On modularity of rigid and nonrigid Calabi-Yau varieties associated to the root lattice A_4

0.46 MB·English

by  Klaus Hulek

On modularity of rigid and nonrigid Calabi-Yau
A
varieties associated to the root lattice
4
5 Klaus Hulek and Helena Verrill
0
0
2

n Abstract
a
J We prove the modularity of four rigid and three nonrigid Calabi-
6 Yau threefolds assciated with the A4 root lattice.

]
G 1 Introduction
A
. In this paper we investigate the geometry and arithmetic of a family of
h
t Calabi-Yau threefolds Xa, a = (a1 : : a6) P5, birational to the projec-
a ··· ∈
tive hypersurface in T := P4 X ...X = 0 given by
m 1 5
\{ }
[
a1 a5
5 Xa∩T : (X1 +...+X5) X +...+ X = a6.
v (cid:18) 1 5(cid:19)
9
Our motivation is to find further examples of modular Calabi-Yau varieties,
6
1 i.e.,ofCalabi-YauvarietieswhicharedefinedovertherationalsandwhoseL-
4 series can be described in terms of modular forms. This is motivated by the
0
Fontaine-Mazur conjecture on the modularity of two dimensional ℓ-adic Ga-
3
0 lois representations comingfromgeometry [FM], whichis ageneralization of
h/ the Taniyama-Shimura-Weil conjecture on the modularity of elliptic curves,
t proved by Wiles et al, [Wi], [BCDT]. More precisely, Fontaine and Mazur
a
m [FM] define the notions of a “geometric Galois representation”, and a “Ga-
: lois representation coming from geometry”. They conjecture that geometric
v
Galois representations are precisely the Galois representations coming from
i
X
geometry(suchastheonesweconsider)([FM,conjecture1]),andcombining
r this with classical conjectures (see e.g. [Se2]) leads them to the conjecture
a
that two-dimensional irreducible geometric Galois representations are mod-
ular up to a Tate twist ([FM, conjecture 3c]). From these two conjectures,
oneobtains theconjecture thattwo dimensionalirreducibleGalois represen-
tations coming from geometry are modular up to a Tate twist. We will use
the term “modular ”to mean “modular up to a Tate twist ”, and even to
denote direct sums of modular Galois representations.
RigidCalabi-Yauthreefolds(definedoverQ)areexpectedtobemodular,
since they have 2-dimensional middle cohomology. One expects the L-series
1
of the Galois action on the middleℓ-adic cohomology to betheMellin trans-
form of a weight 4 elliptic modular form. Although recently Dieulefait and
Manoharmayum [DM] proved that a rigid Calabi-Yau threefold is modular
provided it has good reduction at 3 and 7, or at 5 and another suitable
prime (in fact most of our examples have bad reduction at 3 and 5, so this
result does not apply, and in general does not determine the exact modular
form), it is still the case that relatively few examples of modular Calabi-Yau
threefolds are explicitly known. Most currently known examples are given
in Yui’s survey articles [Y1], [Y2]. Otherrecent examples aregiven by [CM].
Modularity hasbeenalso conjecturedforcertain nonrigidexamples, e.g.,
[CM]. What is new in this paper is the proof of modularity for several
nonrigid examples. Note that we mean modularity in the sense of Fontaine-
Mazur, i.e., the semisimplification of the Galois representation is a sum of 2
dimensional pieces. There are few examples of other kinds of modularity of
nonrigid Calabi-Yau threefolds known. Consani and Scholten [CS] consider
an example correspondingto a Hilbert modular form for which they provide
evidence for the modularity and Livn´e and Yui [LY] very recently gave some
cases involving weight 2 and 3 forms. Their examples and techniques are
quite different from ours.
We shall study a certain 5-dimensional family Xa, a P5, of (singu-
∈
lar) Calabi-Yau threefolds, associated to the root lattice A , by means of
4
Batyrev’s construction [Ba] of Calabi-Yau varieties as toric hypersurfaces.
The rigid cases are X , X , X , and X , where X :=
1 9 (1:1:1:1:4:4) (1:1:1:4:4:9) t
X . These have 40, 35, 37 and 35 nodes respectively, and their (big)
(1:1:1:1:1:t)
resolutions have 2-dimensional middle cohomolgy. We will show that the
L-series is the Mellin transform of a modular form of weight 4, and level 6,
6, 12 and 60 respectively. The first few terms of their q-expansions are
f = q 2q2 3q3+4q4+6q5+6q6 16q7 8q8+..., (1)
6
− − − −
f = q+3q3 18q5+8q7+9q9+36q11 10q13+..., (2)
12
− −
f = q 3q3 5q5 28q7+9q9 24q11 70q13+15q15 +..., (3)
60
− − − − −
where f has level N. Although the middle cohomology of X and X have
N 1 9
the same L-series, we will see that they are not birational to each other,
thoughbyaconjectureofTate, oneexpectsacorrespondencebetweenthem.
The nonrigid examples we consider are X , X and X .
25 (1:1:1:9:9:9) (1:1:4:4:4:16)
In these cases we show that L-series of the middle cohomology of the big
resolutions are
L(f ,s)L(g ,s 1)4, L(f′ ,s)L(g ,s 1)2, L(f ,s)L(g ,s 1), (4)
30 30 − 30 30 − 90 30 −
respectively, where L(h) denotes the Mellin transform of the function h,
and the functions g ,f ,f′ and f are cuspidal Hecke eigen newforms,
30 30 30 90
g having weight 2, the others weight 4, and f having level 90, the others
30 90
2
level 30. (The level 30 has, to our knowledge, not previously appeared in
examples of this kind.) The first few terms of the q-expansions are
g (q) = q q2+q3+q4 q5 q6 4q7 q8+q9+q10+q12+...(5)
30
− − − − −
f (q) = q 2q2+3q3+4q4+5q5 6q6+32q7 8q8+9q9+... (6)
30
− − −
f′ (q) = q+2q2+3q3+4q4 5q5+6q6 4q7+8q8+9q9+... (7)
30 − −
f (q) = q 2q2+4q4 5q5 4q7 8q8+10q10 12q11 +... (8)
90
− − − − −
Given expression (4), one would expect, by the Tate conjecture, that there
is a geometric reason for the occurrence of the weight 2 modular form g ,
30
which is the Mellin transform of the L-series of a certain elliptic curve. We
will see that this is indeed the case.
In both rigid and nonrigid cases, we use the powerful theorem due to
Faltings, SerreandLivn´e[Li],whichpermitsonetodetermine2-dimensional
Galois representations from a finite set of data. In practice this means
counting the number of points modulop for a given finite number of primes,
a task which can be done easily by computer.
In 2 we consider the toric geometry set up. In 3 we discuss the reso-
§ §
lution of singularities of the singular subfamily Xa. In 4 we show that Xa
§
is birational to a fibre product of families of elliptic curves, which allows us
to apply results of Schoen [Sc]. In 5 we study a certain elliptic surfaces
§
contained in Xa, and in 6 we count points and apply Livn´e’s method to
§
determine the L-series of the 7 cases of Xa mentioned above.
Figure1givesaschematicdiagramofthe5-dimensionalfamilyXawhich
we shall study, and some of its subfamilies (a complete list is given in Ta-
ble 1). The diagram gives the dimension of these subfamilies and the value
of h12 of the big resolution Xa of the general member of the subfamily. Val-
ues of a where Xa is modular are marked with points, and those which are
rigidwithcircled points. Ine5wewillseethatfortwoequalindicesa = a ,
i j
§
there is a correspeonding elliptic surface in Xa. We call the piece of H3 cor-
responding to these elliptic surfaces Wa. When dimH3 dimWa = 2, i.e.,
−
2h12 = dimWa, one expects Xa to be modular, which we will see is the case
for all 7 marked points in the diagram.
3
FX1a:hIafs √3a1i=0,
rIf=a0=,1(,19:,1:t1h:e`rn`:r:`r), th1=r2``2=5`4 ````t=re9 (cid:8)(cid:25)(cid:8) sainndguhItlf1a=2arPi(0t≥X=ie,e1as(,,1)9:≤,1:t4h1e:n1:1:t),
iddfii6mmr=WWaa9,==(th84e?(,,nhh11(22==(52.((((((((rhr1=29=2 rerea=(1:1t:1=re`(:`11:(`4:4(`)(`(cid:19)/(cid:19)``dt`i66=(cid:2)m(cid:14)(cid:2)r2W`5,a`t=he8n,ihf1a2ls=o5
(cid:14)(cid:2) ra=(1:1:1:4:4:9) t=0
(cid:2) a=(4:4:4:1:16:1) degenerate
F10:Ifa=(1:1X:ba2h:ba2s:(≥ban+3d51h)s2i1n2:g((ubXel+aar)i1t)≤i2es)1, (cid:27)  sFbin86=g:u0I,lfa1ra,itX=ieas(,1ha:na1sd:1h≥:1b223(4:X4a:()b≤+11)2),
Figure 1: Values of the parameter a for certain members and subfaemilies of
the family of Calabi-Yau threefolds Xa, with a for modular Xa marked by
a point, which is circled if Xa is rigid.
Finally we would like to point out that the family of Calabi-Yau vari-
eties which we are considering in this paper has recently also appeared in
a different context. C. Borcea has studied these varieties in the context of
configuration spaces of planar polygons (see [Bo] where these varieties are
called Darboux varieties).
Acknowledgments
We are grateful to the following institutions for support: to the DFG for
grant Hu 337/5-1 (Schwerpunktprogramm “Globale Methoden in der kom-
plexen Geometrie”) and to the University of Essen and M. Levine for hos-
pitality during a stay supported by a Wolfgang Paul stipend. We are also
greatly indebted to V. Batyrev, W. Fulton, J. Kolla´r and P.H.M. Wilson
whosecomments on intersection theory andCalabi-Yau manifolds werevery
helpful. We also thank N. Fakhruddin for useful remarks on ℓ-adic Galois
representations and M. Schu¨tt for pointing out some misprints.
Notation
In this paper we consider projective Calabi-Yau varieties defined by poly-
nomial equations with coefficients in Z. We work over the field k, where
k = C, Q, Q, F or F . Further notation is as follows.
p p
4
M The A root lattice, as a sublattice of Z5.
A4 4
N (M )∨, identified with a sublattice of M Q.
A4 A4 A4 ⊗
ε Point in M at e e .
ij A4 i− j
∆ Polytope in M R with vertices at ε .
A4 A4 ⊗ ij
Σ Fan in N R given by all faces of the Weyl chambers.
A4 A4 ⊗
Σ3 3-dimensional cones in Σ .
A4
Pe Smooth toric variety defined by Σ .
A4
Ae4 Affine piece of P correspeonding to σ Σ3.
σ ∈
Te Torus (k∗)4 P. e
⊂
T Orbit under theetorus action correspondeing to σ Σ ;
σ ∈ A4
P = T . e
σ∈Σ σ
X For u P20, a Calabi-Yau threefold defined in P deefined by ∆ .
Xau Feor aF∈ Pe5, a member of a 5 dimensional family of singular A4
∈
Calabi-Yaus in P, with 30 nodes on Xa\T if e5i=1ai 6= 0.
Xa A Calabi-Yau given by taking a specific choice of small projective
Q
resolution of thee30 singularities on Xa T.
\
Xa Big resolution of remaining singularities on Xa.
X X .
t (1:1:1:1:1:t)
e
X Big resolution of X = Xu,Xa,Xa or Xt, for X irreducible.
X A choice of small projective resolution of X, if one exists.
e
2b Toric varieties
Batyrev [Ba] constructs Calabi-Yau varieties as hypersurfaces in a toric va-
riety defined by a reflexive polytope ∆ ([Ba, p.510]), in a lattice M, as
follows.
The pair (M,∆) gives rise to a fan Σ in the dual space NR = MR∨,
and a strictly convex support function h on NR. Let (P, P(1)) be the
O
correspondingpolarized toricvariety. IngeneralP is asingularFanovariety
with Gorenstein singularities and (1) is the anticanonical bundle.
P
O
Batyrev shows that the general element in the anticanonical system is
a Calabi-Yau variety, with canonical singularities exactly at the singular
points of P. A desingularization P of P can be constructed by taking the
maximalprojectivetriangulation∆∗ ofthedualpolytope∆∗. Wedenotethe
corresponding fan by Σ. The strictetransforms of the anticanonical divisors
on P define a family of Calabi-Yafu threefolds on P whose general element
X is smooth. e
u
e
2.1 The polytopes ∆ , ∆∗ and ∆∗
A4 A4 A4
We now describe the lattice and polytope to which we apply Batyrev’s con-
g
struction. More generally, in the following construction one may replace A
4
by A , defined similarly; more details can be found in [DL], [P] and [Lu].
n
5
Tetrahedral face Prism face
F+ := (x = 1) ∆ F+ := (x +x = 1) ∆
1 1 ∩ 15 1 5 ∩
ε12 ε12 ε13
(cid:19)(cid:19)(cid:5)(cid:5)@@ QQ (cid:17)(cid:17)
(cid:19) (cid:5) @ εQ(cid:17)
(cid:19) (cid:5) @ 14
ε15 JJ (cid:5)(cid:5) """ε13 ε52QQ (cid:17)(cid:17)ε53
"
J"(cid:5) Q(cid:17)
ε14 ε54
there are 10 such faces there are 20 such faces
Figure 2: Three dimensional faces of ∆
Let M be the A root lattice, given as the following sublattice of Z5:
4
M = M = (x ,x ,x ,x ,x ) x Z, x = 0 Z5.
A4 1 2 3 4 5 i ∈ i ⊂
n (cid:12) X o
TheinnerproductonM isinducedbyth(cid:12)(cid:12)estandardinnerproductonZ5,and
we identify the dual lattice N := (MA4)∨ with a sublattice of MR := M⊗R.
Definition 2.1. The polytope ∆ = ∆A4 in MR is defined to be the convex
hull of the roots ε := e e ,1 i,j 5,i = j of M, where e ,...,e is
ij i j 1 5
− ≤ ≤ 6
the standard basis for Z5. In [V, p.427] it is shown that ∆ is reflexive.
It is a simple combinatorial exercise to enumerate the faces of ∆ , ∆∗
A4 A4
and ∆∗ . We have the following results.
A4
Lemma 2.2. The polytope ∆ has 20 vertices, 60 edges, 30 square faces, 40
g
triangular faces, and 30 three dimensional faces, given by
Fε := (x = ε1) ∆, Fε := (x +x = ε1) ∆,
i i ∩ ij i j ∩
for 1 i,j 5, i = j, and ε= . Two of these faces are shown in Figure 2.
≤ ≤ 6 ±
Proof. See [Lu, Lemma 1.18 and Korollar 1.19].
Lemma 2.3. The dual polytope ∆∗ has 30 vertices and 20 three dimensional
cubical faces, Θ for 1 i,j 5, i = j. The vertices of Θ are shown in
ij 15
≤ ≤ 6
Figure 3, and Θ = σΘ , where σ S with σ : 1,5 i,j.
ij 15 5
∈ 7→
Proof. See [Lu, Lemma 1.46].
Lemma 2.4. The subdivded polytope ∆∗ has 120 three dimensional faces,
which are translations under S of Θ given in Figure 3.
5 12345
f
Proof. See [Lu, Beispiel 2.34].
6
Cubical face Θ = ε∗ of ∆∗ Face Θ of ∆∗
15 15 12345
15(3=,3,F−+2∗,\−2,−2) −15\(−=4F,1+,1∗,1,1) F\C1+2∗ (cid:17)(cid:17)(cid:1)\fF1+∗
12 \\ F4−5∗ \\ F11+3∗ CCC\\(cid:0)(cid:17)(cid:0)(cid:0)F(cid:0)(cid:17)(cid:0)(cid:0)4(cid:1)(cid:0)−(cid:0)5(cid:1)(cid:0)∗(cid:1) \\
F−∗ F+∗ CC (cid:0)(cid:0)(cid:0)(cid:1)
35 \ S14 \ C(cid:0)(cid:0)(cid:1) \
F5−\\∗ SS F2−5∗ \\C(cid:1)(cid:1) \\
= 1(1,1,1,1,1, 4) = 1(2, 3,2,2, 3) F−∗
5 − 5 − − 5
Figure 3: Three dimensional faces of ∆∗ and ∆∗
In order to apply Batyrev’s formulae for h11 and h12 wfe need to count
the number of lattice points in the interior of the faces of ∆ and ∆∗. We
make use of the following easy result.
Lemma 2.5. If w ,w ,w ,w is a basis for a lattice L, and Θ is a polytope
1 2 3 4
with vertices 0,w1,...,w4, then the only lattice points in Θ are its vertices.
By taking appropriate subdivisions of the faces Fε and Θ , we have
ij ij
Lemma 2.6. No proper face of ∆ or ∆∗ contains an interior lattice point,
and the only lattice point in the interior of ∆ or ∆∗ is the origin.
2.2 The toric variety P defined by the fan Σ
The fan Σ in NR consists of cones given by the 120 Weyl chambers
e e
5
σ =e (α ,α ,α ,α ,α ) R5 α = 0, α α α α α ,
ijklm 1 2 3 4 5 v i j k l m
∈ ≥ ≥ ≥ ≥
( )
(cid:12) Xv=1
(cid:12)
where i,j,k,l,m = 1,2,3,4,5 ,(cid:12) together with all their subfaces. E.g.,
{ } { }
σ is the cone on Θ (see Figure 3). The dual cones are given by
12345 12345
σ∨ := R (e e )+R (e e )+R (e e )+R (e e ).
ijklm ≥0 i− j ≥0 j − k ≥0 k − l ≥0 l − m
We will consider Calabi-Yau threefold hypersurfaces in the toric variety
P defined by Σ. We first fix choices of local and global coordinates for P.
We identify the torus T = (k∗)4 P with P4 ( 5 X = 0), and use
∼ ⊂ \ i=1 i
tehe projectiveecoordinates X ,...,X of P4 when considering points in Te.
1 5
Q
For the affine piece A4 := Spec(k[ς(σe∨ )]) P, where ς S , we use
ς 12345 ⊂ ∈ 5
coordinates x ,y ,z ,w corresponding to the basis ε ,ε ,ε ,ε
ς ς ς ς ς(12) ς(23) ς(34) ς(45)
of ς(σ∨ ). Usually we just write x,y,z,w. e
12345
The identification of T and A4 (x y z w = 0) is given by
ς \ ς ς ς ς
x = X /X , y = X /X ,
ς ς(1) ς(2) ς ς(2) ς(3)
z = X /X , w = X /X .
ς ς(3) ς(4) ς ς(4) ς(5)
7
HH P 1 × P 1 H Hl
(cid:0)(cid:0) (cid:2) ll There are 10 copies of TFiε∗ in P \ T,
(cid:2) two being x=0 and w =0 . The
(cid:12)(cid:12)(cid:0)P2 (cid:2)(cid:2) P2 l(cid:2)(cid:2)TT fcalocseuorfesFoεf,Tc{oΘn∗s,ifsot}rsΘofa8tcwo{poiedsimofe}nPes2ioannadl
J(cid:12)J@(cid:0)@P(cid:0)e1(cid:0)(cid:0)×@P1@@@(cid:0)(cid:24)e(cid:24)(cid:24)(cid:24)(cid:24)(cid:2)\(cid:2)P\1(cid:3)×(cid:3)(cid:3)(cid:3)P1 6icnodcroricepasipteesodnjodifnPtot1hP×is2,Pfig1a.unrdTe;shqheueseaxraiegnotfaenrcaseelescfactcoaerss-
,,
@ (cid:0) , respond to P1 P1.
@(cid:0)H(cid:0) P2(cid:16)(cid:16), ×e
HH(cid:16)(cid:16)
e
Figure 4: The threefold TFiε∗ in P \T.
e
This relationship between the coordinates of P4 and of the affine pieces of
P is explained by the following lemma.
Lemma 2.7. The variety P is the graph of the Cremona transformation
e
X 1/X of P4. Thus P is obtained from P4 by blowing up successively the
i i
7→
(strict transforms of the) poeints (1 : 0 : 0 : 0),(0 : 1 : 0 : 0)...,(0 : 0 : 0 : 1),
lines and planes spannedeby any subset of these points.
Proof. See [DL, Lemma 5.1].
2.2.1 Toric orbits in P
There is a decomposition P = T , where T is the toric orbit of P
e σ∈Σ σ σ
corresponding to σ Σ. Since Σ is given by taking cones on the faces of
∆∗, we use the notat∈ion TΘe:= TFR+Θewhere Θ is a face of ∆∗. By standared
methods of toric geomeetry we haeve
e e
TFiε∗ ∼= PA3,
TFiεj∗ ∼= PeA2 ×PA1 ∼= P2×P1,
where P is the toric variety coerresponeding teo the root lattice A , and P2
An n
is P2 blown up in 3 points. These are sketched in Figures 4 and 5.
In teerms of local coordinates x,y,z,w for A4 , we have e
id
T = T = x = 0= yzw ,
F4−∗ 15(1,1,1,1,−4) { 6 }
T = T = y = 0= xzw ,
F4−5∗ 15(2,2,2,−3,−3) { 6 }
T = T = z = 0 = xyw ,
F1+2∗ 15(3,3,−2,−2,−2) { 6 }
T = T = w = 0 = xyz .
F1+∗ 15(4,−1,−1,−1,−1) { 6 }
The intersections of the closures these hypersurfaces is sketched in Figure 6.
8
There are 20 copies of TFε∗ in P, two being
l(cid:8)l(cid:8)(cid:8)(cid:8) P2 (cid:8)(cid:8)(cid:8)ll(cid:8) {Θya=tw0}oadnimde{nzs=ion0a}l.fTacheeiocjfloFsiuεjr,eecsoonfsTisΘts∗,offo2r
lP1×P1P1e×P1 P1(cid:8)×P(cid:8)1 scpoponiedsinogf tPo2,thaendhe6xacgoopnisesanodf Psq1u×arPes1,recsoprerce--
l (cid:8)(cid:8)
tively in tehis figure.
Figure 5: The threefold TFiεj∗ in P \T.
e
PP a The closures of x = 0 , y = 0
,,,@@(cid:1)(cid:1)PlPl(cid:16)(cid:16)(cid:16)aZZ and {z = 0} inte{rsect as}ind{icated by}
(cid:2) (cid:1) (cid:1) ZZ the intersections of the corresponding
(cid:2) (cid:1)XyX=0(cid:1)(cid:1) (cid:2)(cid:2)TT polyhedra. A polyhedron correspond-
(cid:2)(cid:2) (cid:0) x=0 ing to w = 0 meets these polyhedra
\\(cid:0)(cid:0) @ (cid:2) (cid:3) inthel{abeledf}aces. Wheretwopolyhe-
(cid:0)(cid:0)@z=0 @@ (cid:2)ee(cid:3)(cid:3)(cid:3) dra meet in a hexagon the correspond-
@(cid:0)@@%X%@@X%`(cid:0)bb`(cid:0)`(cid:0)(cid:24)(cid:24)(cid:24)(cid:24),,, wsinphgoentrhderintehegefotyhldrmseeehfeoatlvdiensiinanttseeqrrsuseeaccrtteioitnnhPePe12c,oarPrne1d-.
×
Figure 6: How x = 0 , y =0 and z = 0 P T meet.
{ } { } { } ⊂ \
e
3 The Calabi-Yau varieties
Following Batyrev, we define a family of hypersurfaces in P, given by ele-
mentsX K . Thegeneralmemberofthefamilyisgivenbyanequation
∈ |− P|
containing exactly the monomials corresponding to the lattiece points of ∆.
Inourcase, ∆e has21lattice points,andsogives a20dimensionalfamily.
The general member, when restricted to the open torus T, has an equation
X : u X X−1 =t for u= (u :u : : u :t) P20 (9)
u ij i j 12 13 ··· 45 ∈
1≤i,j≤5,i6=j
X
in terms of the homogenous coordinates for T P4.
⊂
Given the above analysis of ∆ and ∆∗, we can now prove the following.
Proposition 3.1. For every smooth member X of the family of Calabi-Yau
u
threefolds (9), we have
(i) The Euler number e(X ) = 20.
u
(ii) The Hodge numbers of X are given by h00 = h33 = h30 = h03 =
u
1, h10 = h01 = h20 =h02 = 0, h11 =26 and h12 = h21 = 16.
9
Proof. Since X is smooth and Calabi-Yau the only Hodge numbers to be
u
computed are h11 and h21. By [Ba, p.521] we have
h11(X )= l(∆∗) 5 l∗(Θ∗)+ l∗(Θ∗)l∗(Θ),
u
− −
codimΘ =1 codimΘ =2
X∗ X∗
h21(X ) = l(∆) 5 l∗(Θ)+ l∗(Θ)l∗(Θ∗),
u
− −
codimΘ=1 codimΘ=2
X X
where l(Θ) denotes the number of lattice points in Θ and l∗(Θ) denotes the
number of interior lattice points of Θ, for any face Θ of ∆.
Lemmas 2.2, 2.3 and 2.6 imply that l(∆) = 21,l(∆∗) = 31, l∗(∆∗) = 1
andl∗(Θ∗)= 0forallproperfacesΘof∆∗. Henceh11(X ) = 26, h21(X ) =
u u
16. The Euler characteristic is then given by e(X ) = 2h11 2h12 = 20.
u
−
3.1 Resolution of singular Calabi-Yau threefolds
We consider elements X K which have s nodes, but no other sin-
∈ |− P|
gularities. We denote the big resolution of X, obtained by blowing up the
nodes, by X. We also have 2s smeall resolutions of X, where each node is
replaced by a P1. It is not clear whether there are any small projective res-
olutions asethese could all contain null homologous lines. By X we denote
a small projective resolution, when one exists.
Let Xu bea smooth member of the family (9), and let Xa beban element
of the family with s nodes, but no other singularities. Then we have
Proposition 3.2. Let hpq = hpq(Xa), resp. hpq = hpq(Xa) be the Hodge
numbers of the big resolution Xa of Xa, resp. a small projective resolution
of Xa. Then the followieng holds e b b
e
(i) e(Xa) = e(Xu)+s, e(Xa) = e(Xu)+2s, e(Xa)= e(Xu)+4s
(ii) h30 =h03 = hˆ30 = hˆ03 = 1
b e
(iii) h˜10 =h˜01 = hˆ10 = hˆ01 = 0, h˜20 = ˜h02 = hˆ20 = hˆ02 = 0
e e
(iv) h˜11 h˜12 = 1e(Xa), hˆ11 hˆ12 = 1e(Xa).
− 2 − 2
Proof. (i) These formulae are well known, cf. [C, 1] or [We, Kapitel II].
e b
§
(ii) Since hp0 = h0p are birational invariants, it is enough to prove the
assertion forh˜30. LetQ1,...,Qs betheexceptional quadricsinXa andnote
that their normal bundleis ( 1, 1). Since Xa is a Calabi-Yau variety with
− −
s e
s nodes it follows that ω = Q and h˜30 = h0(ω )= 1.
Xa OXa i X˜a
(cid:18)i=1 (cid:19)
(iii) We consider the sequence P
e e
0 ( K ) 0.
→ OP − P → OP → OXa →
e e e
10