그로모브-위튼 불변량과 그의 응용


  • Published : 2004.07.01


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.


그로모브-위튼 불변량;모듀라이공간;미니마리티;퀀텀 코호몰로지;허비츠수;심프렉틱 합;불변량의 합공식


  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
  5. Ann. of Math. v.87 no.2 The index of elliptic operations. II. M. F. Atiyah;G. B. Segal
  6. Ann. of Math. v.87 no.2 The index of elliptic operations. III. M. F. Atiyah;I. Singer
  7. Gromov-Witten invariants in algebraic geometry K. Behrend
  8. Invent. Math. v.54 Necessary conditions for the existence of Branched curverings N. Brand
  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
  12. Topology Appl. v.62 Equivariant metric for smooth moduli spaces Y. S. Cho
  13. J. Aust. Math. Soc. (series A) v.66 Finite group actions on 4-manifolds Y. S. Cho
  14. Trans. Amer. Math. Soc. v.323 Finite group actions on the moduli space of self-dual connections I Y. S. Cho
  15. Michigan Math. J. v.37 Finite group actions on the moduli space of self-dual connections II Y. S. Cho
  16. Acta Math. Hungar. v.84 no.1;2 Finite group actions on SpinC bundles Y. S. Cho
  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
  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
  22. Glasg. Math. J. v.45 Seiberg-Witten invariants and Anti-Symplectic Involution Y. S. Cho;Y. H. Hong
  23. Acta Math. Sin. (Engl. Ser.) Discreteness of Flux Group Y. S. Cho;M. I. Lim
  24. Arnold. Inventiones Mathematicae v.73 The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. C. Conley;E. Zehnder
  25. Publ. Math. IHES v.36 The irreducibility of the space of curves of given gemnus P. Deligne;D. Mumford
  26. J. Differential Geom. v.18 An appilcayion of gauge theory to four-manifold theory S. Donaldson
  27. Bull. Amer. Math. Soc. v.33 The Seiberg-Witten equations and 4-manifold topology S. Donaldson
  28. Topology v.29 Polynomial invariants for four manifolds S. K. Donaldson
  29. Geometry for four Manifolds S. K. Donaldson;P. Kronheimer
  30. Internat. J. Math. v.9 no.8 Smooth group actions on 4-manifolds and Seiberg-Witten invariants F. Fang
  31. Ann. of Math. v.122 Pseudofree orbifolds R. Fintushel;R. Stern
  32. J. Differential Geom. v.20 SO(3)-connections of topology of 4-manifolds R. Fintushel;R. Stern
  33. J. Differential Geom. v.28 Morse theory for Lagrangian intersections A. Floer
  34. Comm. Pure Appl. Math. v.43 The unregularized gradient flow of the symplectic action A. Floer
  35. J. Differential Geom. v.30 Witten's complex and infinite dimensional Morse theory A. Floer
  36. J. Differential Geom. v.17 The topology of four dimensional manifolds M. Fredman
  37. J. Differential Geom. v.27 On the diffeomorphism type of certain algebraic surface I R. Fredman;J. Morgan
  38. M.S.R.I. Pub. v.1 Instantons and Four-Manifolds D. S. Freed;K. K. Uhlenbeck
  39. Topology v.38 no.5 Arnold conjecture and Gromov-Witten invariant K. Fukaya;K. Ono
  40. Monopole equation and the 11/8 conjecture M. Furuta
  41. Invent. Math. v.82 A new construction of symplectic curves in symplectic manifolds R. Gompf
  42. A proof of conjecture for the number of ramified covering of the sphere by the torus I. P. Goulden;D. M. Jackson
  43. Proc. Amer. Math. Soc. v.125 no.1 Transitive factorisations into transpositions and holomorphic mapping on the sphere I. P. Goulden;D. M. Jackson
  44. 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
  45. Invent. Math. v.82 Pseudo-holomorphic curves in symplectic manifolds M. Gromov
  46. Global Analysis(Papers in Honors of K. Kodaira) The signature of ramified coverings F. Hirzebruch
  47. Math. Ann. v.39 Uber Riemann'sche Flachen mit gegebenen Verzweigungspunkten A. Hurwitz
  48. Ann. of Math. v.142 A new construction of symplectic manifolds A. Hurwitz
  49. ICM v.2 Symplectic sums and Gromov-Witten Invariants E. N. Ionel
  50. Ann. of Math. Relative Gromov-Witten Invariants E. Ionel;T. H. Parker
  51. math. SG/0010217 The symplectic Sum Formula for Gromov-Witten Invariants E. Ionel;T. H. Parker
  52. Trans. Amer. Math. Soc. v.330 Intersection theory of moduli space of stable n-pointed curves of genus zero S. Keel
  53. Problems in low-dimensional topology R. Kirby
  54. hepth/9405035 Enumeration of Rational Curves Via Torus Actions M. Kontsevich
  55. hepth/9402147 Gromov-Witten classes, quantum cohomology and enumerative geometry M. Kontsevich;Y. Manin
  56. Comm. Math. Phys. v.164 Gromov-Witten classes, Quantum cohomology and Enumerative Geometry M. Kontsevich;Y. Manin
  57. On irreducible four-manifolds D. Kotschick
  58. Math. Res. Lett. v.2 four-manifolds without symplectic structures but with nontrivial Seiberg-Witten invariants D. Kotschick;S. Morgan;C. Taubes
  59. Math. Res. Lett. v.I no.2 The geneus of embedded surfaces in the projective plane P. Kronheimer;T. Mrowka
  60. Topology v.3 The homotopy type of the unitary group of Hilbert space N. H. Kuiper
  61. Conference on Complex Analysis New proof for the existence of local free complete families of complex structures Mo Kuranishi
  62. Math. Ann. v.208 The quotient space of ${\mathbb{CP}}^2$ by the complex conjugation is the 4-sphere Mo Kuranishi
  63. Math. alg-geom/ 9803036 Symplectic surgery and Gromov-Witten Invariants of Calabi-Yau 3-folds A. M. Li;Y. Ruan
  64. Turkish J. Math v.26 Minimality of certain connected sums T. J. Li;A. I. Stipsicz
  65. Comm. Math. Phys. v.213 The Number of Ramified covering of a Riemann Surface by Riemann Surface A. M. Li;G. Zhao;Q. Zheng
  66. Invent. Math. v.89 Example of symplectic structures D. McDuff
  67. J. Amer. Math. Soc. v.3 The structure of rational and ruled symplectic 4-manifolds D. McDuff
  68. Univ. Lecture Ser. v.6 J-holomorphic curves and Quantum cohomology D. McDuff;D. Salamon
  69. J-holomorphic curves and Quantum cohomology D. McDuff;D. Salamon
  70. Soviet Mathematics Doklady v.24 Multivalued functions and functionals-an of the Morse theory S. Novikov
  71. Comm. Math. Phys. v.85 Gauge theories on four dimensional Riemannian manifolds T. Parker
  72. J. Geom. Anal. v.3 Pseudo holomorphic maps and bubble trees T. Parker;J. Wolfson
  73. Proc. Sympos. Pure Math. v.32 Pseudo equivalence of G-manifolds T. Petrie
  74. Duke Math. J. v.83 Topological sigma model and Donaldson type invariant in Gromov theory Y. Ruan
  75. J. Differential Geom. v.42 A mathematical theory of quantum cohomogy Y. Ruan;G. Tian
  76. J. Math. Soc. Japan v.9 The Gauss-Bonnet theorem for V-manifolds I. Satake
  77. Lecture Notes in Math. v.638 The Atiyah-Singer Index Theorem P. Shanahan
  78. Global Analysis The curvature of 4-dimensional Einstein spaces I. Singer;J. Thorpe
  79. Amer. J. Math. v.87 An infinite dimensional version of Sard's theorem S. Smale
  80. Counting pseudo holomorphic submanifolds in dimension 4 C. Taubes
  81. Math. Res. Lett. v.2 More constraints on symplectic forms from Seiberg-Witten invariants C. Taubes
  82. J. Differential Geom. v.19 Self-dual connections on manifolds with indefinite intersection matrix C. Taubes
  83. J. Differential Geom. v.17 Self-dual connections on non-self-dual 4-manifolds C. Taubes
  84. The Geometry of the Seiberg-Witten invariants C. Taubes
  85. Math. Res. Lett. v.2 The Seiberg-Witten and te h Gromov invariants C. Taubes
  86. Math. Res. Lett. v.I The Seiberg-Witten invariants and symplectic forms C. Taubes
  87. Geometry and Physics (Aarhus, 1995) (Lecture Notes in Pure and Appl. Math., 184) Seigerg-Witten and Gromov invariants C. Taubes
  88. Oxford Univ. Thesis Gauge theory and involutions S. Wang
  89. Math. Res. Lett. v.1 Monopoles and four-manifolds E. Witten
  90. J. Differential Geom. v.117 Supersymmetry and Morse theory E. Witten
  91. Communications in Mathematical Topological sigma model E. Witten
  92. J. Differential Geom. v.24 Connections. cohomology and the intersection forms of 4-manifolds E. Witten
  93. 대우총서 432 다양체의 미분위상수학 조용승
  94. 이론물리의 수학적 접근 수학에서 게이지 이론 조용승
  95. Commun. Korean Math. Soc. v.15 no.3 심프렉틱 다양체의 불변량 조용승
  96. Proc. Amer. Math. Soc. v.130 Anti-Symplectic Involution with langrangian Fixed Loci and their Quotient Y. S. Cho;D. S. Joe
  97. 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
  98. Bull. Amer. Math. Soc. v.23 Elliptic methods in symplectic geometry D. McDuff
  99. Ann. of Math. v.93 no.2 The index of elliptic operations, IV.V. E. Witten