canoni calbundle, Proceedings of Symposia in Pure Mathematics: Complex Geometry and Lie Theory, Vol.53, American Mathematical Society, Providence, RI, 1984, 103–134.

12. A. Fujiki, S. Nakano; Supplement to ”On the inverse of monoidal transformation”, Publ. Res. Inst. Math. Sci. 7 (1970/71), 637–644.

13. W. Fulton; Intersection Theory, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete 3, vol. 2, Springer-Verlag, Berlin, 1998.

14. P. Griffith, W. Schmid; Recent developments in Hodge theory: a discussion of techniques and results, in Proc. Internat. Colloq. on Discrete Subroups of Lie Groups, Bombay, (1975), 31–127.

15. M. Gross; Deforming Calabi-Yau threefolds, Math. Ann. 308 (1997), 187–220.

16. P.S. Greene, T. Hübsch; Calabi–Yau manifolds as complete intersections in products of complex projective spaces, Comm. Math. Phys. 109 (1987), 99–108.

17. P.S. Greene, T. Hübsch; Connetting moduli spaces of Calabi–Yau threefolds, Comm. Math. Phys. 119 (1988), 431–441.

18. P.S. Green, T. Hübsch, C.A. Lütken; All the Hogde numbers of all Calabi-Yau complete intersections, Class. Quantum Gravity 6 (1989), 105–124.

19. R. Hartshorne; Algebraic geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York, 1977.

20. S. Katz, D.R. Morrison; Gorenstein threefold singularities with small resolutions via invariant theory of Weyl groups, J. Alg. Geom. 1 (1992), 449–530.

21. Y. Kawamata; Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. 127 (1988), 93–163.

22. Y. Kawamata; Small contractions of four dimensional algebraic manifolds, Math. Ann. 284 (1989), 595–600.

23. J. Kollár; Flops, Nagoya Math. J. 113 (1989), 15–36.

24. J. Kollár; Flips, flops, minimal models, etc, Surv. in Diff. Geom. 1 (1991), 113–199.

25. J. Kollar, S. Mori; Classification of the three-dimensional flips, J. Amer. Math. Soc. 5 (1992), 533–703.

26. J. Kollar, S. Mori; Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics. 134, Cambridge University Press, Cambridge, 1998.

27. H. Laufer; On CP 1 as an exceptional set, Recent developments in several complex variables, Ann. of Math. Stud. 100, Princeton University Press, 1981, 261–275.

28. R. Lazarsfeld; Positivity in Algebraic Geometry, I & II, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 48 & 49, Springer-Verlag, berlin, 2004.

29. Y.-P. Lee; H.-W. Lin; C.-L. Wang; Flops, motives and invariance of quantum rings, Ann. of Math. (2) 172 (2010), no. 1, 243-290.

30. A. Li, Y. Ruan; Symplectic surgery and Gromov-Witten invariants of Calabi--Yau 3-folds, Invent. Math. 145 (2001), no. 1, 151-218.

31. J. Lipman, A. J. Sommese; On blowing down projective spaces in singular varieties, J. Reine. Angew. Math. 362 (1985), 51–62.

32. R. Miranda; The basic theory of elliptic surfaces, Dottorato di ricerca in matematica, ETS Editrice, Pisa (1989).

33. D.R. Morrison; The birational geometry of surfaces with rational double points, Math. Ann. 271 (1985), 415–438.

34. M. Nakamaye; Stable base loci of linear series, Math. Ann, 318 (2000), 837–847.

35. S. Nakano; On the inverse of monoidal transformation, Publ. Res. Inst. Math. Sci. 6 (1970/71), 483–502.

36. Y. Namikawa and J.H.M. Steenbrink; Global smoothing of Calabi-Yau threefolds, Invent. Math. 122 (1995), 403–419.

37. Y. Namikawa; On Deformations of Calabi-Yau 3-folds with Terminal Singularities, Topology, 33 (1994), 429–446.

38. Y. Namikawa; Stratified local moduli of Calabi-Yau 3-folds, Topology, 41 (2002), 1219–1237.

39. G. Ottaviani; Varietà proiettive di codimensione piccola, INDAM course, Aracne, Roma 1995.

40. H. Pinkham; Factorization of birational maps in dimension 3, Proc. of A.M.S. Summer Inst. on Singularities, Arcata, 1981. Proc. Symposia in Pure Math., A.M.S., 40 (1983), Part 2, 343–371.

41. M. Reid; Canonical 3-folds, in Journées de Géométrie Algébrique d’Angers, ed. A Beauville, Sijthoff and Noordhoff, Alphen (1980), 273–310.

42. M. Reid; Minimal models of canonical 3-folds, Adv. St. Pure Math. 1 (1983), 131–180.

43. M. Reid; The moduli space of 3-folds with K = 0 may nevertheless be irreducible, Math. Ann. 278 (1987), 329–334.

44. F. Sakai; Enriques classification of normal Gorenstein surfaces, Amer. J. Math. 104 (1982), 1233–1241.

45. F. Sakai; Weil divisors on normal surfaces, Duke. Math. J. 51 (1984), 877–887.

46. F. Sakai; The structue of normal surfaces, Duke. Math. J. 52 (1985), 627–648.

47. M. Schlessinger; Rigidity of quotient singularities, Invent. Math. 14 (1971), 17–26.

48. J. Wahl; Equisingular deformations of normal surface singularities I, Ann. Math. 104 (1976), 325–356.

49. C.-L. Wang; Quantum invariance under flop transitions, Advanced Lectures in Mathematics, Vol 18, Higher Education Press and International Press, 2010.

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