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2022년 7월 7일 (목) 04:17 기준 최신판

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  1. The figure shows part of the real points (of real dimension 2) in a certain complex K3 surface (of complex dimension 2, hence real dimension 4).[1]
  2. An (algebraic) K3 surface over any field means a smooth proper geometrically connected algebraic surface that satisfies the same conditions.[1]
  3. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero.[1]
  4. Together with two-dimensional compact complex tori, K3 surfaces are the Calabi–Yau manifolds (and also the hyperkähler manifolds) of dimension two.[1]
  5. Abstract: This monograph gives a view, from both dierential and algebraic geometry, of K3 surfaces.[2]
  6. First we build-up to the RiemannRoch theorem on K3 surfaces, the statement of which needs several powerful tools, including divisors and sheaves, which are introduced along the way.[2]
  7. Next we describe in detail the cohomology groups and Hodge decomposition of K3 surfaces, nishing with some current research directions.[2]
  8. Understanding K3 surfaces . . .[2]
  9. The natural pairing S where the cotangent sheaf S of a K3 surface is gives an algebraic symplectic structure.[3]
  10. For a K3 surface, by denition we have q(S) = 0 hence b1(S) = 0, and pg(S) = 1.[3]
  11. Check that the zero set of a generic section of these vector bundles gives a K3 surface.[3]
  12. dN ) hence if we arrange the weights correctly we will get K3 surfaces.[3]
  13. General introduction to K3 surfaces Svetlana Makarova MIT Mathematics Contents 1 Algebraic K3 surfaces 1.1 Denition of K3 surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .[4]
  14. 1 Algebraic K3 surfaces 1.1 Denition of K3 surfaces 1 1 3 9 9 11 12 16 Let K be an arbitrary eld.[4]
  15. One can observe several simple facts for a K3 surface: = TX ; 1. X 2. H2(OX ) = H0(OX ); 3.[4]
  16. There is a notion of complex K3 surfaces.[4]
  17. These notes provide an introduction to the geometry of K3 surfaces and the dynamics of their automorphisms.[5]
  18. In these notes we concentrate on K3 surface automorphisms.[5]
  19. K3 surfaces have several additional features that become handy when looking at their automor- phisms.[5]
  20. This is related to the Hodge structure on the cohomology of a K3 surface and quite a bit about an automorphism can be understood already by looking at its action on cohomology.[5]
  21. huybrech@math.uni-bonn.de N.B.: The copy of Lectures on K3 Surfaces displayed on this website is a draft, pre- publication copy.[6]
  22. Moduli spaces of polarized K3 surfaces 1.[6]
  23. Appendix: Lifting K3 surfaces Chapter 10.[6]
  24. Over the eld of complex numbers a more general notion exists that includes non-algebraic K3 surfaces.[6]
  25. We prove that the Chow motives of twisted derived equivalent K3 surfaces are isomorphic, not only as Chow motives (due to Huybrechts), but also as Frobenius algebra objects.[7]
  26. In particular, K3 surfaces in that family are pairwise non-isogenous but have isomorphic rational Hodge algebras.[7]
  27. ‘K3 surfaces play something of a magical role in algebraic geometry and neighboring areas.[8]
  28. They arise in astonishingly varied contexts, and the study of K3 surfaces has propelled the development of many of the most powerful tools in the field.[8]
  29. K 3 K3 K3 surfaces can be considered as a 2-dimensional analogue of an elliptic curve, and the theory of periods – called the Torelli-type theorem for K 3 K3 K3 surfaces – was established around 1970.[9]
  30. Since then, several pieces of research on K 3 K3 K3 surfaces have been undertaken and more recently K 3 K3 K3 surfaces have even become of interest in theoretical physics.[9]
  31. The theory of lattices and their reflection groups is necessary to study K 3 K3 K3 surfaces, and this book introduces these notions.[9]
  32. Thus the result for at four-tori does not carry over to K3 surfaces.[10]
  33. The rst author would also like to thank Mark Gross for many useful conversa- tions about K3 surfaces.[10]
  34. Preliminaries In this section we review basic results in Kahler geometry and the geom- etry of K3 surfaces that will be used in the proof of our result.[10]
  35. Let X be a K3 surface, that is, X is a compact, complex, simply connected surface with trivial canonical bundle.[10]
  36. These algebraic classes can be used to construct counter-examples to the Hasse principle on K3 surfaces via Brauer-Manin obstructions.[11]
  37. I omitted several active research topics due to time constraints, notably rational curves on K3 surfaces, modularity questions, and Mordell-Weil ranks of elliptic K3 surfaces over number elds.[11]
  38. I am particularly grateful to Brendan Hassett, Sho Tanimoto, and Bianca Viray; our joint projects and innumerable conversations have shaped my understanding of the arithmetic of K3 surfaces.[11]
  39. A polarized K3 surface is a pair (X, h), where X is an algebraic K3 surface and h H2(X, Z) is an ample class.[11]
  40. (For those who dont know, a K3 surface is a (smooth) surface X which is simply connected and has trivial canonical bundle.[12]
  41. In this second approach, we construct X rst instead of C. What we want is a K3 surface X together with two curves H and C on X. (H is the hyperplane section from the embedding X P3).[12]
  42. The rst step is to check that the topology of a K3 surface permits curves with these numerical properties to exist.[12]
  43. In addition, 4 DAVID R. MORRISON the signature of the pairing (the number of +1 and 1 eigenvalues) can be computed as (3,19) for a K3 surface.[12]
  44. Apr 22, MIT, 2-132, 1-3pm: Barbara Bolognese (NEU): Examples of compact hyperkahler manifolds as moduli spaces of sheaves on K3 surfaces, notes.[13]
  45. Apr 29, NEU, WVG 102, 12.30-2.30pm: Isabel Vogt (MIT): Deformation types of moduli spaces of stable sheaves on a K3 surface, notes; preceded by a short address by Emanuele on Elliptic K3 surfaces.[13]
  46. A Torelli theorem for Kähler-Einstein K3 surfaces, Springer Lect.[14]
  47. All K3 surfaces are Kähler manifolds.[15]
  48. It is known that there is a coarse moduli space for K3 surfaces, of dimension 20.[15]
  49. There is a period mapping and Torelli theorem for complex K3 surfaces.[15]
  50. Compactification on a K3 surface preserves one half of the original supersymmetry.[15]
  51. The Problem Problem: Find rational curves in a complex K3 surfaces.[16]
  52. Main questions: given a compact, complex K3 surface S, Q1 : does S contain a rational curve?[16]
  53. The Conjecture (on Q3) Conjecture: Any smooth complex K3 surface S contains innitely many rational curves.[16]
  54. A complex elliptic K3 surface contains innitely many rational curves.[16]
  55. We study the relationship between an elliptic (cid:12)bration on an ellip- tic K3 surface and its Jacobian surface.[17]
  56. Then we use the description to prove that certain K3 surfaces do not admit a non-Jacobian (cid:12)bration.[17]
  57. Some examples of K3 surfaces are discussed.[17]
  58. Introduction By a K3 surface we mean a simply connected projective complex surface with trivial canonical bundle.[17]
  59. Introduction U(k) In this paper we study the existence of correspondences between K3 surfaces X(k, m, n) with k, m, n N, Picard number 17, and transcendental lattices .[18]
  60. In Section 4 we consider a generic genus-2 curve and we show the existence of a correspondence between the Jacobian of the curve and a K3 surface with isomorphic transcendental lattice.[18]
  61. Since this construction involves a second K3 surface whose transcendental lattice has quadratic form multiplied by 2, in Sections 5 and 6 we generalize this rst example.[18]
  62. First, we construct K3 surfaces twisting each direct summand of the transcendental lattice of the Jacobian by natural numbers.[18]
  63. I will explain how wall-crossing with respect to Bridgeland stability conditions provides a new upper bound for the number of global sections of sheaves on K3 surfaces.[19]
  64. This, in particular, extends and completes a program proposed by Mukai to reconstruct a K3 surface from a curve on that surface.[19]
  65. Furthermore, the upper bound characterizes special vector bundles on curves on K3 surfaces, which have the maximum number of global sections for the minimum degree.[19]
  66. Therefore, it gives an explicit expression for Clifford indices of curves on K3 surfaces.[19]
  67. A key player in understanding the monodromy of families of polarized K3s is the symplectic mapping class group of a polarized K3 surface.[20]
  68. A K3 surface is a smooth two-dimensional algebraic variety which is simply connected and has trivial canonical class.[21]
  69. We will then give a few examples of K3 surfaces as well as a dimension count of various hypersurfaces in the moduli space of K3s.[21]
  70. In the examples given below, we will use this result as well as some arithmetic of divisors to determine when certain spaces have KX = 0, showing that they are K3 surfaces.[21]
  71. Thus, a degree 4 hypersurface in P3 is a K3 surface since it has trivial canonical class.[21]
  72. What can we say about K3 surfaces, topologically?[22]
  73. 2 Complex Algebraic Structure of K3 Surfaces Let X be a K3 surface.[22]
  74. It is well known that we have a Hodge decomposition for compact, complex Kahler manifolds of which K3 surfaces are an example.[22]
  75. Well say that (X, ) and (Y, ) are isomorphic as marked K3 surfaces if there exists a biholomorphism f X Y which also satises f = .[22]
  76. Lattices Lattice theory is an important tool when studying K3 surfaces.[23]
  77. If g Aut(S) we will often consider H 2(S, Z)g = {x H 2(S, Z) | gx = x} the invariant lattice The problem Study properties of automorphisms of nite order on K3 surfaces.[23]
  78. Some main problems: classify nite groups G that can act on a K3 surface and study their action on cohomology.[23]
  79. Most of the K3 surfaces have an innite automorphism group, only a nite number have nite automorphism group and the Picard lattices are known by results of Nikulin and Vinberg.[23]
  80. P. We have a similar result on the uniqueness of this expression of the K3 surface (see (4.1)).[24]
  81. The main step is to show that isogenies between Kuga-Satake abelian varieties induce isogenies between K3 surfaces, in the context of integral models of Shimura varieties.[25]
  82. Shafarevich dened an isogeny between two complex algebraic K3 surfaces X, X (cid:48) to be a Hodge isometry H 2(X (cid:48), Q) H 2(X, Q) (cf.[25]
  83. Starting with a complex algebraic K3 surface, one can easily construct an isogenous one by prescribing a dierent Z lattice in its rational Hodge structure.[25]
  84. Let X be a complex algebraic K3 surface.[25]
  85. Modular compactications of moduli spaces for polarized K3 surfaces are con- structed using the tools of logarithmic geometry in the sense of Fontaine and Illusie.[26]
  86. Polarized log K3 surfaces 6.[26]
  87. The stack of polarized log K3 surfaces 7.[26]
  88. Log K3 surfaces with level structure 8.[26]

소스

  1. ↑ 1.0 1.1 1.2 1.3 K3 surface
  2. ↑ 2.0 2.1 2.2 2.3 A brief survey of k3 surfaces
  3. ↑ 3.0 3.1 3.2 3.3 A lecture on k3 surfaces
  4. ↑ 4.0 4.1 4.2 4.3 General introduction to k3 surfaces
  5. ↑ 5.0 5.1 5.2 5.3 An introduction to k3 surfaces and their
  6. ↑ 6.0 6.1 6.2 6.3 Lectures on k3 surfaces
  7. ↑ 7.0 7.1 A motivic global Torelli theorem for isogenous K3 surfaces
  8. ↑ 8.0 8.1 Lectures on K3 Surfaces
  9. ↑ 9.0 9.1 9.2 $K3$ Surfaces
  10. ↑ 10.0 10.1 10.2 10.3 Area minimizers in a k3 surface and
  11. ↑ 11.0 11.1 11.2 11.3 Arithmetic of k3 surfaces
  12. ↑ 12.0 12.1 12.2 12.3 The geometry of k3 surfaces
  13. ↑ 13.0 13.1 Moduli of sheaves on K3 surfaces
  14. ↑ Surfaces $K3$
  15. ↑ 15.0 15.1 15.2 15.3 Academic Kids
  16. ↑ 16.0 16.1 16.2 16.3 Rational curves on k3 surfaces
  17. ↑ 17.0 17.1 17.2 17.3 Transactions of the
  18. ↑ 18.0 18.1 18.2 18.3 Michigan math. j. 52 (2004)
  19. ↑ 19.0 19.1 19.2 19.3 Workshop K3 surfaces
  20. ↑ Home page Eduard Looijenga
  21. ↑ 21.0 21.1 21.2 21.3 Introduction to k3 surfaces
  22. ↑ 22.0 22.1 22.2 22.3 Basic properties of k3 surfaces
  23. ↑ 23.0 23.1 23.2 23.3 Old and new on the symmetry groups of k3 surfaces
  24. ↑ Algebraic geometry and commutative algebra
  25. ↑ 25.0 25.1 25.2 25.3 Isogenies between k3 surfaces over ¯fp
  26. ↑ 26.0 26.1 26.2 26.3 Semi-stable degenerations and period spaces for

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