Table Of ContentTracts in Mathematics 32 S Tracts in Mathematics 32
h
i
g
e
y
u
k
i
K
o
n
d
o¯
Shigeyuki Kondo¯
Shigeyuki Kondo¯
K 3 Surfaces
K 3 surfaces are a key piece in the classification of complex analytic or K K 3 Surfaces
algebraic surfaces. The term was coined by A. Weil in 1958 – a re sult 3
of the initials Kummer, Kähler, Kodaira, and the mountain K2 found in
Karakoram. The most famous example is the Kummer surface d iscovered
S
in the 19th century.
u
K 3 surfaces can be considered as a 2-dimensional analogue of an elliptic r
curve, and the theory of periods – called the Torelli-type theorem for f
K 3 surfaces – was established around 1970. Since then, several pieces a
of research on K 3 surfaces have been undertaken and more rece ntly K 3 c
surfaces have even become of interest in theoretical physics. e
s
The main purpose of this book is an introduction to the Torelli-type
theorem for complex analytic K 3 surfaces, and its applications. The theory
of lattices and their reflection groups is necessary to study K 3 surfaces,
and this book introduces these notions. The book contains, as well as
lattices and reflection groups, the classification of complex analyt ic
surfaces, the Torelli-type theorem, the subjectivity of the period map,
Enriques surfaces, an application to the moduli space of plane quartics,
finite automorphisms of K 3 surfaces, Niemeier lattices and the Mathieu
group, the automorphism group of Kummer surfaces and the Leech lattice.
The author seeks to demonstrate the interplay between several sorts
of mathematics and hopes the book will prove helpful to researchers in
algebraic geometry and related areas, and to graduate students with a
basic grounding in algebraic geometry.
ISBN 978-3-03719-208-5
www.ems-ph.org
Kondo Cover | Font: Nuri_Helvetica Neue | Farben: Pantone 116, Pantone 287 | RB 25? mm
EMS Tracts in Mathematics 32
EMS Tracts in Mathematics
Editorial Board:
Michael Farber (Queen Mary University of London, Great Britain)
Carlos E. Kenig (The University of Chicago, USA)
Michael Röckner (Universität Bielefeld, Germany, and Purdue University, USA)
Vladimir Turaev (Indiana University, Bloomington, USA)
Alexander Varchenko (The University of North Carolina at Chapel Hill, USA)
This series includes advanced texts and monographs covering all fields in pure and applied
mathematics. Tracts will give a reliable introduction and reference to special fields of
current research. The books in the series will in most cases be authored monographs,
although edited volumes may be published if appropriate. They are addressed to graduate
students seeking access to research topics as well as to the experts in the field working
at the frontier of research.
For a complete listing see our homepage at www.ems-ph.org.
14 Steffen Börm, Efficient Numerical Methods for Non-local Operators. 2-Matrix
Compression, Algorithms and Analysis
15 Ronald Brown, Philip J. Higgins and Rafael Sivera, Nonabelian Algebraic Topology.
Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids
16 Marek Janicki and Peter Pflug, Separately Analytical Functions
17 Anders Björn and Jana Björn, Nonlinear Potential Theory on Metric Spaces
18 Erich Novak and Henryk Woz´niakowski, Tractability of Multivariate Problems.
Volume III: Standard Information for Operators
19 Bogdan Bojarski, Vladimir Gutlyanskii, Olli Martio and Vladimir Ryazanov, Infinitesimal
Geometry of Quasiconformal and Bi-Lipschitz Mappings in the Plane
20 Hans Triebel, Local Function Spaces, Heat and Navier–Stokes Equations
21 Kaspar Nipp and Daniel Stoffer, Invariant Manifolds in Discrete and Continuous
Dynamical Systems
22 Patrick Dehornoy with François Digne, Eddy Godelle, Daan Kramer and Jean Michel,
Foundations of Garside Theory
23 Augusto C. Ponce, Elliptic PDEs, Measures and Capacities. From the Poisson Equation
to Nonlinear Thomas–Fermi Problems
24 Hans Triebel, Hybrid Function Spaces, Heat and Navier-Stokes Equations
25 Yves Cornulier and Pierre de la Harpe, Metric Geometry of Locally Compact Groups
26 Vincent Guedj and Ahmed Zeriahi, Degenerate Complex Monge–Ampère Equations
27 Nicolas Raymond, Bound States of the Magnetic Schrödinger Operator
28 Antoine Henrot and Michel Pierre, Shape Variation and Optimization. A Geometrical Analysis
29 Alexander Kosyak, Regular, Quasi-regular and Induced Representations of Infinite-
dimensional Groups
30 Vladimir G. Maz’ya, Boundary Behavior of Solutions to Elliptic Equations in General Domains
31 Igor V. Gel‘man and Vladimir G. Maz’ya, Estimates for Differential Operators in Half-space
Shigeyuki Kondo¯
K 3 Surfaces
Author:
Shigeyuki Kondo¯
Graduate School of Mathematics
Nagoya University
Chikusa-ku
464-8602 Nagoya
Japan
[email protected]
2010 Mathematical Subject Classification (primary; secondary): 14J28; 14C34, 14J10, 14J15, 14J50,
32G20
Key words: K 3 surface, Enriques surface, Kummer surface, Torelli-type theorem, period, lattice,
reflection group, automorphism group
Translated from the Japanese by the author
Authorized translation from the Japanese language edition from chapter 0 to chapter 10, entitled
K3 kyokumen by Shigeyuki Kondo, Sugaku no Kagayaki series 5, published by Kyoritsu Shuppan Co., Ltd.
All rights reserved
© Shigeyuki Kondo 2015
ISBN 978-3-03719-208-5
The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and
the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broad-
casting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use
permission of the copyright owner must be obtained.
© 2020 European Mathematical Society
Contact address:
European Mathematical Society Publishing House
TU Berlin Mathematikgebäude, Room MA266
Straße des 17. Juni 136
10623 Berlin
Germany
Email: [email protected]
Homepages: ems-ph.org / ems.press
Typeset using the author’s TEX files: Alison Durham, Manchester, UK
Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany
∞ Printed on acid free paper
9 8 7 6 5 4 3 2 1
To Mizuho
Preface to the English translation
This book is an extended English version of the author’s book “K3 surfaces” in
Japanese,whichformsvolume5oftheseries“SuugakunoKagayaki”,publishedin
2015 by Kyoritsu Shuppan, Japan. Chapters 0–10 are an English translation of the
above book by the author himself. Chapters 11 and 12 are new and added to this
Englishversionbytheauthor.
InChapter11wediscussfinitegroupsofsymplecticautomorphismsof K3sur-
faces. Inparticularwestudyarelation,discoveredbyS.Mukai,betweenthesegroups
and the Mathieu group, a finite sporadic simple group. We give a lattice-theoretic
proofofMukai’sresultbyusingtheclassificationofevendefiniteunimodularlattices
ofrank24,calledNiemeierlattices.
In Chapter 12, we study the Kummer surface associated with the Jacobian of a
curveofgenus2. ItisknownthattheautomorphismgroupoftheKummersurface
is discrete and infinite. During the 19th century and at the beginning of the 20th
century, geometers constructed many automorphisms of the Kummer surface. We
show that these generate the automorphism group of a generic Kummer surface.
Roughlyspeaking,werealize32non-singularrationalcurvesontheKummersurface
forming a beautiful configuration, called the (166)-configuration, in terms of 32
octads appearing in the Steiner system. Here we use essentially a description of
the fundamental domain of the reflection group of the even unimodular lattice of
signature(1,25)discoveredbyJ.H.Conway.
InthisbookwerestrictourselvestocomplexK3andEnriquessurfaces. Moreover,
the last three chapters are based on the author’s work. References are not complete
andarekepttoanecessaryminimum. WereferthereadertothebookonK3surfaces
byHuybrechts[Huy1]andasurveypaperonEnriquessurfacesbyDolgachev[Do3]
forthecaseofpositivecharacteristicandforrecentprogress.
Finally,GerardvanderGeerkindlyencouragedtheauthortopublishanEnglish
version and, moreover, he read the manuscript and pointed out many misprints.
MatthiasSchüttalsoreadthemanuscriptandsuggestedseveralimprovementstothe
author. The editors of the series “Suugaku no Kagayaki” and Kyoritsu Shuppan
willingly accepted the English translation. In taking this opportunity the author
wouldliketothankthemheartily.
Preface
ThemainthemeofthisbookistheTorelli-typetheoremforK3surfaces. AK3surface
isaconnectedcompact2-dimensionalcomplexmanifoldthatissimplyconnectedand
whosecanonicallinebundleistrivial. Itisdifficulttorelatethename“K3surface”
to its definition, but A. Weil invented the name, with K3 resulting from the initials
of Kummer, Kähler, Kodaira, as well as from a mountain in Karakoram called K2,
whichwasunclimbedandmysteriousatthattime. Themostfamousexampleisthe
Kummer surface, discovered in the 19th century. In the case of elliptic curves, that
is, 1-dimensional compact complex tori, the period of an elliptic curve determines
its isomorphism class. For K3 surfaces one can define the notion of periods, and
the claim that the isomorphism class of a K3 surface is determined by its period
is the Torelli-type theorem for K3 surfaces. In the 1970s, the Torelli-type theorem
was proved and then many results were established using this theorem. Since the
1990s, K3 surfaces have become of interest in mathematical physics. Nowadays,
K3surfacesarestillmysterious: forexample,afewyearsagophysicistsdiscovered
MathieuMoonshinewhichclaimsarelationbetweentheellipticgenusofK3surfaces
and one of the sporadic finite simple groups called the Mathieu group. The theory
oflatticesandtheirreflectiongroupsisnecessarytostudyK3surfaces. Inthisbook
we start to explain these notions and give a proof of the Torelli-type theorem and
its applications. We hope that this book shows the interplay between several sorts
ofmathematics. Inparticular,theauthorwouldbehappyifthisbookweretoprove
helpfultotheresearchofyoungpeople.
Finally, Hisanori Ohashi and the referee pointed out many misprints in the
manuscript and gave the author useful comments. The author would like to take
thisopportunitytothankthem.
