Calabi-Yau Threefolds Fibred by Kummer Surfaces Associated to Products of Elliptic Curves

0.36 MB·English

by  Charles F. Doran

CALABI-YAU THREEFOLDS FIBRED BY KUMMER SURFACES
ASSOCIATED TO PRODUCTS OF ELLIPTIC CURVES
5
1 CHARLESF.DORAN,ANDREWHARDER,ANDREYY.NOVOSELTSEV,
0 ANDALANTHOMPSON
2

l Abstract. We study threefolds fibred by Kummer surfaces associated to
u
products ofellipticcurves,thatariseasresolvedquotientsofthreefoldsfibred
J
by certain lattice polarized K3 surfaces under a fibrewise Nikulin involution.

8 We present a general construction for such surfaces, before specializing our
results to study Calabi-Yau threefolds arising as resolved quotients of three-

] folds fibred by mirror quartic K3 surfaces. Finally, we give some geometric
G
properties of the Calabi-Yau threefolds that we have constructed, including
A expressions forHodgenumbers.
.
h
t
a
m 1. Introduction
[
Building on earlier work by Shioda, Inose [24][12], Nikulin [21] and Morrison

2 [18], Clingher and Doran [2][3] exhibited a duality between K3 surfaces admitting
v a lattice polarization by the lattice
4
2 M :=H E8 E8
⊕ ⊕
0
and Kummer surfaces associated to products of elliptic curves, that closely relates
4
the geometryofthe surfacesoneachside. This duality is easyto describe: anyM-
0
. polarized K3 surface admits a canonically defined Nikulin involution, the resolved
1
quotient by which is a Kummer surface associated to a product of elliptic curves
0
and, conversely, a Kummer surface associated to a product of elliptic curves also
5
1 admits a Nikulin involution, the resolved quotient by which is isomorphic to an
: M-polarized K3 surface. Moreover, applying this process twice returns us to the
v
surface we started with.
i
X
This duality was exploited in [4], to obtain certain geometric properties of a
r Calabi-Yau threefold admitting a fibration by M-polarized K3 surfaces. In that
a
case, it was proven that Clingher’s and Doran’s construction could be performed
fibrewise, giving rise to a new Calabi-Yau threefold that was fibred by Kummer
surfaces associated to products of elliptic curves. As it turns out, the geometry of
the Kummer fibred threefold thus obtained was easier to study, and could be used
to derive geometric properties of the original threefold fibred by M-polarized K3
surfaces.
2010 Mathematics Subject Classification. Primary14D06,Secondary 14J28,14J30, 14J32.
C. F. Doran and A. Y. Novoseltsev were supported by the Natural Sciences and Engineering
ResourceCouncilofCanada(NSERC),thePacificInstitutefortheMathematicalSciences(PIMS),
andtheMcCallaProfessorshipattheUniversityofAlberta.
A.Harderwassupported byanNSERCPost-Graduate Scholarship.
A.Thompsonwassupported byaFields-Ontario-PIMSPostdoctoral Fellowshipwithfunding
providedbyNSERC,theOntarioMinistryofTraining,CollegesandUniversities,andanAlberta
AdvancedEducationandTechnologyGrant.
1
2 C.F.DORAN,A.HARDER,A.Y.NOVOSELTSEV,ANDA.THOMPSON
Themainaimofthispaperistoinvestigatetowhatextentthisconstructioncan
be generalized to arbitrary threefolds fibred by M-polarized K3 surfaces.
More precisely, suppose that is a threefold fibred by K3 surfaces and that
X
the restriction of this fibration to the locus of smooth fibres is an M-polarized
U
X
family of K3 surfaces,in the sense of [10, Definition 2.1]. Then results of [10] show
that the canonical Nikulin involution on the fibres of extends to the entire
U
X
threefold, so we may quotient and resolve singularities to obtain a threefold fibred
by Kummer surfaces . We may then ask whether can be compactified to a
U U
Y Y
threefold and, if so, what properties this new threefold has.
Y
One case that is of particular interest is when the threefold is Calabi-Yau, as
X
occurred in [4]. In this case, one would like to know whether (if it exists) is also
Y
Calabi-Yau and, if so, how its properties relate to those of .
X
In the latter part of this paper we address this second question in a special
case, where the Calabi-Yau threefolds are very well-understood. Specifically,
X
we consider the setting where is one of the Calabi-Yau threefolds fibred by
g
X X
mirror quartic K3 surfaces constructed in [9]. Note that this is a special case of
the construction above as, by definition, a mirror quartic K3 surface is polarized
by the lattice M :=H E E 4 , which clearly contains M as a primitive
2 8 8
⊕ ⊕ ⊕h− i
sublattice. These threefolds encompass many well-known examples, including
g
X
the quintic mirror threefold, and provide a useful illustration of our methods in a
concrete setting.
In this special case, we show that we can explicitly construct Kummer surface
fibred threefolds that are related to the by a fibrewise quotient-resolution
g g
Y X
procedure as above. Moreover, the are Calabi-Yau in most cases and have
g
Y
geometric properties that are closely related to those of the . This gives a new
g
X
perspective from which to study the geometry of the threefolds , amongst them
g
X
the quintic mirror.
Finally,wenotethattheconstructionofthethreefolds issomewhatinteresting
g
Y
initsownright,astheyareallconstructedfromasingle,rigidCalabi-Yauthreefold.
ThisrigidCalabi-Yauthreefoldisinturnbuiltfromawell-knownextremalrational
ellipticsurface,usingamethodoriginallyduetoSchoen[22]thatwaslaterextended
by Kapustka and Kapustka [13].
Thestructureofthispaperisasfollows. InSection2wereviewsomebackground
material, mostly taken from [2] and [3], about M-polarized K3 surfaces, and de-
scribethe threefolds thatarefibredby them. Then,inSection3,wedevelopthe
X
theory required to construct the associated threefolds fibred by Kummer surfaces
, and describe their constructionin generalterms. This constructionproceeds by
Y
first undoing the Kummer construction, as originally described in [10, Section 4.3],
then running a generalized version of the forward construction from [4, Section 7].
Finally,inSection4,wespecializetheentirediscussiontothecasewhere isone
X
of the Calabi-Yau threefolds fibred by quartic mirror K3 surfaces constructed
g
X
in [9]. In this case we can construct the associated threefolds fibred by Kummer
surfaces completelyexplicitlyaspull-backsofaspecialthreefold . Thisspecial
g 2
Y Y
threefoldisconstructedinturnasaresolvedquotientofarigidCalabi-Yauthreefold
, which is described in Section 4.2. The properties of are then studied in
2 2
A A
Section 4.3, after which we carefully describe the quotient-resolution procedure
used to obtain from it in Sections 4.4 and 4.5. The method for constructing
2
Y
CALABI-YAU THREEFOLDS FIBRED BY KUMMER SURFACES 3
the from is detailed in Section 4.6, and some of the properties of the are
g 2 g
Y Y Y
computed in Section 4.7.
2. Background material
We begin by setting up some notation. Let be a smooth projective threefold
X
that admits a fibration B by K3 surfaces over a smooth curve. Let N :=
X →
NS(X )denotetheNéron-Severigroupofthefibreof overageneralpointp B.
p
X ∈
Suppose that there exists a primitive lattice embedding M ֒ N of the lattice
→
M :=H E E into N (we will assume that such an embedding has been fixed
8 8
⊕ ⊕
in what follows).
Denotetheopensetoverwhichthefibresof aresmoothK3surfacesbyU B
X ⊂
and let U denote the restriction of to U. Suppose further that U
U U
X → X X →
is an N-polarized family of K3 surfaces, in the sense of [10, Definition 2.1].
Remark 2.1. Nineteen such fibrations are known on Calabi-Yau threefolds with
X
h2,1( ) = 1; these are summarized by [10, Table 5.1]. Moreover, a large class of
X
additionalexamplesofCalabi-YauthreefoldsfibredbyK3surfacespolarizedbythe
lattice M := H E E 4 are constructed in [9]; we will return to these
2 8 8
⊕ ⊕ ⊕h− i
examples in Section 4.
2.1. M-polarized K3 surfaces. By assumption, a general fibre X of is an
p
X
M-polarized K3 surface. We recall here some basic properties of M-polarized K3
surfaces,thatwillbeusedrepeatedlyinwhatfollows. Inthissectionwewilldenote
an M-polarized K3 surface by (X,i), where X is a K3 surface and i is a primitive
lattice embedding i: M ֒ NS(X).
→
Building upon workof Inose [12], Clingher, Doran,Lewis and Whitcher [5] have
shown that M-polarized K3 surfaces have a coarse moduli space given by the lo-
cus d = 0 in the weighted projective space WP(2,3,6) with weighted coordinates
6
(a,b,d). Thus, by normalizing d=1, we may associate a pair of complex numbers
(a,b) to an M-polarized K3 surface (X,i).
The first piece of structure that we need on (X,i) comes from the work of Mor-
rison [18], who showed that the composition of i with the canonical embedding
E E ֒ M defines a canonical Shioda-Inose structure on (X,i) (named for Sh-
8 8
⊕ →
iodaandInose[24],whowerethefirsttostudysuchstructures). Bydefinition,such
astructureconsistsofaNikulininvolutionβ onX,suchthatthe resolvedquotient
Y = X/β is a Kummer surface and there is a Hodge isometry T =T (2), where
Y ∼ X
T and T denote the transcendental lattices of X and Y respectively, and T (2)
X Y X
indicagtes that the bilinear pairing on T has been multiplied by 2.
X
By[2,Theorem3.13]1,weseethatinoursettingY isisomorphictotheKummer
surface Kum(A), where A=E E is an Abelian surface that splits as a product
of elliptic curves. By [2, C∼oroll1a×ry 42.2]1 the j-invariants of these elliptic curves are
given by the roots of the equation
j2 σj+π =0,
−
where σ and π are given in terms of the (a,b) values associated to (X,i) by σ =
a3 b2+1 and π = a3. Label the exceptional ( 2)-curves in Y arising from the
− −
resolution of the singularities of X/β by F ,...,F .
1 8
{ }
1We note that equivalent results to those attributed to Clingher and Doran [2] here were
provedindependentlybyShioda[23],usingaslightlydifferentcharacterizationofM-polarizedK3
surfacesasellipticallyfibredK3surfaces withsectionandtwofibres oftypeII∗.
4 C.F.DORAN,A.HARDER,A.Y.NOVOSELTSEV,ANDA.THOMPSON
Thereisonemorepieceofstructureon(X,i)thatwewillneedinourdiscussion.
By [2, Proposition 3.10], the K3 surface X admits two uniquely defined elliptic
fibrations Θ : X P1, the standard and alternate fibrations. We will be mainly
1,2
→
concerned with the alternate fibration Θ . This fibration has two sections, one
2
singular fibre of type I and, if a3 = (b 1)2, six singular fibres of type I [2,
1∗2 6 ± 1
Proposition 4.6].
The alternate fibration Θ is preserved by the Nikulin involution β, so induces
2
an elliptic fibration Ψ: Y P1 on the Kummer surface Y. The two sections of Θ
2
→
are identified to give a sectionS of Ψ, and Ψ has one singularfibre of type I and,
6∗
if a3 =(b 1)2, six I ’s [2, Proposition 4.7].
2
6 ±
Remark 2.2. Asnotedintheintroduction,thisconstructioniscompletelyreversible.
Clingher and Doran [3, Section 1] identify a second distinguished section S of the
′
fibrationΨ,alongwithauniquelydefinedNikulininvolutionβ onY thatpreserves
′
]
Ψ and takes S to S . The resolved quotient Y/β is then isomorphic to X, and Ψ
′ ′
induces the alternate fibration Θ on X.
2
The locations of the I fibres in Ψ are given by [2, Proposition 4.7]. They occur
2
at the roots of the polynomials (P(x) 1), where P is the cubic equation
±
(2.1) P(x):=4x3 3ax b,
− −
(a,b)arethe modularparametersassociatedto X,andx is anaffine coordinateon
P1 chosen so that the I fibre occurs at x= .
6∗ ∞
Finally, using this information we may identify some of the ( 2)-curves F in
i
−
Y. By the discussion in [10, Section 4.3], F ,F ,F (resp. F ,F ,F ) are the
3 4 5 6 7 8
{ } { }
( 2)-curves in the I fibres lying over the roots of (P(x) 1) (resp. (P(x)+1))
2
− −
that do not meet the section S (this labelling may seem arbitrary, but in fact is
chosen to match with that used in [10, Section 4.3]).
3. Threefolds fibred by Kummer surfaces
We will now apply this theory to study the K3-fibredthreefold B. Via the
X →
embeddingM ֒ N,weseethatageneralfibre of is anM-polarizedK3surface.
→ X
Thus, by the discussion in Section 2.1, there is a canonical Shioda-Inose structure
on such a fibre, which defines a Nikulin involution on it. This involution extends
uniquely to all fibres of by [10, Corollary 2.12].
U
X
Let U denote the family obtained by taking the quotient of by this
U U
Y → X
involution and resolving the resulting singularities. The discussion from Section
2.1 shows that the fibres of are Kummer surfaces Kum(E E ) associated to
U 1 2
Y ×
products ofelliptic curvesE E . Furthermore,the alternate fibrationΘ onthe
1 2 2
×
fibres of induces a uniquely defined elliptic fibration Ψ on the fibres of .
U U
X Y
Remark 3.1. AsintheK3surfacecase,thisconstructionturnsouttobereversible.
Let β be the Nikulin involution on a general fibre of , as described in Remark
′ U
Y
2.2. Bythe descriptionofthe actionofmonodromyin U from[10,Section4.3]and
the description of β from [3, Section 1], it can be shown that the action of β and
′ ′
the action of monodromy on the Néron-Severi lattice of a general fibre commute.
So,by[10,Proposition2.11],β extendstoainvolutionon ,theresolvedquotient
′ U
Y
by which is isomorphic to .
U
X
CALABI-YAU THREEFOLDS FIBRED BY KUMMER SURFACES 5
Our aim is to explicitly construct K3-fibred threefolds over B, so that the
Y
restriction of to the open set U B is isomorphic to , and to study their
U
Y ⊂ Y
properties.
3.1. Undoing the Kummer construction. In order to do this, the first step is
to undo the Kummer construction for , i.e. to find a family of Abelian surfaces
U
Y
U whichgivesriseto uponfibrewiseapplicationoftheKummerconstruc-
U U
A → Y
tion. To dothis,wewillusethe resultsfrom[10,Section4.3]. However,inorderto
applytheseresultsweneedtomakethefollowingassumption[10,Assumption4.6];
unlessotherwisestated,wewillmakethis assumptionthroughoutthe remainderof
this section.
Assumption 3.2. The fibration Ψ on a general fibre Y of has six singular
p U
Y
fibres of type I .
2
Remark 3.3. Note that each I fibre in the fibration Ψ on Y arises as the total
2 p
transform of an I fibre in the alternate fibration Θ on X . Thus to check that
1 2 p
Assumption 3.2 is satisfied, it is equivalent to show that the alternate fibration Θ
2
on a general fibre X of has six singular fibres of type I
p U 1
X
This latter condition is easy to check numerically from the (a,b) parameters
associatedto X . Indeed,the locationsofthe I fibresinΘ aregivenbythe roots
p 1 2
of the polynomials (P(x) 1), where P(x) is defined by Equation (2.1), which are
±
all distinct if and only if a3 =(b 1)2.
6 ±
Unfortunately, by the discussion in [10, Section 4.3], it is not always possible to
undo the Kummer construction on directly. Instead, we must pull everything
U
Y
back to a cover f: C B.
→
Thiscoverisconstructedbythemethoddescribedin[10,Section4.3]. Letp U
∈
be a point and consider the six divisors F ,...,F in the fibre Y of over p.
3 8 p
{ } Y
Monodromy in U preserves the fibration Ψ along with its section S (as both are
induced from the structure of the alternate fibration Θ on the fibres of ), so
2 U
X
must act to permute the F . We thus have a homomorphism ρ: π (U,p) S ;
i 1 6
→
call its image G. Then define an unramified G-fold cover f: V U as follows:
| | →
the preimages of p U are labelled by permutations in G and, if γ is a loop in
∈
U, monodromy around f 1(γ) acts on these labels as composition with ρ(γ). This
−
cover extends uniquely to a cover f: C B, with ramification over the points in
→
B U.
−
Let denote the pull-back of to V. Then [10, Theorem 4.11] shows that
YV′ YU
we can undo the Kummer construction for , so there exists a family of Abelian
YV′
surfaces V that gives rise to under fibrewise applicationof the Kummer
AV → YV′
construction.
We have the following diagram:
❴Ku❴mm❴er❴// // oo❴N❴iku❴lin❴ (cid:31)(cid:127) //
AV YV′ YU XU X
V(cid:15)(cid:15) V(cid:15)(cid:15) f //U(cid:15)(cid:15) U(cid:15)(cid:15) (cid:31)(cid:127) // B(cid:15)(cid:15)
3.2. The forward construction. Our next aim is to construct threefolds ,
′
A Y
and thatagreewith , and overV andU respectively. Thisconstruction
Y AV YV′ YU
will generalize the forward construction of [4, Section 7].
6 C.F.DORAN,A.HARDER,A.Y.NOVOSELTSEV,ANDA.THOMPSON
We begin by constructing a threefold fibred by Abelian surfaces C that
A →
agrees with over V. The first step is to identify some special divisors on the
V
A
fibres of .
YV′
Recall that a fibre of is isomorphic to Kum(E E ), where E and E
YV′ 1 × 2 1 2
are elliptic curves. There is a special configuration of twenty-four ( 2)-curves on
−
Kum(E E ) arising from the Kummer construction, that we shall now describe
1 2
×
(here we note that we use the same notation as [2, Definition 3.18], but with the
roles of G and H reversed).
i j
Let x ,x ,x ,x and y ,y ,y ,y denote the twosets ofpoints ofordertwo
0 1 2 3 0 1 2 3
{ } { }
on E and E respectively. Denote by G and H (0 i,j 3) the ( 2)-curves
1 2 i j
≤ ≤ −
on Kum(E E ) obtained as the proper transforms of E y and x E
1 2 1 i j 2
× ×{ } { }×
respectively. Let E be the exceptional ( 2)-curve on Kum(E E ) associated
ij 1 2
− ×
to the point (x ,y ) of E E . This gives 24 curves, which have the following
j i 1 2
×
intersection numbers:
G .H =0,
i j
G .E =δ ,
k ij ik
H .E =δ .
k ij jk
Definition 3.4. The configuration of twenty-four ( 2)-curves
−
G ,H ,E 0 i,j 3
i j ij
{ | ≤ ≤ }
is called a double Kummer pencil on Kum(E E ).
1 2
×
With this in place, we can prove an analogue of [4, Lemma 7.4 and Proposition
7.5].
Proposition 3.5. V is isomorphic over V to a fibre product of
V 1 C 2
A → E × E
minimal elliptic surfaces C with section. Furthermore, the j-invariants of the
i
E →
elliptic curves E and E forming the fibres of and over a point p C are
1 2 1 2
E E ∈
given by the roots of the equation
(3.1) j2 σ(p)j+π(p)=0,
−
where σ(p) and π(p) are the σ and π invariants associated to the M-polarized K3
surface fibre X of over f(p).
f(p) U
X
Remark 3.6. Thus, using the expressions for (σ,π) in terms of (a,b) from Section
2.1, we find that the discriminant of Equation (3.1) is
σ2 4π =a6 2a3b 2a3+b4 2b2+1=(a3 (b 1)2)(a3 (b+1)2).
− − − − − − −
But Assumption 3.2 and Remark 3.3 imply that, for generic p C, this does not
∈
vanish, so the roots of Equation (3.1) are generically distinct.
Proof of Proposition 3.5. We begin by showing that the fibration V has a
V
A →
section s. Construct a double Kummer pencil G ,H ,E on the fibre Y of
{ i j ij} p′ YV′
over p V as described in [10, Section 4.3]. By [10, Theorem 4.11], is NS(Y )-
∈ YV′ p′
polarized, so the divisors in this pencil are invariant under monodromy around
loopsinV. Inparticular,the curveE isinvariant. SoE sweepsoutadivisorin
11 11
, which intersects each smooth fibre in a ( 2)-curve. Passage to contracts
YV′ − AV
this divisor to a curve which intersects each smooth fibre in a single point, i.e. a
section over V.
CALABI-YAU THREEFOLDS FIBRED BY KUMMER SURFACES 7
Now, let denote the fibre of over p, which is isomorphic to a product
p V
A A
E E of elliptic curves. We may identify E (resp. E ) with the preimages of
1 2 1 2
×
the curve G (resp. H ) in the double Kummer pencil on Y . As G and H are
1 1 p′ 1 1
invariantunder monodromyaroundloops in V, they sweepout two divisorsin Y .
V′
Uponpassageto thesetwodivisorsbecomeapairofellipticsurfaces, V
V 1,V
A E →
and V, which intersect along the section s. By [19, Theorem 2.5], there
2,V
E →
are unique extensions of V to minimal elliptic surfaces C over C,
i,V i
E → E →
for i = 1,2. By construction, we have an isomorphism over V between and
V
A
, as required.
1 C 2
E × E
Finally, the statement about the j-invariants is an easy consequence of the dis-
cussion in Section 2.1. (cid:3)
Using this, we may construct a threefold C that is isomorphic to over
Y′ → YV′
V byapplyingtheKummerconstructionto fibrewise. Tofurtherconstruct
1 C 2
E × E
a model for , we need to know how the group G defining the cover f acts on
U
Y
.
1 C 2
E × E
Lemma 3.7. Let ϕ denote the action of a permutation in G S on C. Then
6
⊂
either ϕ induces automorphisms on and , or ϕ induces an isomorphism
1 2 1
E E E →
.
2
E
Proof. Note first that ϕ induces an automorphism ϕˆ of . Furthermore, ϕˆ pre-
V
A
serves the section s as, by [10, Lemma 4.5], the curve E in a general fibre Y of
11 p
is invariant under monodromy in U.
U
Y
As in the proof of Proposition 3.5, we identify and with the elliptic
1,V 2,V
E E
surfacesin sweptout by the preimagesofthe curvesG andH . These elliptic
V 1 1
A
surfaces intersect along the section s.
As ϕˆ preserves s, we see that ϕˆ( ) is an elliptic surface in that contains
1,V V
E A
s as a section. It must therefore either be or . The same holds for .
1,V 2,V 2,V
E E E
Thus,we see that ϕ either induces automorphismson and , or induces an
1,V 2,V
E E
isomorphism .
1,V 2,V
E →E
Thus ϕ induces either a birational automorphism on and , or a birational
1 2
E E
map . But, by [17, Proposition II.1.2], a birational map between minimal
1 2
elliptiEc s→urfEaces is an isomorphism. (cid:3)
It is easy to determine which case of Lemma 3.7 occurs:
Lemma 3.8. Let ϕ denote the action of a permutation in G S on C. Then ϕ
6
⊂
induces automorphisms on and (resp. ϕ induces an isomorphism ) if
1 2 1 2
E E E →E
and only if the action of ϕ preserves (resp. exchanges) the roots of Equation (3.1)
(which are generically distinct by Remark 3.6).
Proof. ByLemma3.7,weknowthateitherϕinducesautomorphismson and ,
1 2
E E
or ϕ induces an isomorphism . To see which occurs, we study the action
1 2
E → E
on a general fibre of .
1
E
So let E denote the fibre of overa generalpoint p V and let E denote the
i Ei ∈ i′
fibreof overϕ(p)(fori 1,2 ). ByProposition3.5,weseethatthej-invariants
i
E ∈{ }
of E ,E are equal to those of E ,E . Thus, either
{ 1 2} { 1′ 2′}
j(E )=j(E )thenj(E )=j(E ),soE =E andE =E andϕinduces
• 1 1′ 2 2′ 1 ∼ 1′ 2 ∼ 2′
automorphisms on and , or
1 2
E E
• j(E1)=j(E2′), so E1 ∼=E2′ and ϕ induces an isomorphism E1 →E2.
8 C.F.DORAN,A.HARDER,A.Y.NOVOSELTSEV,ANDA.THOMPSON
But the j-invariants of E and E are given by the roots of Equation (3.1), so
i i′
the first (resp. second) case occurs if and only if the action of ϕ preserves (resp.
exchanges) these roots. (cid:3)
Let H G denote the subgroup of G that preserves the decomposition of
⊂
F ,...,F into subsets F ,F ,F and F ,F ,F . Then we can say more
3 8 3 4 5 6 7 8
{ } { } { }
about the action of the subgroup H on and .
1 2
E E
Proposition 3.9. (See [4, Lemmas 7.6 and 7.7]) Let τ be any permutation in
H S and let ϕ denote the corresponding map on C. Then
6
⊂
If τ is an odd permutation, then ϕ induces an isomorphism .
1 2
• E →E
If τ is an even permutation, then ϕ induces automorphisms of and
1 2
• E E
Proof. Suppose first that τ is an odd permutation. Let γ denote a path in V that
connects a point p V to ϕ(p). We will show that as we move along γ, the j-
∈
invariantsofE andE areswitched. Todothisweusef topusheverythingdown
1 2
to B. The image f(γ) is a loop in U starting and ending at f(p). By Lemma
3.8, we therefore need to show that monodromy around f(γ) switches the roots of
Equation (3.1).
Monodromyaroundf(γ)actsonthesetofdivisors F ,...,F inthefibreY
3 8 f(p)
{ }
of over f(p) (see Section 2.1) as the permutation τ. Furthermore, as H is the
Y
subgroup of G that preserves the sets F ,F ,F and F ,F ,F , monodromy
3 4 5 6 7 8
{ } { }
around f(γ) must also preserve these sets.
As these divisors are permuted if and only if the roots of the cubic polynomials
(P(x) 1) (see Equation (2.1)) are permuted and as τ is an odd permutation, we
±
see that the product of the discriminants of these cubics must vanish to odd order
inside γ. This product is given by
∆:=a6 2a3b2+b4 2a3 2b2+1,
− − −
where a and b are the (a,b)-parameters associated to the M-polarized fibre X
f(p)
of over f(p).
X
Now, monodromy around f(γ) switches the roots of Equation (3.1) if and only
if its discriminant (σ2 4π) vanishes to odd order inside γ. However, by Remark
−
3.6, we find that this discriminant is given exactly by ∆. So monodromy around
f(γ) switches the roots of Equation (3.1) and thus, by Lemma 3.8, it induces an
isomorphism .
1 2
E →E
This completes the proof in the case when τ is an odd permutation. The proof
when τ is an even permutation is similar. (cid:3)
To construct a model for , our starting data consists of the cover f: C B
U
Y →
and the two elliptic surfaces C. This data must satisfy the condition that
1,2
E →
the decktransformationgroupGofthe coverf shouldactasautomorphismsonor
isomorphisms between the elliptic surfaces and , in a way that is compatible
1 2
E E
with its action on C.
We begin by constructing a model for by performing the Kummer construc-
YV′
tion fibrewise on to obtain a new threefold , which is isomorphic to
1 C 2 ′
E × E Y
over V. Then, to obtain a model for , we perform a quotient of by G.
YV′ YU Y′
However, the action of G is not the obvious one induced by the action of G on
and (if it were, we would be able to undo the Kummer construction on ,
1 2 U
E E Y
whichisnotpossibleingeneral). Instead,wecomposethisactionwiththefibrewise
CALABI-YAU THREEFOLDS FIBRED BY KUMMER SURFACES 9
automorphism induced by the action of G (as a subset of S ) on the set of curves
6
F ,...,F .
3 8
{ }
Remark 3.10. AsthefibrewiseKummerconstructiondefinesanaturaldoubleKum-
merpencilonsmoothfibresof ,wecanusetheresultsofKuwataandShioda[16,
F
Section5.2]to define the elliptic fibrationΨ onthe smoothfibres of . The curves
F
F ,...,F are then the components of the I fibres that do not meet a chosen
3 8 2
{ }
section.
Once the curves F ,...,F are known, the automorphisms permuting them
3 8
{ }
may be computed explicitly as compositions of the relevant symplectic automor-
phisms from [14, Section 4].
Quotienting by this G-action, we obtain a new threefold B. By construc-
Y →
tion, is isomorphic to over U, as required. We have a diagram:
U
Y Y
❴Ku❴mm❴er❴// // oo❴ N❴ik❴uli❴n❴
E1×OOC E2 YOO′ YOO XOO
(cid:31)? // (cid:31)? // (cid:31)? oo❴ ❴ ❴ ❴ (cid:31)?
AV YV′ YU XU
This construction will be illustrated in the next section.
4. Some Calabi-Yau threefolds fibred by Kummer surfaces
In the remainder of this paper, we will illustrate how these methods can be
usedto constructexplicit examples of Calabi-Yauthreefolds. We note that, in this
paper, a Calabi-Yau threefold will always be a smooth projective threefold with
X
ω = and H1( , ) = 0. We further note that the cohomological condition
X ∼ OX X OX
inthisdefinitionimpliesthatanyfibrationofaCalabi-YauthreefoldbyK3surfaces
must have base curve P1, so from this point we restrict our attention to the case
where B =P1.
∼
Asourstartingpoint,wewilltaketheK3-fibredthreefolds P1 constructed
g
X →
in[9]. Byconstruction,theNéron-Severigroupofageneralfibreinthesethreefolds
is isometric to M = H E E 4 , which admits an obvious embedding
2 8 8
⊕ ⊕ ⊕h− i
M ֒ M , and the restriction U of to the subset U P1 over which the
2 U g
→ X → X ⊂
fibres are smooth is an M -polarized family of K3 surfaces. Thus these threefolds
2
satisfy all of the conditions of Section 2.
4.1. A special family. In [9], the threefolds are constructed as resolved pull-
g
X
backs of a special family over the (compact) 1-dimensional moduli
X2 → MM2
space of M -polarized K3 surfaces, by a map g: P1 . To study the
MM2 2 → MM2
threefolds related to the by the construction detailed above, we will begin by
g
X
studying .
2
X
The family is described in [10, Section 5.4.1]. It is given as the
X2 → MM2
minimal resolution of the family of hypersurfaces in P3 obtained by varying λ in
the following expression
(4.1) λw4+xyz(x+y+z w)=0.
−
This family has also been studied extensively by Narumiya and Shiga [20], we will
use some of their results in the sequel (note, however, that our λ is not the same
as the λ used in [20], instead, our λ is equal to µ4 or u from [20]).
256
10 C.F.DORAN,A.HARDER,A.Y.NOVOSELTSEV,ANDA.THOMPSON
Dolgachev [8, Theorem 7.1] proved that is isomorphic to the compactifi-
MM2
cation of the modular curve Γ (2)+ H. In [10, Section 5.4.1] it is shown that the
0
\
orbifoldpointsoforders(2,4, )onthiscurveoccuratλ=( 1 , ,0)respectively,
∞ 256 ∞
and that the K3 fibres of are smooth away from these three points. Let U
X2 M2
denote the open set obtained from by removing these three points. Then
MM2
the restriction of to U is anM -polarizedfamily of K3 surfaces,and
X2,U X2 M2 2 X2
satisfies all of the conditions of Section 2.
Wenextcomputethe(a,b,d)-parametersofthefibresof (consideredasM-
2,U
X
polarizedK3surfaces,seeSection2.1). Todothis,weusethefactthatthestandard
and alternate fibrations on the K3 fibres are torically induced, so their g and g
2 3
invariants may be computed (in terms of λ) using the toric geometry functionality
of the computer software Sage. These expressions can then be compared to the
correspondingexpressionscomputedforanM-polarizedK3surfaceinnormalform
(see, for instance, [5, Theorem 3.1]). We thus obtain
1 3 1
a=λ+ , b= λ , d=λ3,
2432 8 − 2633
The σ and π invariants for the family are then given in terms of λ by
2,U
X
23 1
σ :=2 + ,
− 263λ 2633λ2
1 3
π := 1+ .
2432λ
(cid:18) (cid:19)
4.2. Undoing the Kummer construction. Let U denote the family
Y2,U → M2
ofKummersurfacesobtainedfrom byquotientingbytheNikulininvolutionand
2
X
resolvingany resulting singularities. From Remark 3.3 and the (a,b,d) parameters
for computed above, we see that Assumption 3.2 is satisfied by . We will
2 2,U
X Y
explicitlyshowhowtoconstructamodelfor fromellipticsurfaces,asdescribed
2,U
Y
in Section 3.2.
Our first step is to undo the Kummer construction for U . To do this,
Y2,U → M2
we proceed to a cover f: C , as computed in [10, Section 5.3.2]. This
M2 → MM2
cover is calculated in three steps. The first step is to take the cover f : Γ (2)
1 0
\
H . This is a double cover ramified over λ 1 , , which is given in
→ MM2 ∈ {256 ∞}
coordinates by
1
λ= µ2+ ,
− 256
where µ is a coordinate on Γ (2) H and the orbifold points of orders (2, , )
0
\ ∞ ∞
occur at µ=( , 1 , 1 ) respectively.
∞ 16 −16
The secondstep is to take the coverf : Γ (4) H Γ (2) H. This is a double
2 0 0
\ → \
cover ramified over µ 1 , , which is given in coordinates by
∈{16 ∞}
1
µ= (µ)2+ ,
′
− 16
whereµ isacoordinateonΓ (4) Handthethreecuspsoccuratµ =(0, 1 , 1 ).
′ 0 \ ′ √8 −√8
Finally,thethirdstepistotakethecoverf : C Γ (4) H. Thisisadouble
3 M2 → 0 \
cover ramified over µ 1 , 1 , which is given in coordinates by
′ ∈{√8 −√8}
1 (1 ν2)
µ′ = − ,
√8(1+ν2)