대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제19권3호
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  • Pages.389-425
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  • 2004
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  • 1225-1763(pISSN)
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  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
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그로모브-위튼 불변량과 그의 응용

  • 조용승 (이화여자대학교 자연과학대학 수학과)
  • 발행 : 2004.07.01
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⟨ 이전 논문 다음 논문 ⟩

초록

심프렉틱 다양체는 미분다양체와 케러다양체 사이에 있는 다양체로서 심프렉틱 구조를 갖는 다양체이다. 케러다양체의 성질들을 얼마나 확장할수 있는지, 미분다양체와 다른 성질은 무엇이 있는지 연구함은 흥미있는 일이다. 심프렉틱 구조로부터 준복소구조가 정의되어 2차원 부분다양체를 나타내는 슈도-호로모르픽 사상이 정의되고, 이들은 모듀라이 공간이 된다. 또한 심프렉틱 구조는 메트릭과 에너지를 정의하여 노비코프환을 정의한다. 여기서 모듀라이 공간의 위상구조가 그로모브-위튼 불변량을 정의한다. 이 불변량은 심프렉틱 다양체 연구에 핵심적인 역할을 한다. 이 논문은 그로모브-위튼 불변량의 여러 가지 성질과 그 응용에 대한 여러 학자들의 결과를 소개하는 해설 논문이다.

키워드

참고문헌

  1. J. Math. Pures Appl. v.36 A unique continuation principle for elliptic differential epuations or inequalities of the second order N. Aronszajn
  2. Three lectures for the GARC homotopy K3 surfaces and gluing results in Seiberg-Witten Theory D. Auckly
  3. Progr. Math. Holomorphic in Symplectic Geometry M. Audin.;J. Lafontaine
  4. Proc. Roy. Soc. London Ser. A v.362 Self duality in four-dimensional Riemannian geometry M. F. Atiyah;N. Hitchin;I. Singer https://doi.org/10.1098/rspa.1978.0143
  5. Ann. of Math. v.87 no.2 The index of elliptic operations. II. M. F. Atiyah;G. B. Segal https://doi.org/10.2307/1970716
  6. Ann. of Math. v.87 no.2 The index of elliptic operations. III. M. F. Atiyah;I. Singer https://doi.org/10.2307/1970717
  7. Gromov-Witten invariants in algebraic geometry K. Behrend
  8. Invent. Math. v.54 Necessary conditions for the existence of Branched curverings N. Brand https://doi.org/10.1007/BF01391172
  9. Introduction to Compact Transformation Groups G. Bredon
  10. Mathematics of AMS v.314 The cohomology ring of S$^2$-fibration contemperary Y. S. Cho
  11. Differential Geom. Appl. v.6 Cyclic group actions on gauge Theory Y. S. Cho https://doi.org/10.1016/0926-2245(96)00009-5
  12. Topology Appl. v.62 Equivariant metric for smooth moduli spaces Y. S. Cho https://doi.org/10.1016/0166-8641(94)00049-9
  13. J. Aust. Math. Soc. (series A) v.66 Finite group actions on 4-manifolds Y. S. Cho https://doi.org/10.1017/S1446788700036612
  14. Trans. Amer. Math. Soc. v.323 Finite group actions on the moduli space of self-dual connections I Y. S. Cho https://doi.org/10.2307/2001625
  15. Michigan Math. J. v.37 Finite group actions on the moduli space of self-dual connections II Y. S. Cho https://doi.org/10.1307/mmj/1029004070
  16. Acta Math. Hungar. v.84 no.1;2 Finite group actions on SpinC bundles Y. S. Cho https://doi.org/10.1023/A:1006602919996
  17. Osaka J. Math. v.34 Seiberg-Witten invariants on non-symplectic 4-manifolds Y. S. Cho
  18. Chinese Ann. Math. v.21B no.1 Genus Minimizing in symplectic 4-manifolds Y. S. Cho;M. S. Cho
  19. Czechoslovak Math. J. v.53 no.128 The Geography of Simply connected symplectic 4-Manifolds Y. S. Cho;M. S. Cho https://doi.org/10.1023/A:1026270916962
  20. Taiwanese J. Math. v.17 no.1 Symplectic Surfaces in Symplectic Four-Manifolds Y. S. Cho;M. S. Cho
  21. Acta Math. Hungar. v.94 no.4 The Cyclic Group Actions on Four-Manifolds Y. S. Cho;Y. H. Hong https://doi.org/10.1023/A:1015647713638
  22. Glasg. Math. J. v.45 Seiberg-Witten invariants and Anti-Symplectic Involution Y. S. Cho;Y. H. Hong https://doi.org/10.1017/S0017089503001344
  23. Proc. Amer. Math. Soc. v.130 Anti-Symplectic Involution with langrangian Fixed Loci and their Quotient Y. S. Cho;D. S. Joe https://doi.org/10.1090/S0002-9939-02-06391-8
  24. Acta Math. Sin. (Engl. Ser.) Discreteness of Flux Group Y. S. Cho;M. I. Lim
  25. Arnold. Inventiones Mathematicae v.73 The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. C. Conley;E. Zehnder
  26. Publ. Math. IHES v.36 The irreducibility of the space of curves of given gemnus P. Deligne;D. Mumford https://doi.org/10.1007/BF02684599
  27. J. Differential Geom. v.18 An appilcayion of gauge theory to four-manifold theory S. Donaldson
  28. Bull. Amer. Math. Soc. v.33 The Seiberg-Witten equations and 4-manifold topology S. Donaldson
  29. Topology v.29 Polynomial invariants for four manifolds S. K. Donaldson https://doi.org/10.1016/0040-9383(90)90001-Z
  30. Geometry for four Manifolds S. K. Donaldson;P. Kronheimer
  31. Internat. J. Math. v.9 no.8 Smooth group actions on 4-manifolds and Seiberg-Witten invariants F. Fang https://doi.org/10.1142/S0129167X98000427
  32. Ann. of Math. v.122 Pseudofree orbifolds R. Fintushel;R. Stern
  33. J. Differential Geom. v.20 SO(3)-connections of topology of 4-manifolds R. Fintushel;R. Stern
  34. J. Differential Geom. v.28 Morse theory for Lagrangian intersections A. Floer
  35. Comm. Pure Appl. Math. v.43 The unregularized gradient flow of the symplectic action A. Floer
  36. J. Differential Geom. v.30 Witten's complex and infinite dimensional Morse theory A. Floer
  37. J. Differential Geom. v.17 The topology of four dimensional manifolds M. Fredman
  38. J. Differential Geom. v.27 On the diffeomorphism type of certain algebraic surface I R. Fredman;J. Morgan
  39. M.S.R.I. Pub. v.1 Instantons and Four-Manifolds D. S. Freed;K. K. Uhlenbeck
  40. Topology v.38 no.5 Arnold conjecture and Gromov-Witten invariant K. Fukaya;K. Ono https://doi.org/10.1016/S0040-9383(98)00042-1
  41. Monopole equation and the 11/8 conjecture M. Furuta
  42. Invent. Math. v.82 A new construction of symplectic curves in symplectic manifolds R. Gompf https://doi.org/10.1007/BF01388806
  43. A proof of conjecture for the number of ramified covering of the sphere by the torus I. P. Goulden;D. M. Jackson
  44. The number of ramified covering of the sphere by the double torus, and a general form for higher genera I. P. Goulden;D. M. Jackson
  45. Proc. Amer. Math. Soc. v.125 no.1 Transitive factorisations into transpositions and holomorphic mapping on the sphere I. P. Goulden;D. M. Jackson https://doi.org/10.1090/S0002-9939-97-03880-X
  46. AG/9902125 The number of ramified covering of the sphere by the torus, and surface of higher genera I. P. Goulden;D. M. Jackson;A. Vainshtain
  47. Invent. Math. v.82 Pseudo-holomorphic curves in symplectic manifolds M. Gromov https://doi.org/10.1007/BF01388806
  48. Global Analysis(Papers in Honors of K. Kodaira) The signature of ramified coverings F. Hirzebruch
  49. Math. Ann. v.39 Uber Riemann'sche Flachen mit gegebenen Verzweigungspunkten A. Hurwitz https://doi.org/10.1007/BF01199469
  50. Ann. of Math. v.142 A new construction of symplectic manifolds A. Hurwitz https://doi.org/10.2307/2118554
  51. ICM v.2 Symplectic sums and Gromov-Witten Invariants E. N. Ionel
  52. Ann. of Math. Relative Gromov-Witten Invariants E. Ionel;T. H. Parker
  53. math. SG/0010217 The symplectic Sum Formula for Gromov-Witten Invariants E. Ionel;T. H. Parker
  54. Trans. Amer. Math. Soc. v.330 Intersection theory of moduli space of stable n-pointed curves of genus zero S. Keel https://doi.org/10.2307/2153922
  55. Problems in low-dimensional topology R. Kirby
  56. hepth/9405035 Enumeration of Rational Curves Via Torus Actions M. Kontsevich
  57. hepth/9402147 Gromov-Witten classes, quantum cohomology and enumerative geometry M. Kontsevich;Y. Manin
  58. Comm. Math. Phys. v.164 Gromov-Witten classes, Quantum cohomology and Enumerative Geometry M. Kontsevich;Y. Manin https://doi.org/10.1007/BF02101490
  59. On irreducible four-manifolds D. Kotschick
  60. Math. Res. Lett. v.2 four-manifolds without symplectic structures but with nontrivial Seiberg-Witten invariants D. Kotschick;S. Morgan;C. Taubes https://doi.org/10.4310/MRL.1995.v2.n2.a1
  61. Math. Res. Lett. v.I no.2 The geneus of embedded surfaces in the projective plane P. Kronheimer;T. Mrowka
  62. Topology v.3 The homotopy type of the unitary group of Hilbert space N. H. Kuiper https://doi.org/10.1016/0040-9383(65)90067-4
  63. Conference on Complex Analysis New proof for the existence of local free complete families of complex structures Mo Kuranishi
  64. Math. Ann. v.208 The quotient space of ${\mathbb{CP}}^2$ by the complex conjugation is the 4-sphere Mo Kuranishi https://doi.org/10.1007/BF01432386
  65. Math. alg-geom/ 9803036 Symplectic surgery and Gromov-Witten Invariants of Calabi-Yau 3-folds A. M. Li;Y. Ruan
  66. Turkish J. Math v.26 Minimality of certain connected sums T. J. Li;A. I. Stipsicz
  67. Comm. Math. Phys. v.213 The Number of Ramified covering of a Riemann Surface by Riemann Surface A. M. Li;G. Zhao;Q. Zheng https://doi.org/10.1007/s002200000254
  68. Bull. Amer. Math. Soc. v.23 Elliptic methods in symplectic geometry D. McDuff https://doi.org/10.1090/S0273-0979-1990-15928-2
  69. Invent. Math. v.89 Example of symplectic structures D. McDuff https://doi.org/10.1007/BF01404672
  70. J. Amer. Math. Soc. v.3 The structure of rational and ruled symplectic 4-manifolds D. McDuff https://doi.org/10.2307/1990934
  71. Univ. Lecture Ser. v.6 J-holomorphic curves and Quantum cohomology D. McDuff;D. Salamon
  72. J-holomorphic curves and Quantum cohomology D. McDuff;D. Salamon
  73. Soviet Mathematics Doklady v.24 Multivalued functions and functionals-an of the Morse theory S. Novikov
  74. Comm. Math. Phys. v.85 Gauge theories on four dimensional Riemannian manifolds T. Parker https://doi.org/10.1007/BF02029127
  75. J. Geom. Anal. v.3 Pseudo holomorphic maps and bubble trees T. Parker;J. Wolfson https://doi.org/10.1007/BF02921330
  76. Proc. Sympos. Pure Math. v.32 Pseudo equivalence of G-manifolds T. Petrie
  77. Duke Math. J. v.83 Topological sigma model and Donaldson type invariant in Gromov theory Y. Ruan https://doi.org/10.1215/S0012-7094-96-08316-7
  78. J. Differential Geom. v.42 A mathematical theory of quantum cohomogy Y. Ruan;G. Tian
  79. J. Math. Soc. Japan v.9 The Gauss-Bonnet theorem for V-manifolds I. Satake https://doi.org/10.2969/jmsj/00940464
  80. Lecture Notes in Math. v.638 The Atiyah-Singer Index Theorem P. Shanahan
  81. Global Analysis The curvature of 4-dimensional Einstein spaces I. Singer;J. Thorpe
  82. Amer. J. Math. v.87 An infinite dimensional version of Sard's theorem S. Smale https://doi.org/10.2307/2373250
  83. Counting pseudo holomorphic submanifolds in dimension 4 C. Taubes
  84. Math. Res. Lett. v.2 More constraints on symplectic forms from Seiberg-Witten invariants C. Taubes https://doi.org/10.4310/MRL.1995.v2.n1.a2
  85. J. Differential Geom. v.19 Self-dual connections on manifolds with indefinite intersection matrix C. Taubes
  86. J. Differential Geom. v.17 Self-dual connections on non-self-dual 4-manifolds C. Taubes
  87. The Geometry of the Seiberg-Witten invariants C. Taubes
  88. Math. Res. Lett. v.2 The Seiberg-Witten and te h Gromov invariants C. Taubes https://doi.org/10.4310/MRL.1995.v2.n2.a10
  89. Math. Res. Lett. v.I The Seiberg-Witten invariants and symplectic forms C. Taubes
  90. Geometry and Physics (Aarhus, 1995) (Lecture Notes in Pure and Appl. Math., 184) Seigerg-Witten and Gromov invariants C. Taubes
  91. Oxford Univ. Thesis Gauge theory and involutions S. Wang
  92. Math. Res. Lett. v.1 Monopoles and four-manifolds E. Witten https://doi.org/10.4310/MRL.1994.v1.n6.a13
  93. J. Differential Geom. v.117 Supersymmetry and Morse theory E. Witten
  94. Communications in Mathematical Topological sigma model E. Witten
  95. Ann. of Math. v.93 no.2 The index of elliptic operations, IV.V. E. Witten https://doi.org/10.2307/1970756
  96. J. Differential Geom. v.24 Connections. cohomology and the intersection forms of 4-manifolds E. Witten
  97. 대우총서 432 다양체의 미분위상수학 조용승
  98. 이론물리의 수학적 접근 수학에서 게이지 이론 조용승
  99. Commun. Korean Math. Soc. v.15 no.3 심프렉틱 다양체의 불변량 조용승