Calabi-Yau coverings over some singular varieties and new Calabi-Yau 3-folds with Picard number one

0.22 MB·English

by  Nam-Hoon Lee

CALABI–YAU COVERINGS OVER SOME SINGULAR
VARIETIES AND NEW CALABI-YAU 3-FOLDS WITH
8
0 PICARD NUMBER ONE
0
2
NAM-HOONLEE
n
a
J
Abstract. Thisnoteisareportontheobservationthatsomesingular
4
varieties admit Calabi–Yau coverings. As an application, we construct
1
18 new Calabi–Yau 3-folds with Picard number one that have some

interesting properties.
]
G
A
.
h 0. Introduction
t
a
A Calabi–Yau manifold is a compact Ka¨hler manifold with trivial canon-
m
ical class such that the intermediate cohomologies of its structure sheaf are
[
alltrivial(hi(X,OX)= 0for0 < i< dim(X)). Onehandywayofconstruct-
4 ing Calabi–Yau manifolds is by taking coverings of some smooth varieties
v
such that some multiples of their anticanonical class have global sections.
0
6 Indeed many of known examples of Calabi–Yau 3-folds with Picard num-
0 ber one are constructed in this way (see, for example, Table 1 in [EnSt]).
0
In this note we show that singular varieties with some cyclic singularities
1
also admit Calabi–Yau manifolds as their coverings (Theorem 1.1). We give
6
0 some formula for calculating their invariants by using degeneration method
h/ (Theorem 2.1, Theorem 3.2).
t In his beautiful papers ([Ta1], [Ta2]), H. Takagi classified possible invari-
a
m ants of certain Q-Fano 3-folds of Gorenstein index 2 and constructed some
exotic examples of Q-Fano 3-folds. We apply our theorem to construct
:
v Calabi–Yau 3-folds which are double coverings of Takagi’s Q-Fano 3-folds.
i
X Itturnsoutthat 18 of them are new Calabi–Yau 3-folds with Picard number
r one (Table 1). Although a huge number of Calabi–Yau 3-folds have been
a
constructed, those with Picard number one are still quite rare (for example,
see Table 1 in [EnSt]). Note that they are primitive and play an important
role in the moduli spaces of all Calabi–Yau 3-folds ([Gr]). We show that
some of them are connected by projective flat deformation although they
are of different topological types (Theorem 3.5). It is interesting that three
of them have the invariants which were predicted by C. van Enckevort and
D. van Straten in their paper ([EnSt]).
Letusrecallanotationforcertainsingularities. Leta ,··· ,a beintegers
1 n
and let x ,··· ,x be coordinates on Cn. Suppose that the cyclic group G
1 n
acts on Cn via
2000 Mathematics Subject Classification. 14J32, 14J45, 14D06.
1
2 N.-H.LEE
x 7→ εaix , for all i,
i i
where ε is a primitive rth root of unity for some positive integer r. A sin-
gularity q ∈ X is called a quotient singularity of type 1(a ,··· ,a ) if there
r 1 n
is a neighborhood of q that is isomorphic to a neighborhood of (0,··· ,0) in
Cn/G.
Consider a simple example of covering. Let
X = {x10+y10+z10 +w5 = t2} ⊂ P(1,1,1,2,5),
where x, y, z, w and t are homogeneous coordinates of weights 1, 1, 1, 2
and 5 respectively. Then X is a Calabi–Yau 3-fold. Define a projection
π : X → P(1,1,1,2)
by
(x,y,z,w,t) 7→ (x,y,z,w).
Note that P(1,1,1,2) has a singularity of type 1(1,1,1) at p = (0,0,0,1).
2
Let S = {x10 +y10 +z10 +w5 = 0} ⊂ P(1,1,1,2), then S is smooth. It is
easy to see that X is a double covering of P(1,1,1,2) with the branch locus
S ∪{p}.
Over p, the map π locally looks as the quotient map
C3 → C3/ ∼,
where (x,y,z) ∼ (−x,−y,−z).
In the following section, we prove that a variety with singularities of type
1(1,··· ,1)admitsaCalabi–Yaudoublecoveringwhenthedimensionisodd.
2
Acknowledgments
Prof. HiromichiTakagikindlycalculatedthetopologicalEulernumbersof
hisexamplesofQ-Fano3-folds. Theauthorgotsomeadviceonconstruction
of‘quasi-linebundle’inSection2fromProf. Ja´nosKolla´r. Theauthorisalso
thankfulto the referee for pointing out several typos and unclear arguments
in the previous version. Finally the author would like to express his sincere
thanks to Prof. Igor V. Dolgachev for many suggestions, discussions and
encouragement.
1. Existence of Calabi–Yau covering
For a Cartier divisor D on a variety X, let V(D) denote the total space of
the line bundle, corresponding to D and pr : V(D) → X be the projection.
Note that an element of H0(X,O(D)) can be regarded as a map from X to
V(D).
For any integer r, we can define a natural map, V(D)→ V(rD), by
(x,t) 7→ (x,tr)
in local coordinates, where t ∈ pr−1(x). We denote singularities of types
1(1,1), 1(1,1,1), ··· by 1(1[2]), 1(1[3]), ··· respectively. Note that 1(1,1)
2 2 2 2 2
is the ordinary double point singularity on a surface.
CALABI–YAU COVERINGS OVER SINGULAR VARIETIES 3
Letann-dimensionalvarietyY haveasingularityoftype 1(1[n])atp ∈ Y.
2
Let f : Y → Y bethe blow-up at p. Then thesingularity is resolved and the
exceptioenal divisor E is a copy of Pn−1 with normal bundle O (−2). Note
E
that
∗
2KYe = f (2KY)+(n−2)E.
Theorem1.1. LetY beaprojectiven-dimensionalvarietywithsingularities
of type only 1(1[n]). Suppose that n is odd and that the complete linear
2
system |−2K | contains a smooth (n−1)-fold S. Then there is a smooth
Y
projective n-fold X with K = 0 that is a double covering of Y with the
X
branch locus
S ∪Sing(Y).
Furthermore assume that
hi(Y,O ) =0 for 0< i < n,
Y
then X is a Calabi–Yau n-fold.
Proof. For simplicity, let Sing(Y)= {p} be composed of a single point. The
proof for general case is similar. Let f : Y → Y be the blow-up of Y at
p. Then Y is smooth. As mentioned beforee, 2KYe = f∗(2KY)+(n−2)E,
e
where E is the exceptional divisor. We can set n =2m+1 for some positive
integer m because n is odd. Then we have
f∗(−2KY)+E = −2KYe +(n−1)E = 2(−KYe +mE).
Let D = −KYe +mE, then 2D = f∗(−2KY)+E. Note that S′ = f∗(S)
is smooth and disjoint from E. Clearly there exists a global section ρ ∈
H0(Y,O (2D)) such that div(ρ) = S′+E. Note that ρ is also interpreted
Y
as aemap from Y to V(2D). Let X = φ−1(imρ), where φ : V(D) → V(2D)
e
is the natural map. Then X is smooth. Let π˜ : X → Y be the restriction
e e e
of the projection V(D) → Y to X, then π˜ is a double covering map. The
e e
canonical class of X is:
e
∗ ∗
(1.1) Ke = π˜ (Ke +D) =π˜ (mE) = 2mF,
X Y
where F = π˜−1(E). Note that F is isomorphic to Pn−1 because E is. Let
H be the unique ample generator of Pic(F), then
−nH = KF = (KXe +F)|F = (2mF +F)|F = (2m+1)F|F = nF|F.
Since Pic(F) is torsion-free, F| = −H. Therefore F can be smoothly
F
contracted to a point. Let g : X → X be the contraction. Note that X is
e
smooth. Since
KXe = g∗(KX)+(n−1)F,
we have
g∗(KX)= KXe −(n−1)F = 2mF −(n−1)F = 0.
4 N.-H.LEE
Because the map g∗ : Pic(X) → Pic(X) is injective, the canonical class of X
e
is trivial: K = 0. Furthermore since Y is projective, so are Y, X and X.
X
e e
So the first assertion is proved. Now we assume that
hi(Y,O ) = 0 for 0 < i < n
Y
and show that hi(X,O ) = 0 for 0< i < n. Firstly we note that
X
hi(X,Oe) = 0 for 0 < i < n.
X
e
Consider the following exact sequence,
0 → OYe(kE) → OYe((k+1)E) → OE((k+1)E) → 0,
where k is a non-negative integer. Note that
O ((k+1)E) = O (−2(k+1)).
E E
We have an exact sequence
Hi−1(E,OE(−2(k+1))) → Hi(Y,OYe(kE)) → Hi(Y,OYe((k+1)E)) →
Hi(E,Oe (−2(k+1))). e
E
Note that
Hi−1(E,O (−2(k+1))) = 0
E
and
Hi(E,O (−2(k+1))) = 0
E
for 0< i < n, k =0,1,··· ,m−1, which implies that
hi(Y,Oe((k+1)E)) = hi(Y,Oe(kE)).
Y Y
e e
Therefore we have
hi(Y,Oe(mE)) = hi(Y,Oe((m−1)E)) = ··· = hi(Y,Oe) = 0
Y Y Y
e e e
for 0< i < n. Note that the map π˜ is finite and
π˜∗(OX˜)≃ OYe ⊕OYe(−D)= OYe ⊕OYe(KYe −mE).
Finally we have
Hi(X,OXe) ≃ Hi(Y,π˜∗(OX˜))
e e
≃ Hi(Y,Oe)⊕Hi(Y,Oe(Ke −mE))
Y Y Y
e e
≃ Hi(Y,Oe)⊕Hi(Y,Oe(mE))∗
Y Y
e e
= 0
for 0< i < n, which leads to:
hi(X,OX) = hi(X,OXe) = 0
e
for 0 < i < n. In conclusion, X is a Calabi–Yau n-fold. There is a rational
mapπ : X 99K Y suchthatthefollowingdiagram(Figure1)iscommutative.
But it is not hard to see that π is actually a double covering morphism with
branch locus S ∪{p}. (cid:3)
CALABI–YAU COVERINGS OVER SINGULAR VARIETIES 5
π˜ //
X Y
e e
g f
(cid:15)(cid:15) (cid:15)(cid:15)
_ π_ _//
X Y
Figure 1.
2. Hodge numbers of the Calabi–Yau covering
For a double covering with dimension higher than two, it is a non-trivial
task to calculate the Hodge numbers even in the case that the base of the
covering is smooth. They are calculated for some special cases (see, for
example, [Cl2], [Cy] and [CySt]). In this section, we give a formula for
h1,1(X) when the complete linear system |− K | contains a variety with
X
a certain mild singularities. In three-dimensional case, the other Hodge
numbers of a Calabi–Yau 3-fold are determined by its topological Euler
number, which is easily calculable.
Theorem 2.1. Let X be the Calabi–Yau double covering of Y in Theorem
1.1. Suppose that the linear system |−K | contains a variety D such that:
Y
(1) The variety D has its singularities at those of Y, i.e. Sing(D) =
Sing(Y).
(2) The singularity types of Sing(D) are all 1(1[n−1]) and C = S∩D is
2
smooth.
(3) h1(O ) =0.
D
Then h1,1(X) = 2h2(Y)−k, where k is the dimension of the image of the
restriction map H2(Y,Q) → H2(D,Q).
To prove this theorem, we make a semistable degeneration of X and use
the well-known Clemens–Schmid exact sequence.
Note that the anticanonical divisor −K is not Cartier unless Sing(Y) =
Y
∅. So there is no line bundle corresponding to −K a priori. Let F =
Y
O (K ) and
Y Y
F[i] = (F⊗i)∗∗
be the double dual of F⊗i. Consider a sheaf of algebras on Y,
∞
A = MF[i],
i=0
where we use the multiplication
F[i]⊗F[j] ≃ F[i+j].
Let V(−K ) = Spec (A). Note that it is a generalization of the construc-
Y Y
tion of a line bundle for a Cartier divisor (cf. Definition 5 in [Ko]). Let
pr :V(−K )→ Y
Y
6 N.-H.LEE
be the projection. Let
∞
I = MF[i].
i=1
The map
A → A/I = O
Y
induces the embedding
Spec (O )֒→ Spec (A),
Y Y Y
i.e.
(2.1) i :Y ֒→ V(−K ),
Y
underwhichweconsiderY asasubvarietyofV(−K ). NotethatV(−2K ) =
Y Y
Spec(A(2)), where
∞
A(2) = MF[2i].
i=0
The map
Spec (A)→ Spec (A(2))
Y Y
is denoted by
(2.2) ψ :V(−K )→ V(−2K ).
Y Y
Let Y∗ = Y \Sing(Y), then we have
pr−1(Y∗)≃ V(−KY∗).
Note that −KY∗ is Cartier on Y∗.
This kind of construction was studied in a more general setting under
the name of ‘Seifert G -bundle’ ([Ko]). See Section 2 of [Ko] for relevant
m
discussion. The variety V(−K ) has its singularities at those of Y and
Y
the singularities in V(−K ) are all of type 1(1[n+1]), where we consider Y
Y 2
as a subvariety of V(−K ) (equation 2.1). Over singular points of Y, the
Y
projection pr: V(−K ) → Y locally looks like the following map:
Y
φ :V → U ,
n n
where
V = Cn+1/(x ,··· ,x ,t) ∼ (−x ,··· ,−x ,−t)
n 1 n 1 n
U = Cn/(x ,··· ,x )∼ (−x ,··· ,−x )
n 1 n 1 n
and the projection φ maps:
(x ,··· ,x ,t) 7→ (x ,··· ,x ).
1 n 1 n
We call V(−K ) with the projection pr : V(−K ) → Y a ‘quasi-line
Y Y
bundle’ corresponding to −K .
Y
A section ρ of this quasi-line bundle is a map ρ: Y → V(−K ) such that
Y
pr◦ρ = id . Note that the map pr| : imρ → Y is an isomorphism. We
Y imρ
obtain the double coverings in Theorem 1.1 in this setup.
CALABI–YAU COVERINGS OVER SINGULAR VARIETIES 7
Proposition 2.2. Let Y be a projective n-dimensional variety Y with sin-
gularities of type only 1(1[n]). Suppose that n is odd and that the complete
2
linear system |−2K | contains a smooth (n−1)-fold S. Then there is a
Y
smooth projective n-fold X with K = 0 that is a double covering of Y with
X
the branch locus
S ∪Sing(Y).
Proof. Let ρ be a section of O (−2K ) such that div(ρ) = S. Let X =
Y Y
ψ−1(imρ)(see equation 2.2 for ψ). We want to show that X is smooth.
Firstly, the open set ψ−1(ρ(Y∗)) of X is smooth. Let us look at what
happens near ψ−1(ρ(Sing(Y))). Locally it looks like:
{(x ,··· ,x ,t)∈ V t2 = g(x ,··· ,x )},
1 n n(cid:12) 1 n
(cid:12)
where g : U → C is a function such that g(0,··· ,0) 6= 0 (∵ S is smooth, so
n
disjointfromSing(Y)). By directcalculation, onecan show thatthis variety
is smooth. Therefore X is smooth. The map π = pr| :X → Y is a double
X
covering, branched along S ∪Sing(Y). Note that
∗
Kπ−1(Y∗) = π|π−1(Y∗)(KY∗ +(−KY∗)) = 0
Since the set
X \π−1(Y∗) = π−1(Sing(Y))
is finite, we have K = 0. (cid:3)
X
It is not hard to see that the Calabi–Yau manifold in the above theorem
is the same Calabi–Yau manifold in Theorem 1.1 for given Y.
Now we construct a degeneration of X, assuming that the linear system
|−K | contains a variety D such that Sing(D) = Sing(Y), the singularity
Y
types of Sing(D) are all 1(1[n−1]) and D ∩ S is smooth (the condition in
2
Theorem 2.1).
Let α be a section of OY∗(−KY∗) such that
∗
D∩Y =div(α).
It is not hard to see that there is a unique extension of α to a section
δ : Y → V(−K ) of the quasi-line bundle. Let −δ be the extension of −α.
Y
We also set Y = im(δ) and Y = im(−δ). Then Y is a copy of Y. If we
1 2 i
regard Y as a subvariety of V(−K ) (equation 2.1), we have
Y
Y ∩Y = Y ∩Y = D
1 2 i
and
X ∩Y = S.
We have a section α2 of OY∗(−2KY∗) and it has a unique extension to a
section of O (−2K ). Let us denote it by δ2. Note that δ2+tρ is a section
Y Y
of O (−2K ) for t ∈ C. Let ∆ be a small open disk which is centered at
Y Y
the origin in C. We define a variety:
Z = [ ψ−1(im(δ2 +tρ)) ⊂ V(−KY)×∆.
t∈∆
8 N.-H.LEE
Let
f :Z → ∆
be the projection and Z = f−1(t). Then the central fiber is:
t
Z = Y ∪Y .
0 1 2
and a generic fiber (t 6= 0) Z is a Calabi–Yau n-fold which is a deformation
t
of X (Note that X = ψ−1(im(ρ)) = f−1(∞)).
By direct calculation, we have:
Proposition 2.3. If ∆ is sufficiently small, then
Sing(Z) = ((D∩S)⊔Sing(Y))×{0}.
Along the smooth C =D∩S, Z is locally the product of an (n−2)-fold
and a three-dimensional ordinary double point singularity. If we blow up
Z along C, then the exceptional locus is a P1 ×P1-bundle over C. It is a
usual procedure to contract one of the ruling of the bundle smoothly to get
Z′ (see, for example, Proposition II.1. in [Lee]). We can choose the ruling
of the contraction such that
′ ′ ′
Z = Y ∪Y ,
0 1 2
where Y′ is isomorphic to Y and Y′ is the blow-up of Y along C. Now the
1 1 2 2
remaining singularities of Z are those which correspond to Sing(Y)×{0}
and they are all singularities of type 1(1[n+1]). In summary, we have:
2
Proposition2.4. LetX betheCalabi–Yau doublecoveringofY inTheorem
1.1. Assume the existence of D as in Theorem 2.1. Then we can construct
a degeneration Z′ → ∆ such that:
(1) A generic fiber is a deformation of the Calabi–Yau double covering
X of Y.
(2) The central fiber is a union of Y′ and Y′ such that Y′ is a copy of
1 2 1
Y and Y′ is the blow-up of Y along D∩S. Y′∩Y′ = D.
2 1 2
(3) The total space Z′ has its singularity at Sing(Y′) =Sing(Y′) and the
1 2
singularity types are all 1(1[n+1]).
2
By analyzing the well-known Clemens–Schmid exact sequence([Cl1]), we
have the following lemma (Theorem III.1 in [Lee]):
Lemma 2.5. Let W be central compact simple normal crossing fiber in a
0
semistable degeneration and W (t 6= 0) be the smooth compact Ka¨hler fiber
t
with h2(O ) = 0. Then we have
Wt
h1,1(W )= h2(W ) = h2(W )−r+1,
t t 0
where r is the number of components of W and h2(W ) = dim H2(W ,Q),
0 0 Q 0
etc.
For detailed proof, we refer to Chapter III of [Lee].
Now we are ready to prove Theorem 2.1.
CALABI–YAU COVERINGS OVER SINGULAR VARIETIES 9
proof of Theorem 2.1. For simplicity, let Sing(Y) be a single point. The
proof for the general case will be given later. Since the Hodge numbers
are invariant under deformation, h1,1(X) is the same with that of a generic
fiber of the degeneration Z′ → ∆ in Proposition 2.4. Since Z′ has a singular
point, the degeneration is not semistable. Let W → Z′ be the blow-up
at the singular point. Then we have another degeneration, W → ∆: the
composite of W → Z′ and Z′ → ∆. Note that the generic fiber is not
affected and the central fiber W = V ∪V ∪E is a reduced normal crossing
0 1 2
variety, whereV is thepropertransformation of Y′ and E is theexceptional
i i
divisor. So W → ∆ is a semistable degeneration. Note that D := V ∩V
1 2
is a smooth (n−1)-fold that is the proper transformation of Ye′ ∩Y′ ≃ D.
1 2
Let H := V ∩ E, then it is a copy of Pn−1. Note also that H ∩ D and
i i i
H ∩H are copies of Pn−2. Recall that h2(O ) = h2(O ) = 0 foret 6= 0.
1 2 Wt X
By Lemma 2.5,
(2.3) h2(W ) = h2(W )−3+1 = h2(W )−2.
t 0 0
Notethath1(OD)impliesh1(ODe) = 0. ByHodgedecompositiontheorem,
we have H1(D,Q)= 0. The Mayer–Vietoris sequence gives:
e
dimH2(H ∪H ,Q)= 1 and dimH1(H ∪H ,Q) = 0.
1 2 1 2
Let W′ = V ∪V and consider the following exact sequence:
0 1 2
0 → QW0 → QW0′ ⊕QE → QH1∪H2 → 0
to derive
(2.4)
0= H1(H ∪H ,Q) → H2(W ,Q) → H2(W′,Q)⊕H2(E,Q) → H2(H ∪H ,Q) → .
1 2 0 0 1 2
Since H2(E,Q) → H2(H ∪H ,Q) is not a zero map, neither is
1 2
H2(W′,Q)⊕H2(E,Q) → H2(H ∪H ,Q).
0 1 2
Because dimH2(H ∪H ,Q) = 1, the above map is actually surjective. So
1 2
the sequence (2.4) becomes a short exact sequence:
(2.5) 0→ H2(W ,Q) → H2(W′,Q)⊕H2(E,Q) → H2(H ∪H ,Q)→ 0.
0 0 1 2
Accordingly we have
(2.6) h2(W )= h2(W′)+h2(E)−h2(H ∪H ) = h2(W′).
0 0 1 2 0
10 N.-H.LEE
But with H1(D,Q)= 0, again the Mayer–Vietoris sequence gives:
e
h2(W′) =h2(V )+h2(V )−dim(im(H2(V ,Q)⊕H2(V ,Q) → H2(D,Q)))
0 1 2 1 2
e
=h2(Y′)+1+h2(Y′)+1−dim(im(H2(V ,Q)⊕H2(V ,Q)→ H2(D,Q)))
1 2 1 2
e
=h2(Y)+1+h2(Y)+2− dim(im(H2(Y′,Q)⊕H2(Y′,Q) → H2(D,Q)))+1
(cid:0) 1 2 (cid:1)
=h2(Y)+1+h2(Y)+2−(k+1)
(2.7)
=2h2(Y)+2−k.
So we have
(2.8)
h1,1(X) =h2(X) = h2(W ) = h2(W )−2 =h2(W′)−2 = 2h2(Y)−k.
t 0 0
Let us consider general case that m = #Sing(Y) is not necessarily one.
Let W → Z′ be the blow-up at singular points. Let E ,··· ,E be the
1 m
exceptional divisors and H = V ∩E . We note that E ∩E = ∅. Then
ij i j i j
the equation (2.3) becomes
h2(W )= h2(W )−(2+m)+1= h2(W )−m−1,
t 0 0
the short exact sequence (2.5) becomes
0→ H2(W0,Q)→ H2(W0′,Q)⊕MH2(Ei,Q)→ MH2(H1j∪H2j,Q) → 0,
i j
we still have the equation (2.6):
h2(W0) = h2(W0′)+Xh2(Ei)−Xh2(H1j∪H2j) = h2(W0′)+m−m = h2(W0′),
i j
the equation (2.7) becomes:
h2(W′) = 2h2(Y)+2m+1−(k+m) = 2h2(Y)+m+1−k
0
and finally the equation (2.8) becomes:
h1,1(X) = h2(W ) = h2(W )−m−1= 2h2(Y)+m+1−k−m−1= 2h2(Y)−k.
t 0
(cid:3)
3. Calabi–Yau 3-folds as coverings of Takagi’s Q-Fano 3-folds
In[Ta1]and[Ta2],H.TakagiclassifiedpossibleinvariantsofsomeQ-Fano
3-folds of Gorenstein index 2 and constructed some examples of Q-Fano 3-
folds. They (Y’s) have the following properties (Corollary 3.4 in [Ta2]):
(1) The singularities are all isolated, type of 1(1,1,1).
2
(2) The class group is: Cl(Y) = hK i.
Y
(3) The divisor −2K is very ample.
Y
(4) A general element of | − K | is a singular K3 surface with only
Y
ordinary double points at Sing(Y). Say D is such a divisor.