K3 Surfaces and Their Moduli

411 Pages·2016·4.39 MB·English

by  Carel Faber

Progress in Mathematics
315
Carel Faber
Gavril Farkas
Gerard van der Geer
Editors
K3 Surfaces
and Their
Moduli

Progress in Mathematics
Volume 315
Series Editors
Hyman Bass, University of Michigan, Ann Arbor, USA
Jiang-Hua Lu, The University of Hong Kong, Hong Kong SAR, China
Joseph Oesterlé, Université Pierre et Marie Curie, Paris, France
Yuri Tschinkel, Courant Institute of Mathematical Sciences, New York, USA
More information about this series at http://www.springer.com/series/4848
Carel Faber • G avril Farkas • Gerard van der Geer
Editors
K3 Surfaces and Their Moduli
Editors
Carel Faber Gavril Farkas
Mathematisch Instituut Institut für Mathematik
Universiteit Utrecht Humboldt Universität Berlin
Utrecht, The Netherlands Berlin, Germany

Gerard van der Geer
Korteweg-de Vries Instituut
Universiteit van Amsterdam
Amsterdam, The Netherlands
ISSN 0743-1643 ISSN 2296-505X (electronic)
Progress in Mathematics
ISBN 978-3-319-29958-7 ISBN 978-3-319-29959-4 (eBook)
DOI 10.1007/978-3-319-29959-4

Library of Congress Control Number: 2016934933
M athematics Subject Classification (2010): primary: 14J28, 14J15, 14J10, secondary: 14J32, 14J33,
14J50, 14N35

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CONTENTS
Introduction vii
The automorphism group of the Hilbert scheme of two points on a
generic projective K3 surface 1
Samuel Boissi`ere, Andrea Cattaneo, Marc Nieper-Wisskirchen,
and Alessandra Sarti
Orbital counting of curves on algebraic surfaces and sphere packings 17
Igor Dolgachev
Moduli of polarized Enriques surfaces 55
V. Gritsenko and K. Hulek
Extremal rays and automorphisms of holomorphic symplectic varieties 73
Brendan Hassett and Yuri Tschinkel
An odd presentation for W(E ) 97
6
Gert Heckman and Sander Rieken
On the motivic stable pairs invariants of K3 surfaces 111
S. Katz, A. Klemm, and R. Pandharipande,
with an appendix by R. P. Thomas
The Igusa quartic and Borcherds products 147
Shigeyuki Kond¯o
Lectures on supersingular K3 surfaces and the crystalline Torelli theorem 171
Christian Liedtke
On deformations of Lagrangian fibrations 237
Daisuke Matsushita
Curve counting on K3×E, the Igusa cusp form χ , and descendent
10
integration 245
G. Oberdieck and R. Pandharipande
Simple abelian varieties and primitive automorphisms of null entropy
of surfaces 279
Keiji Oguiso
v
vi CONTENTS
The automorphism groups of certain singular K3 surfaces and an
Enriques surface 297
Ichiro Shimada
Geometry of genus 8 Nikulin surfaces and rationality of their moduli 345
Alessandro Verra
Remarks and questions on coisotropic subvarieties and 0-cycles of
hyper-K¨ahler varieties 365
Claire Voisin
INTRODUCTION
K3 surfaces and their moduli represent today a central subject in algebraic
and arithmetic geometry that enjoyed considerable attention and progress in
the last few decades. It goes back at least to 1864, when Kummer discovered
a special class of quartic surfaces in projective three-space with 16 nodes and
linked their intricate geometry to the theory of theta functions associated to
Riemann surfaces of genus 2. Important work by Kummer, Klein, Plu¨cker and
othersfollowedandtopicsstillrelevanttodayinthetheoryofKummersurfaces,
like classification, singularities, automorphisms, or the 16 configuration were
6
verypopularatthetime.Averyreadableaccountofthistheorycanbefoundin
thebooksofHudsonandJessoponquarticsurfaces,publishedatthebeginning
ofthe20thcentury.Aroundthesametime,thesubjectwasbroadenedbyItalian
geometers, who found and studied regular surfaces in Pn of geometric genus 1
whose hyperplane sections are canonical curves of genus n.
Such surfaces were baptized K3 surfaces in 1958 by Andr´e Weil in his
famous report on his Air Force contract. The symbol K3 referred to the three
mathematicians Kummer, Ka¨hler, and Kodaira, and to the mountain K2 on
the China-Pakistan border, the second highest mountain in the world, which
featured prominently in the news at that time. The report drew attention to
several problems concerning the moduli of K3 surfaces, in particular to the
questions whether all K3 surfaces are diffeomorphic and deformations of each
otherandwhethertheperiodsofaK3surfacedeterminetheisomorphismclass.
Infact,AndreottiandWeilindependentlyconjecturedthatallK3surfacesform
one connected family, that these surfaces all are Ka¨hler and that the period
map is surjective. They also conjectured the global Torelli theorem. These
conjectures were all settled in the following decades by the newly developed
techniques of algebraic and complex-analytic geometry. Kodaira proved that
all complex K3 surfaces are deformations of each other and Siu proved that
theyareallKa¨hler.TheTorellitheoremforalgebraicK3swasfirstattackedby
Piatetski-Shapiro and Shafarevich; an extension to the K¨ahler case was given
by Burns and Rapoport; and the general case was resolved by Todorov, who
provedthesurjectivityoftheperiodmap.Theproofusedinadecisivewaythe
methods developed by S.-T. Yau for his proof of the Calabi conjecture. With
these results established, lattice theory and powerful group-theoretic methods
couldbeusedtostudyconcretequestionsonK3surfaces,likedeterminingtheir
automorphism groups, and this has been carried out to great effect in works of
Mukai, Nikulin, and many others.
vii
viii INTRODUCTION
Mukai discovered in the 1980’s that moduli spaces of sheaves on K3 sur-
faces carry a symplectic structure. In particular, one can define a dual K3 sur-
face as a moduli space of sheaves on the original one. This has opened the way
to mainstream topics in algebraic geometry today, such as the Fourier-Mukai
transform, or the study of the higher dimensional analogues of K3 surfaces:
irreducible holomorphic symplectic varieties. K3 surfaces have also proved to
be amazingly effective in the study of curves and their moduli. To give two
influential examples: using curves on K3 surfaces, Lazarsfeld has proved the
Brill-NoetherTheorem,whereasVoisinhasestablishedtheGenericGreenCon-
jecture on syzygies of canonical curves.
In the 1970’s, K3 surfaces and their moduli in positive characteristic
started to attract attention and in work of Shafarevich and Rudakov, Artin,
and others, new phenomena were discovered, leading to new invariants such as
the height and the Artin number. Deligne proved a lifting theorem and Ogus
developedacrystallineanaloguetothecomplexanalytictheory.Likethemod-
uliofabelianvarieties,themoduliofK3surfacesinpositivecharacteristiccarry
interesting stratifications that have no analogues in characteristic zero.
As a result of all these developments, the K3 moduli became more acces-
sible, similarly to what happened with the moduli of abelian varieties and of
curves. As in those cases, we now know for example the Kodaira dimensions of
the moduli of polarized K3 surfaces, due to work of Kond¯o, Mukai, Gritsenko,
Hulek, and Sankaran, except for a few intriguing unsettled cases.
The enumerative geometry of curves on K3 surfaces has also received
considerable attention. Yau and Zaslow made an amazing prediction for the
numberofrationalcurvesonapolarizedK3surface,whichhasbeenconfirmed
by Beauville and others. This study has then been amplified in the highly
sophisticatedtheoryofGromov-WitteninvariantsofK3surfaces,wheredueto
workofMaulik,Pandharipande,Thomas,andothers,surprisingconnectionsto
modular forms and Noether-Lefschetz invariants of automorphic nature have
been discovered.
The attention that K3 surfaces now attract comes not only from mathe-
maticiansbutalsofromtheoreticalphysicists;thesesurfacesplayanimportant
role in string theory and mirror symmetry. K3 surfaces and their moduli have
intricate connections to the moduli of abelian varieties and of curves, but per-
hapssurprisingly,thetopicalsoconnectstoseveralotherdirectionsofresearch
where there is a lot of activity, such as the study of derived categories and sta-
bility conditions, Gromov-Witten theory, and dynamical systems. These links
areleadingtonewideasandmethods,andprogressinthisfieldoftencomesby
mixing sophisticated techniques coming from algebraic geometry, lattice the-
ory, number theory, and dynamical systems; sometimes, it is steered by the
intuitionfromtheoreticalphysics.Slowly,alsothearithmetictheoryofK3sur-
faces comes off the ground, as number theorists are beginning to extend the
arithmetic theory of elliptic curves to their higher-dimensional analogues.
INTRODUCTION ix
The subject also got new impetus by recent breakthroughs on the Tate
conjectureforK3surfaces,byMaulik,Madapusi-Pera,andCharles.Thisillus-
trates the fact that K3 surfaces are a good testing ground for various conjec-
tures; Deligne was able to prove the Weil conjectures for K3 surfaces before he
dealt with the general case.
At the same time, the field has been widened: Calabi-Yau varieties play
a central role in string theory and the theory of irreducible holomorphic sym-
plectic varieties has become a mainstream subject in algebraic geometry. K3-
like mathematical structures are governing research on hot topics that are not
clearly related to K3 surfaces, such as that of the rationality of cubic hyper-
surfaces.
All these recent developments formed the motivation to dedicate one of
the Texel and Schiermonnikoog Island conferences to K3 surfaces and their
moduli. It was held in May 2014.
Thepresentvolumeiseditedontheoccasionofthisconference.Itcontains
contributions by several experts in the field and gives an overview of recent
developments and new results.
We would like to take the opportunity to thank the participants and the
speakers,whomadetheconferenceasuccess.WealsoliketothanktheFounda-
tion Compositio Mathematica that financed the lion’s share of the conference.
November 2015 Carel Faber
Gavril Farkas
Gerard van der Geer