Brauer Groups and Obstruction Problems : Moduli Spaces and Arithmetic

251 Pages·2017·3.62 MB·English

by  Asher Auel

Progress in Mathematics
320
Asher Auel
Brendan Hassett
Anthony Várilly-Alvarado
Bianca Viray
Editors
Brauer Groups
and Obstruction
Problems
Moduli Spaces and Arithmetic

Progress in Mathematics
Volume 320
Series Editors
Antoine Chambert-Loir, Université Paris-Diderot, Paris, France
Jiang-Hua Lu, The University of Hong Kong, Hong Kong SAR, China
Yuri Tschinkel, Courant Institute of Mathematical Sciences, New York, USA
More information about this series at http://www.springer.com/series/4848
Asher Auel • Brendan Hassett
Anthony Várilly-Alvarado • Bianca Viray
Editors
Brauer Groups
and Obstruction Problems
Moduli Spaces and Arithmetic
Editors
Asher Auel Brendan Hassett
Department of Mathematics Department of Mathematics
Yale University Brown University
New Haven, Connecticut, USA Providence, Rhode Island, USA
Anthony Várilly-Alvarado Bianca Viray
Department of Mathematics MS-136 Department of Mathematics
Rice University University of Washington
Houston, Texas, USA Seattle, Washington, USA
ISSN 0743-1643 ISSN 2296-505X (electronic)
Progress in Mathematics
ISBN 978-3-319-46851-8 ISBN 978-3-319-46852-5 (eBook)
DOI 10.1007/978-3-319-46852-5
Library of Congress Control Number: 2017930680
Mathematics Subject Classification (2010): 14F05, 14F22, 14E08, 14G05, 14J28, 14J35, 14J60

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Contents
Nicolas Addington
The Brauer Group Is Not a Derived Invariant .......................1
Benjamin Antieau
Twisted Derived Equivalences for A(cid:14)ne Schemes ...................7
Kenneth Ascher, Krishna Dasaratha, Alexander Perry, and Rong Zhou
Rational Points on Twisted K3 Surfaces and Derived Equivalences .13
Asher Auel, Jean-Louis Colliot-Th(cid:19)el(cid:18)ene, and Raman Parimala
Universal Unrami(cid:12)ed Cohomology of Cubic Fourfolds Containing
a Plane ...........................................................29
Fedor Bogomolov and Yuri Tschinkel
Universal Spaces for Unrami(cid:12)ed Galois Cohomology ...............57
Brendan Hassett and Yuri Tschinkel
Rational Points on K3 Surfaces and Derived Equivalence ..........87
Colin Ingalls, Andrew Obus, Ekin Ozman, and Bianca Viray
Unrami(cid:12)ed Brauer Classes on Cyclic Covers of the Projective
Plane ............................................................115
Mart(cid:19)(cid:16) Lahoz, Emanuele Macr(cid:18)(cid:16), and Paolo Stellari
Arithmetically Cohen{Macaulay Bundles on Cubic Fourfolds
Containing a Plane ...............................................155
Kelly McKinnie, Justin Sawon, Sho Tanimoto, and
Anthony V(cid:19)arilly-Alvarado
Brauer Groups on K3 Surfaces and Arithmetic Applications ......177
Alena Pirutka
On a Local-Global Principle for H3 of Function Fields of
Surfaces over a Finite Field ......................................219
Alexei N. Skorobogatov
Cohomology and the Brauer Group of Double Covers .............231
v
Preface
ThisvolumegrewoutofaworkshopthatweorganizedattheAmericanInsti-
tute of Mathematics from February 25 through March 1, 2013. The meeting
brought together experts from two di(cid:11)erent (cid:12)elds: number theorists inter-
estedinrationalpoints,andcomplexalgebraicgeometersworkingonderived
categories of coherent sheaves. We were motivated by fresh developments
in the arithmetic of K3 surfaces, which suggest that cohomological obstruc-
tions to the existence and distribution of rational points on K3 surfaces can
be fruitfully studied via moduli spaces of twisted sheaves. Our aim was to
encourage cross-pollination between the two (cid:12)elds and to explore concrete
instances when the derived category of coherent sheaves on a variety over a
number (cid:12)eld determines some of its arithmetic. Within this framework, we
aim to extend to K3 surfaces a number of powerful tools for analyzing ratio-
nalpointsonellipticcurves:isogeniesamongcurves,torsionpoints,modular
curves, and the resulting descent techniques.
Let S and S denote complex K3 surfaces. An isomorphism
1 2
(cid:19): H2(S ;Z)(cid:0)(cid:24)!H2(S ;Z)
1 2
thatiscompatiblewiththecupproductandHodgestructurescausesS and
1
S to be isomorphic by the Torelli theorem. We might weaken this by asking
2
onlythattheselatticesbestablyisomorphic,orequivalently,thatthereisan
isomorphism between the lattices of transcendental classes
(cid:24)
T(S )(cid:0)!T(S ): (1)
1 2
Orlovhasshownthatthisrelationcoincideswiththeapriorialgebraicnotion
of derived equivalence, i.e., an equivalence of the bounded derived categories
of coherent sheaves
Db(S )(cid:0)(cid:24)!Db(S )
2 1
as triangulated categories over C. Such equivalences manifest themselves as
interpretations of S as a moduli space of sheaves over S and vice versa.
2 1
The manuscript by Hassett and Tschinkel explores the implications for K3
surfaces over (cid:12)nite and local (cid:12)elds. Antieau’s contribution takes a broader
perspective, contrasting the local and global features of derived equivalence
by looking at twisted derived equivalences among a(cid:14)ne varieties.
On a K3 surface S, we arrive at the notion of cyclic isogeny by weaken-
ing (1) further to an inclusion
T(X),!T(S) (2)
with cokernel isomorphic to Z=nZ, for some projective variety X. Such an
isogeny arises from, and gives rise to, an element of the Brauer group Br(S)
of S, which can be lifted to an element of the second cohomology of S with
Z=nZ coe(cid:14)cients. These cohomology classes on K3 surfaces are the analog
of torsion points for elliptic curves.
In 2005, van Geemen exploited the interplay between Brauer elements
and cyclic isogenies to give a complete description of the order 2 Brauer
vii
Preface viii
classesonadegree2K3surfacewithPicardrank1.ThepaperofMcKinnie,
Sawon, Tanimoto, and V(cid:19)arilly-Alvarado generalizes van Geemen’s work to
higher order Brauer classes of higher degree K3 surfaces. Their paper uses
lattice-theoretic techniques to study the components of moduli spaces of K3
surfaces with level structure coming from the Brauer group. This is an es-
sential step toward a comprehensive geometric interpretation for the second
cohomology of a K3 surface, with applications to the structure of rational
points.
VanGeemen’sworkisgeneralizedindi(cid:11)erentdirectionsinthecontribu-
tionsbySkorobogatovandbyIngalls,Obus,Ozman,andViray.Skorobogatov
undertakes a systematic study of 2-torsion Brauer elements on double covers
of rational surfaces; Ingalls, Obus, Ozman, and Viray look at higher-degree
cyclic covers of P2. These cases should play a central role in the study of ex-
plicitexamplesinthefuture.Bothcontributionshandlethecaseofarbitrary
Picard rank.
The isogeny (2) suggests that X provides a geometric representative of
an element (cid:11) 2 Br(S)[n]. Similarly, one expects an equivalence between the
derived category of twisted sheaves Db(S;(cid:11)) and an admissible subcategory
of Db(X). An important motivating example is when X is a smooth cubic
fourfoldcontainingaplane,inwhichcaseX isbirationaltoaquadricsurface
bundleoverP2 whoserelativeLagrangianGrassmannianhasthestructureof
an(cid:19)etalelocallytrivialP1-bundleoveraK3surfaceS ofdegree2,givingriseto
aBrauerclass(cid:11)2Br(S)[2].Inthiscase,thereisaninclusionT(X),!T(S),
which has index 2 when (cid:11) 2 Br(S) is nontrivial, and Kuznetsov has shown
thatDb(S;(cid:11))isequivalenttoanadmissiblesubcategoryofDb(X).Thecubic
fourfold X is rational as soon as (cid:11)2Br(S)[2] becomes trivial. Using derived
categorytechniques,Lahoz,Macr(cid:18)(cid:16),andStellariaddtothisgeometricpicture
by providing an elegant geometric reconstruction of the K3 surface S as a
moduli space of vector bundles on the cubic fourfold X.
Given the tight connection between Brauer groups and derived equiv-
alences for K3 surfaces, one could ask whether the Brauer group is invari-
antunderderivedequivalences.Addington’scontributionanswersthisinthe
negative by showing that the Brauer group fails to be invariant under de-
rivedequivalencesofCalabi{Yauthreefolds.ThepaperofAscher,Dasaratha,
Perry, and Zong also demonstrates the limitations of derived equivalence by
showing that naive guesses on the relation between the structure of rational
points and twisted derived equivalence for K3 surfaces are incorrect.
The key role of the Brauer group naturally leads to more systematic
analysis of higher degree unrami(cid:12)ed cohomology. Motivated by the rational-
ity problem for cubic fourfolds, Auel, Colliot-Th(cid:19)el(cid:18)ene, and Parimala obtain
the universal triviality of the unrami(cid:12)ed cohomology of degree 3 for cubic
fourfolds containing a plane. More generally, inspired by a result of Merkur-
jev, they develop the notion of the universal triviality of the Chow group of
0-cycles as a possible obstruction to rationality, which has since been instru-
mental to recent breakthroughs in the (stable) rationality problem. Pirutka
ix Preface
establishes a local-global criterion for the decomposability of elements in de-
gree3cohomology{decompositionsinvolvingBrauerelements{forsurfaces
over (cid:12)nite (cid:12)elds. Finally, Bogomolov and Tschinkel o(cid:11)er a broad classifying
space framework for unrami(cid:12)ed cohomology in higher degrees for varieties
over the algebraic closure of a (cid:12)nite (cid:12)eld.
Acknowledgments. WearegratefultotheAmericanInstituteofMathematics
for their support of this workshop and the collaborative environment that
made this work possible. The (cid:12)rst editor was partially supported by NSF
grantDMS-0903039andbyNSAYoungInvestigatorgrantH98230-13-1-0291.
The second editor was partially supported by NSF grant DMS-1551514. The
third editor was partially supported by NSF CAREER grant DMS-1352291.
And the fourth editor was partially supported by NSA Young Investigator
grant H98230-15-1-0054.
The Brauer Group Is Not a Derived
Invariant
Nicolas Addington
Abstract. In this short note we observe that the recent examples of
derived-equivalentCalabi{Yau3-foldswithdi(cid:11)erentfundamentalgroups
also have di(cid:11)erent Brauer groups, using a little topological K-theory.
MathematicsSubjectClassi(cid:12)cation(2010). 14F05, 14F22, 14J32.
Keywords. Brauer groups, derived equivalence, Calabi{Yau threefolds.
Some years ago Gross and Popescu [12] studied a simply-connected
Calabi{Yau 3-fold X (cid:12)bered in non-principally polarized abelian surfaces.
They expected that its derived category would be equivalent to that of
the dual abelian (cid:12)bration Y, which is again a Calabi{Yau 3-fold but with
(cid:25) (Y) = (Z )2, the largest known fundamental group of any Calabi{Yau 3-
1 8
fold. This derived equivalence was later proved by Bak [2] and Schnell [23].
Ignoring the singular (cid:12)bers it is just a family version of Mukai’s classic de-
rived equivalence between an abelian variety and its dual [19], but of course
the singular (cid:12)bers require much more work. As Schnell pointed out, it is a
bit surprising to have derived-equivalent Calabi{Yau 3-folds with di(cid:11)erent
fundamental groups, since for example the Hodge numbers of a 3-fold are
derived invariants [22, Cor. C].
GrossandPavanelli[11]showedthatBr(X)=(Z )2,thelargestknown
8
Brauer group of any Calabi{Yau 3-fold. In this note we will show that the
(cid:12)nite abelian group H (X;Z)(cid:8)Br(X) is a derived invariant of Calabi{Yau
1
3-folds; thus in this example we must have Br(Y)=0, and in particular the
Brauergroupaloneisnotaderivedinvariant.Thistooisabitsurprising,since
the Brauer group is a derived invariant of K3 surfaces: if X is a K3 surface
then Br(X) (cid:24)= Hom(T(X);Q=Z) [7, Lem. 5.4.1], where T(X) = NS(X)? (cid:26)
H2(X;Z) is the transcendental lattice, which is a derived invariant by work
of Orlov [20].
Since an earlier version of this note (cid:12)rst circulated, Hosono and Takagi
[13] have found a second example of derived-equivalent Calabi{Yau 3-folds
with di(cid:11)erent fundamental groups. Their X and Y are constructed from
spacesof5(cid:2)5symmetricmatricesinwhatislikelyaninstanceofhomological
© Springer International Publishing AG 2017 1
A. Auel (eds.) et al., Brauer Groups and Obstruction Problems,
Progress in Mathematics 320, DOI 10.1007/978-3-319-46852-5_1