Holdings: Arithmetic and geometry of K3 surfaces and Calabi-Yau threefolds

Arithmetic and geometry of K3 surfaces and Calabi-Yau threefolds:

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Bibliographic Details
Format:	Book
Language:	English
Published:	New York, NY [u.a.] Springer 2013
Series:	Fields Institute communications 67
Subjects:	K 3- Fläche Calabi-Yau-Mannigfaltigkeit Konferenzschrift > 2011 > Toronto
Online Access:	Inhaltsverzeichnis
Item Description:	Literaturangaben
Physical Description:	XXVI, 602 S. Ill.
ISBN:	1461464021 9781461464020

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adam_text	Contents Part I Introductory Lectures КЗ and Enriques Surfaces ..................................... 3 Shigeyuki Ко η dö 1 Introduction ........................................... З 2 Lattices ............................................... 4 2.1 Definition ........................................ 4 2.2 Examples ....................................... 5 2.3 Un.imodu.Iar Lattices ............................ 5 2.4 Proposition ....................................... 5 2.5 Proposition ...................................... 5 2.6 Discriminant Quadratic Form ....................... 6 2.7 Overlattices ...................................... 6 2.8 Proposition ....................................... 7 2.9 Examples ........................................ 7 2.10 Primitive Embeddings .............................. 8 2.11 Proposition ....................................... 8 2.12 Corollary ........................................ 8 2.13 Example ......................................... 9 2.14 Corollary ........................................ 9 2.15 Corollary ..................................... . . . 9 2.16 Theorem. ([30]) ................................... 10 3 Periods of КЗ and Enriques Surfaces ......................... 10 3.1 Periods of КЗ Surfaces ............................. 10 3.2 Periods of Enriques Surfaces ........................ 11 3.3 Remark .......................................... 12 3.4 Remark .......................................... 13 3.5 Example ......................................... 14 4 Automorphisms ........................................... 15 4.1 Torelli Type Theorem and the Group of Automorphisms for an Algebraic КЗ Surface ........................ 15 xvi Contents 4.2 Theorem ([35]) ................................... 16 4.3 Theorem ([35])................................... 16 4.4 Corollary ........................................ 17 4.5 The Leech Lattice and the Group of Automorphisms of a Generic Jacobian Kummer Surface ............... 17 4.6 Theorem (f 10], Chap. 27)........................... 18 4.7 Proposition ([4]).................................. 18 4.8 Finite Groups of Automorphisms of A 3 Surfaces ....... 19 4.9 Proposition ([29])................................. 19 4.10 Proposition ([29])................................. 20 4.11 Theorem ([28J) ................................... 20 4.12 Automorphisms of Enriques Surfaces ................. 21 4.13 Theorem ([2, 31], Theorem 10.1.2)................... 21 5 Borcherds Products ........................................ 22 5.1 Theorem ([7]) .................................... 23 5.2 Example ([5])..................................... 23 5.3 Example ([6, 7]).................................. 24 5.4 Example ([15, 25])................................ 25 References ..................................................... 27 Transcendental Methods in the Study of Algebraic Cycles with a Special Emphasis on Calabi-Yau Varieties 29 James U. Lewis 1 Introduction .............................................. 29 2 Notation ................................................. 31 3 Some Hodge Theory ....................................... 31 3.4 Formalism of Mixed Hodge Stxuctures ................ 33 4 Algebraic Cycles .......................................... 36 4.6 Generalized Cycles ................................ 38 5 A Short Detour via Мил or Ä -Theory ......................... 40 5.1 The Gersten-Milnor Complex ....................... 41 6 Hypercohomology ........................................ 42 7 Deligne Cohomology ...................................... 43 7.6 Alternate Take on Deligne Cohomology ............... 46 7.8 Deligne-Beihnson Cohomology ..................... 47 8 Examples οίΉΓτ3^{Χ,Κ^χ) and Corresponding Regulators ....... 51 8.1 Case m ^ 0 and CY Threefolds ...................... 51 8 5 Deligne Cohomology and Normal Functions ........... 55 8.9 Case m ^ 1 and КЗ Surfaces ........................ 56 8.18 Torsion Indécomposables ........................... 59 8.2.1 Case m - 2 and Elliptic Curves ...................... 60 8.2.2 Constructing K2( X) Classes on Elliptic Curves X ....... 61 References ..................................................... 68 Contents xvi IVo Lectures on the Arithmetic of КЗ Surfaces 71 Matthias Schutt 1 Introduction .............................................. 71 2 Motivation: Rational Points on Algebraic Curves ............... 72 3 КЗ Surfaces and Rational Points............................. 73 4 Elliptic КЗ Surfaces ....................................... 74 5 Picard Number One ....................................... 76 6 Computation of Picard Numbers ............................. 77 7 КЗ Surfaces of Picard Number One .......................... 79 7.1 van Luijk s Approach .............................. 80 7.2 Kloosterman s Improvement ........................ 80 7.3 Ejsenhans-Iahneľs Work. ........................... 80 7.4 Outlook .......................................... 81 7.5 Feasibility ........................................ 81 8 Hasse Principle for КЗ Surfaces ............................. 82 9 Rational Curves on КЗ Surfaces ............................. 83 10 Isogeny Notion for КЗ Surfaces ............................. 83 1.1 Singular КЗ Surfaces ...................................... 84 і 1.1. Torelli Theorem for Singular КЗ Surfaces ............. 85 11.2 Surjecti vity of the Period Map ....................... 86 11.3 Singular Abehan Surfaces ........................ 86 12 Shioda-lnose Structures .................................... 87 13 Mordell-Weil Ranks of Elliptic КЗ Surfaces ................... 89 14 Fields of Definition of Singular КЗ Surfaces ................... 91 14.1 Mordeü -Weil Ranks Over Q ....................... 93 15 Modularity of Singular КЗ Surfaces .......................... 93 References ..................................................... 96 Modularity of Calabi-Yau Varieties: 2011 and Beyond ...................................................101 Noriko Yur 1 Introduction ..............................................102 1.1 Brief History Since 2003 ...........................102 1.2 Plan of Lectures ...................................102 1.3 Disclaimer .......................................103 1.4 Calabi-Yau Varieties: Definition .....................103 2 The Modularity of Galois Representations of Calabi- Yau Varieties (or Motives) Over Q ...............................105 3 Results on Modularity of Galois Representations ...............110 3.1 Two-DimensionaJ Galois Representations Arising from Calabi-Yau Varieties Over Q ...................110 3 2 Modularity of Higher Dimensional Galois Representations Arising from КЗ Surfaces Over Q ......112 xviii Contents 3.3 The Modularity of Higher Dimensional Galois Representations Arising from Calabi-Yao Threefolds Over Q ..........................................117 4 The Modularity of Mirror Maps of Calabi-^Yau Varieties, and Mirror Moonshine .....................................126 5 The Modularity of Generating Functions of Counting Some Quanü ties on Calabi-Yau Varieties ..................... 129 6 Future Prospects ..........................................130 6.1 The Potential Modularity ...........................130 6.2 The Modularity of Moduli of Families of Calabi-Yau Varieties ............................131 6.3 Congruences,Forma] Groups ........................131 6.4 The, Griffiths Intermediate Jacobians of Calabi-Yau Threefolds ...........................131 6.5 Geometric Realization Problem (the Converse Problem) ............................132 6.6 Modular Forms and G romo v -Witten Invariants .........133 6.7 Auiomorphic Black Hole Entropy .................... L33 6.8 Af^-Moonshine.................................... 133 References .....................................................137 Part II Research Articles: Arithmetic and Geometry of КЗ, Enriques and Other Surfaces Explicit Algebraic Coverings of a Pointed Torus 143 Ane ST. Anema and Jaap Top і Introduction ..............................................143 2 The Coverings ............................................144 3 The Proofs ...............................................145 3.1 2 -Torsion .................,.......................145 3.2 3-Torsion .........................................146 3.3 ¿-Torsion with і > 5...............................147 4 Intermediate Coverings ....................................150 4.1 All x-Coordinates .................................150 4.2 One Point ........................................151 4.3 One x-Coordinare .................................151 References .....................................................152 Elliptic Fibrations on the Modular Surface Associated to Ι ι (8) 153 M.F Bertin and O. Lecacheux 1 Introduction ..............................................153 2 Definitions ...............................................155 3 D і seri nun an t Forms .......................................156 4 Root Lathees .............................................156 4.1 A /An ...........................................157 4.2 Df/Di ...........................................157 4.3 Ej/Es ...........................................158 Contents 6 1 The Primitive Embeddings of Lattices .................. 6. 2 Generators of L/Lj-oot ..... 6. 3 ¿root - - hgD]fj ............ 6. .4 ¿root= = E -, D i o ............ 6 5 ¿root ~ ~ Ει Α ι ν ............. 6. 6 ¿root= -D] ............... 6. ¿root - -DcjA^ ............ 6.8 ¿root - -Ą ■_■■■........... 6. 9 ¿root - 6 10 ¿rootr - ¿^ ............... 6. 11 ¿root - l)6aI ............. 6 12 = p2ai 4.4 EÎ/E, ...........................................158 4.5 Щ/Ба ...........................................158 Elliptic Eibrations .........................................159 5.1 КЗ Surfaces and Elliptic Fibrations ...................159 5.2 Nikulin and Niemeier s Results ......................161 5.3 Nishiyama s Method ...............................162 Elliptic Eibrations of Yi ....................................164 ©A: into Root .....................164 .....................168 .....................168 .....................170 .....................170 .....................171 .....................171 .....................171 ....................1 /J 7 Equations of the Eibrations ................................173 7.1 Equation of the Modular Surface Associated to the Modular Group /Ί(8) .........................174 7.2 Construction of the Graph from che Modular Eibration . . 175 7.3 Two Eibrations ....................................176 7.4 Divisors .........................................177 8 Fibratrons from the Modular Fibration ........................179 9 A Second Set of Eibrations: Gluing and Breaking ..............186 9.1 Classical Examples ................................186 9.2 Fibration with a Singular Fiber of Type /„, η Large ......187 9.3 Eibrations with Singular Fibers of Type 1*.............190 9.4 Breaking .........................................192 10 Last Set ..................................................192 10.1 From Fibration of Parameter ρ ......................193 10.2 From Fibration of Parameter б .......................195 References .....................................................198 Universal Kammer Families Over Shimura Curves 201 Amnon Besser and Ron Livné 1 introduction ..............................................201 2 КЗ Surfaces with Picard Number 19 and Twists of Elliptic Surfaces .................................................204 2.1 Lattices ..........................................204 2.2 Elliptic Surfaces ..................................205 2.3 Quadratic Twists ..................................206 2.4 The Basic Construction .............................207 2.5 The Néron- -Seven and the Transcendental Lattices ......208 Contents 3 The Moduli Map for Discriminants 6 and 15...................211 3.1 Marked Elliptic Fibrations and Moduli Spaces .........212 3.2 Types of Marked КЗ Surfaces .......................213 3.3 Abehan Surfaces with Quaternionic Multiplication .....216 3.4 The Associated Kummer Surface ....................218 3.5 The Basic Isomorphism ............................220 3.6 Local Monodromies ...............................221 3.7 Case Number 1 on the List ..........................223 3.8 Case Number 3 on the List ..........................225 4 Isogenies Between Abelian Surfaces and Discriminant Forms ... 226 4.1 The Theory of Discriminant Forms ...................226 4.2 Rank 4 Lattices and Discriminant Forms ..............228 4.3 Applications to Isogenies of Abelian Varieties ..........231 4.4 Abelian Varieties with Multiplication by EichJer Orders ...........................................233 4.5 Further Analysis ..................................234 5 Isogcnies Related to Abelian Surfaces with Quaternionic Multiplication ...........................................235 5.1 A Special Subgroup ...............................236 5.2 The Integral Cohomology of a QM Abehan Surface .....236 5.3 The Type of the Special Subgroup ....................237 5.4 Discriminant Forms Associated w.ith QM Abelian Surfaces .........................................238 6 A Special Isogeny .........................................238 6 1 The Isogeny ......................................238 6.2 A Converse Theorem ..............................240 6.3 Level Structures ...................................241 7 Isogeni.es and the Morrison Correspondence ...................242 7.1 The Morrison Correspondence .......................242 7.2 NikultnMarkings .................................243 7.3 The Néron- Seven Lattice of the Quotient Surface ......244 7.4 The Precise Correspondence ........................245 7.5 A Transcendental Description .......................250 8 Explicit Computations .....................................252 8.1 Number 5 on the List ..............................253 8.2 Case Number 9 on the List ..........................254 8 3 Number 10 on the List .............................256 8.4 Number 6 on the List ..............................256 8.5 Number 2 on the List ..............................257 8.6 Number 7 on the List ..............................258 8.7 Number 8 on the List ..............................259 8.8 Number 11 on the List .............................259 Appendix ......................................................260 A.I Rational Invariants of Quadratic Forms Associated with Singular Fibers ...................................................260 A.LI Quadratic Forms Over Qp ..........................260 Contents xxi Λ. 1.2 Quadratic Forms of Singular Fibers ..................261 A. 1.3 Ternary Fornis of Quaternion Algebras ................263 References .....................................................264 Numerical Trivial Automorphisms of Enriques Surfaces in Arbitrary Characteristic ......................................................267 Igor V. DoJgachev 1 Introduction ..............................................267 2 Generalities ..............................................268 3 Lefschetz Fixed Point Formala ..............................270 4 Cohomologically Trivial Automorphisms .....................271 5 Numerically Trivial Automorphisms .........................275 6 Examples ................................................276 7 Extra Special Enriques Surfaces .............................281 References .....................................................283 Picard-Fuchs Equations of Special One-Parameter Families of Invertible Polynomials ............................................285 S wan tie G a h r s 1 Introduction .............................................285 2 Preliminaries on Invertibile Polynomials .......................287 3 The Picard-Fuchs Equation for Invertible Polynomials and Consequences .............................290 3.1 The GKZ System for Invertible Polynomials ...........291 3.2 The Picard-Fuchs Equation .........................296 3.3 Statements on the Cohomology of the Solution Space . . . 299 3.4 The Case of Arnold s Strange Duality ................301 3.5 Relations to the Poincaré Senes and Monodromy .......305 References .....................................................309 A Structure Theorem for Fibratioas on Delsarte Surfaces 311 Bas Heyne and Retake Kloosterman 1 Introduction ..............................................311 2 Delsarte Surfaces .........................................313 3 IsotriviaJ Fibratioas ........................................323 References .....................................................332 Fourier Mukai Partners and Polarised КЗ Surfaces 333 K. Hulek and D. PI oog 1 Review Fourier-Mukai Partners of КЗ Surfaces ................334 1.1 Hislory: Derived Categories in Algebraic Geometry .....334 1.2 Derived Categories as Invariants of Varieties ...........335 1.3 Fourier- Muk ai Partners ............................336 1.4 Derived ami Birational Equivalence ..................338 xxii Contents 2 Lattices ..................................................338 2.1 Gram Matrices ....................................339 2.2 Genera ..........................................340 3 OverlatLices ..............................................342 3.9 Overlattices from Primitive Embeddmgs ..............346 4 K.3 Surfaces ..............................................350 5 Polarised КЗ Surfaces .....................................352 6 Polarisation and FM Partners ...............................356 7 Counting LM Partners of Polarised КЗ Surfaces in Lattice Terms ..........................................358 8 Exam pJ es................................................361 References .....................................................364 On a Family of КЗ Surfaces with У^ Symmetry ........................367 Dagan Karp, Jacob Lewis, Daniel Moore, Dmitri Skjorshammer, and Ursula Wbitcher 1 Introduction ..............................................368 2 Tone Varieties and Senuample Hypersurfaces .................369 2.1 Tone Varieties and Reflexive Polytopes ..............369 2.2 SerniampJe Hypersurfaces and the Residue Map ........371 3 Three Symmetric Families of КЗ Surfaces ....................374 3.1 Symplectic Group Actions on КЗ Surfaces .............^74 3.2 An J^j Symmetry of Polytopes and Hypersurfaces ......374 4 Picard- Fuchs Equations ....................................378 4.1 The Griffiths-Dwork Technique .....................378 4.2 A Picard- Fuchs Equation ...........................379 5 Modularity and Its Geometric Meaning .......................381 5.1 Elliptic Fibrauons on КЗ Surfaces ....................382 5.2 Kummer and Shioda-Inose Structures Associated to Products of Elliptic Curves ........................383 5.3 Modular Groups Associated to Our Families of КЗ Surfaces .........................................384 References ......................................................385 KfA of Elliptically Fibered КЗ Surfaces: A Tale of Two Cycles 387 Matt Kerr 1 Introduction ..............................................387 2 Real and Transcendenta] Regulators .........................388 3 The Ар -éry Family and an Inhomogeneous Picard-Fuchs Equation ................................................391 4 Åi-Polanzed КЗ Surfaces and a Higher Green s Function .......397 5 Proof of the Tauberian Lemma 2.............................405 References .....................................................408 Contents xxiii A Note About Special Cycles on Moduli Spaces of КЗ Surfaces ..........411 Stephen Kudla 1 Introduction.............................................. 411 2 Special Cycles for Orthogonal Groups ........................412 2.1 Arithmetic Quotients ...............................412 2.2 Special Cycles ....................................412 3 Modular Generating Senes .................................414 4 The Case of КЗ Surfaces ...................................416 4.1 Modular Interprétation of the Special Cycles ...........418 4.2 Some Applications ................................418 5 Kuga-Satake AbeLiao Va.net.ies and Special Endomorphisms .....423 5.1 The Kuga-Satake Const.ruct.ion ......................423 5.2 Special Cycles and Specjał Endomorphisms ...........425 References .....................................................426 Enriques Surfaces of IIutchinson-Gopel Type and Mathieu Automorphisms ...................................................429 Shigeru Mukai and Hisanori Ohashi 1 Introduction .............................................429 2 Rational Surfaces and Enriques Surfaces .....................432 3 Abelian Surfaces and Enriques Surfaces ......................434 4 Sexti.c Enriques Surfaces of Diagonal Type ...................439 5 Action of C of Mathieu Type on Enriques Surfaces of ľlutchinson-Gopel Type .................................442 6 Examples of Mathieu Actions by Large Groups ................446 7 The Characterization ......................................451 References .....................................................453 Quartic КЗ Surfaces and Cremona Transformations ...................455 Key і Ogni so 1 Introduction ..............................................455 2 Proof of Theorem 1(1)(2) ...................................456 3 Proof of Theorem 1(3).....................................458 References .....................................................460 Invariants of Regular Models of the Product of Two Elliptic Curves at a Place of Multiplicative Reduction ................................461 Chad Schoen 1 Introduction.............................................. 461 2 Notaüons ................................................463 2.1 Basic Notations ...................................463 2.2 Notations Related to the Closed Fiber, F, of π : í -->■ Γ . . 463 2.3 Notation Related to Componente of V and Then Intersections .............................464 3 The Weil Divisor Class Group of V ...........................464 4 The Homology and Cohomology of V¿ .......................468 xxiv Contents 5 Variation of Lhe Isomorphism Q ass of V ......................471 6 The Picard Group of V .....................................473 7 Senu -sta ble Models and Small Resolutions ....................479 8 The Sheaves R ftZ/n ......................................481 8.1 Notations ........................................481 References .....................................................487 Part Ш Research Articles: Arithmetic and Geometry of Calabi-Yau Threefolds and Higher Dimentional Varieties Dynamics of Special Points on Intermediate Jacobians ..................491 X) Chen and James Ü. Lewis 1 Introduction ..............................................491 2 Some Preliminaries ........................................492 3 Main Results .............................................494 References .....................................................498 Calabi-Yau Coni fold Expansions .....................................499 Sławomir Cynk and Duco van Straten 1 Introduction ..............................................499 2 How to Compute Picard-Fuchs Operators ...................· ■ 502 2.1 The Method of Griffiths Dwork .....................502 2.2 Method of Period Expansion ........................503 3 Double OcLics ............................................506 4 An Algorithm ............................................511 References .....................................................514 Quadratic Twists of Rigid Calabi-Yau Threefolds Over Q ...............517 Fernando Q. Gouvêa, Іал Kiming, and Noriko Yin 1 Introduction ..............................................518 2 Quadratic Twists of Rigid Calabi-Yau Threefolds ..............518 2.1 Easy Examples of Twists ...........................521 2.2 Self-fiber Products of Rational Elliptic Surfaces with Section and Their Twists .......................522 2.3 The Schoen Qui n tic and Its Quadratic Twists ..........523 2.4 Explicit Description for a HoJomorphic 3-Form for a Complete Intersection Calabi-Yau Threefold ......525 2.5 Two Rigid Calabi-Yau Threefolds of Werner and van Geemen ..................................525 2.6 The Rigid Calabi-Yau Threefold of van Geemen and Ny gaard ......................................527 3 Remarks on the Levels of Twists .............................528 4 Final Remarks ............................................528 4.1 An Explicit Unresolved Case ........................528 4.2 The Question About Existence of Geometric Twists .....529 4.3 The Fixed Point Set of the Involution ι ................530 References .....................................................532 Contents xxv Counting Sheaves on Calabi-Yau and Abelian Threefolds 535 Martin G. Gulbrandsen 1 Virtual Counts ............................................536 1.1 Deformation Invariance ............................536 1.2 Virtual .Fundamental Class ..........................536 1.3 Obstruction Theory ................................538 1.4 Behxend s Weighted Euler Characteristic ..............541 2 Abelian Threefolds ........................................542 2.1 Determinants .....................................542 2.2 Translation and Twist ..............................544 .References .....................................................547 The Segre Cubic and Borcherds Products 549 Sh.igeyu.ki Rondo 1 Introduction ..............................................549 2 The Segre Cubic Threefold .................................550 3 Λ Complex Ball Quotieni ..................................551 3.1 A Complex Ball ...................................551 3.2 Roots and Reflections ..............................553 3.3 Ball Quotient and Heegner Divisors ..................554 3.4 Interpreiauon via КЗ Surfaces .......................555 4 Weil Representation .....................................557 5 Borcherds Products ........................................558 6 Grksenko -Borcherds Liftings ...............................561 References .....................................................564 Quasi-modular Forms Attached to Hodge Structures 567 Hossein Movasati 1 Introduction ..............................................567 2 Moduli of Polarized Hodge Structures ........................570 2.1 The Space of Polarized Lattices ......................570 2.2 Hodge. Filtration ..................................571 2.3 Period Domain U .................................572 2.4 An Algebraic Group ...............................573 2.5 Griffiths Period Domain ............................574 3 Period Map ..........................,....................575 3.1 PoincarcDual ....................................575 3 2 Period Matrix .....................................576 3.3 A Canonical Connection on Ĺ .......................576 3 4 Some Functions on Ľ ..............................577 4 Quasi-modular Forms Attached to Hodge Structures ............578 4.1 Enhanced Projective Varieties .......................578 4.2 Period Map .......................................580 4.3 Quasi-modular Forms ..............................580 χ,χνί Contents 5 Examples ................................................581 5.1 Siegel Quasi-modular Forms ........................582 5.2 Hodge Numbers, 1,1,1,1 ..........................584 References .....................................................586 The Zero Locus of the Infinitesimal Invariant ..........................589 G. PearJstein and Ch. Schnell 1 Introduction ..............................................589 2 Proof of the Theorem ......................................591 2.1 AJgebraic Description of the Zero Locus ..............591 2.2 A More Sophisticated Description ....................592 2.3 Zero Loci of Sections of Coherent Sheaves ............595 3 Relation to AJgebraic Cycles ................................598 3.1 Green-Griffiths Program ............................598 References .....................................................601
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genre_facet	Konferenzschrift 2011 Toronto
id	DE-604.BV041128732
illustrated	Illustrated
indexdate	2024-07-10T00:40:14Z
institution	BVB
isbn	1461464021 9781461464020
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-026104603
oclc_num	856813871
open_access_boolean
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physical	XXVI, 602 S. Ill.
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publisher	Springer
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series	Fields Institute communications
series2	Fields Institute communications
spelling	Arithmetic and geometry of K3 surfaces and Calabi-Yau threefolds Radu Laza ..., eds. New York, NY [u.a.] Springer 2013 XXVI, 602 S. Ill. txt rdacontent n rdamedia nc rdacarrier Fields Institute communications 67 Literaturangaben K 3- Fläche (DE-588)4162958-9 gnd rswk-swf Calabi-Yau-Mannigfaltigkeit (DE-588)4440893-6 gnd rswk-swf (DE-588)1071861417 Konferenzschrift 2011 Toronto gnd-content K 3- Fläche (DE-588)4162958-9 s Calabi-Yau-Mannigfaltigkeit (DE-588)4440893-6 s DE-604 Laza, Radu Sonstige oth Erscheint auch als Online-Ausgabe 978-1-4614-6403-7 Fields Institute communications 67 (DE-604)BV035418374 67 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026104603&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Arithmetic and geometry of K3 surfaces and Calabi-Yau threefolds Fields Institute communications K 3- Fläche (DE-588)4162958-9 gnd Calabi-Yau-Mannigfaltigkeit (DE-588)4440893-6 gnd
subject_GND	(DE-588)4162958-9 (DE-588)4440893-6 (DE-588)1071861417
title	Arithmetic and geometry of K3 surfaces and Calabi-Yau threefolds
title_auth	Arithmetic and geometry of K3 surfaces and Calabi-Yau threefolds
title_exact_search	Arithmetic and geometry of K3 surfaces and Calabi-Yau threefolds
title_full	Arithmetic and geometry of K3 surfaces and Calabi-Yau threefolds Radu Laza ..., eds.
title_fullStr	Arithmetic and geometry of K3 surfaces and Calabi-Yau threefolds Radu Laza ..., eds.
title_full_unstemmed	Arithmetic and geometry of K3 surfaces and Calabi-Yau threefolds Radu Laza ..., eds.
title_short	Arithmetic and geometry of K3 surfaces and Calabi-Yau threefolds
title_sort	arithmetic and geometry of k3 surfaces and calabi yau threefolds
topic	K 3- Fläche (DE-588)4162958-9 gnd Calabi-Yau-Mannigfaltigkeit (DE-588)4440893-6 gnd
topic_facet	K 3- Fläche Calabi-Yau-Mannigfaltigkeit Konferenzschrift 2011 Toronto
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026104603&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV035418374
work_keys_str_mv	AT lazaradu arithmeticandgeometryofk3surfacesandcalabiyauthreefolds

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