Complex Manifolds 2014; 1:45–51
Complex Manifolds
Research article
Open Access
Patrick Clarke
A proof of the birationality of certain
BHK-mirrors
Abstract: We generalize and give an elementary proof of Kelly’s refinement [9] of Shoemaker’s result [11] on
the birationality of certain BHK-mirrors. Our approach uses a construction that is equivalent to the Krawitz
generalization [10] of the duality in Berglund-Hübsch [2].
DOI 10.2478/coma-2014-0003
Received March 28, 2014; accepted June 3, 2014.
1 Introduction
We consider certain orbifold quotients of Calabi-Yau hypersurfaces in weighted projective spaces. Such a
Calabi-Yau has a mirror partner according to the Berglund-Hübsch mirror construction [2]. This construction was introduced as a generalization of Greene-Plesser construction of the mirrors of Fermat-type hypersurfaces [7]. Forming the mirror partner to an orbifold quotient of such a hypersuface is best known as
Berglund-Hübsch-Krawitz mirror symmetry since a characterization of the group used to quotient the mirror
hypersurface was given by Krawitz [10].
In [11], Shoemaker proved that BHK-mirrors of distinct Calabi-Yau orbifolds are birational provided the
original Calabi-Yau’s lie in the same weighted projective space and the group used to quotient them is the
same. More recently, Kelly was able to prove this result assuming only that the group was the same [9]; in
particular the assumption that the hypersurfaces lie in the same projective space was dropped.
The present paper proves the result of [9] (and a generalization: Theorem 3.5). The theorem is found to
be an easy consequence of some observations about Fourier transforms and transposed matrices. Our main
tool is the duality introduced in [4]; this duality is a simple construction, and includes the constructions of
[1, 3, 6, 8] as special cases.
BHK-mirror symmetry was shown in [5] to be a special case of the duality of [4] as well. However, since
[5] was never published, we include an appendix proving this.
2 Preliminaries
Matrices multiply from the left. We make the notations
W=
m Y
n
X
p
X i ij : Cn+1 → C,
j=0 i=0
P = (p ij )ij : Zm+1 → Zn+1 ,
and
(+) :
Cm+1
(Z0 , . . . , Z m )
−→
7→
C
Z0 + . . . + Z m
Patrick Clarke: Drexel University Department of Mathematics
© 2014 Patrick Clarke, licensee De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercialNoDerivs 3.0 License.
46 | Patrick Clarke
We assume that the monomials in W are distinct.
Definition 2.1. We consider abelian groups which are isomorphic to finite direct sums whose summands
are algebraic tori and/or finitely generated. For such an abelian group H we have Fourier transform and
transposition functors:
F(H) = Hom(H, C× ),
H τ = Hom(H, Z).
Homomorphisms are assumed to be algebraic (i.e. Hom(C× , C× ) = Z).
Example 2.2. W = X0p00 X1 + X1p11 X2 then

p00

P= 1
0

0

p11  .
1
F(Z) = C× and if we denote Z i : F(Z2 ) → C the map which sends χ 7→ χ(e i ), then F(P) : (C× )3X → (C× )2Z is
given by
(Z0 , Z1 ) = (X0p00 X1 , X1p11 X2 ).
Proposition 2.3.
–
–
–
–
–
For finite H, F(H) ∼
= H (non-canonically),
F is contravariant, and F ◦ F = 1,
F is exact when restricted to finitely generated groups or to affine groups, and it interchanges these two
subcategories,
F(Z`+1 ) = (C× )`+1 and so F((C× )`+1 ) = Z`+1 , and
W restricted to (C× )n+1 is (+) ◦ F(P).
Proof. The first 4 bullets are usual statements about Pontryagin duality with S1 replaced with C× , and are
easily checked. The last bullet is proved by elaborating on Example 2.2.
Proposition 2.4. If W factors through a quotient (C× )n+1 /G of (C× )n+1 by a closed subgroup G, then there is a
unique homomorphism given by the dashed arrow in Figure 1 which makes the diagram commute.
Proof. Denote by R G the Reynolds operator for the action of G on the functions on (C× )n+1 . R G is a projection
and characters are eigenfunctions for the action of G. Characters are also a basis for the space of functions, so
R G evaluated on a function is given by expanding the function in characters and erasing those which are not
invariant under G. Furthermore, a function descends to (C× )n+1 /G if and only if it is sent to itself under R G .
This means that the characters in the expansion of W must be G-invariant, and thus the map exists. Since we
have assumed the monomials are distinct, it is unique.
Corollary 2.5. Quotients of Cn+1 though which W factors correspond to factorizations
P = A ◦ Bτ
where B τ : Zm+1 → M, A : M → Zn+1 , and M = F((C× )n+1 /G). Furthermore, the function to which W descends
is
W G = (+) ◦ F(B τ ) : (C× )n+1 /G → C
Proof. Apply F to the groups in figure 1. A = F(quotient map) and B τ = F(dotted arrow).
Definition 2.6. We call ([Cn+1 /G], W G ) a quotient Landau-Ginzburg model. We assume that such an object
includes the data of the presentation: (Cn+1 , W , G).
A proof of the birationality of certain BHK-mirrors |
/G
1
/ (C× )n+1
F(P)
yr
(C× )m+1
r
r
/ (C× )n+1 /G
r
r
47
/1
WG
(+)
|
C
Fig. 1. G ≤ (C× )n+1 is a closed subgroup such that W descends to the quotient.
Definition 2.7. Given a quotient Landau-Ginzburg model ([Cn+1 /G], W G ), we define the dual quotient
Landau-Ginzburg model ([Cm+1 /G T ], W GT T ) by the data
– P T = P τ , and
– G T = ker F(B) ≤ F((Zm+1 )τ ).
This definition is illustrated in figure 2.
F(G) o
o
Zn+1
O
P
Zm+1
A
=M
{{
{
{
{{ τ
{{ B
F(G T ) o
τ o
(Zm+1
O )
Pτ
B
Mτ
w;
w
w
ww
wwA τ
w
w
(Zn+1 )τ
Fig. 2. The data of a quotient Landau-Ginzburg model and its dual side-by-side.
Remark 2.8. This is the special case of the duality in [4] where the Kähler parameter is set to 0, or equivalently
all the coefficients of the superpotential are equal to 1. The point of view in loc. cit. is that the dual is obtained
by interchanging the roles of the homomorphisms A and B.
3 The theorems of Shoemaker and Kelly
Proposition 3.1. If quotient Landau-Ginzburg models ([Cn+1 /G], W G ) and ([Cn+1 /G], W G0 ) are quotients by the
T
same group G, then W GT T equals W 0 G T on the torus F(M τ ).
Proof. Since G is the same in both cases, A = A0 in the factorization of P and P0 . The definition of the dual
T
and corollary 2.5 imply W GT T = W 0 G T = (+) ◦ F(A τ ).
Definition 3.2. Given a quotient Landau-Ginzburg model ([Cn+1 /G], W G ), a weight vector is a homomorphism q : M → Z such that qB τ (e j − e k ) = 0 for all j, k.
q
Definition 3.3. Given a weight vector q, W G = 0 defines a hypersurface Z W
,G in the quotient of the quotient
q
× n+1
×
×
[(C ) /G/C ]; where the action of C is through the homomorphism F(q)(C× ) → (C× )n+1 /G. Z W
,G is called
the associated non-linear sigma-model.
Remark 3.4. The associated non-linear sigma model defined here is likely wrong for the purposes of mirror
symmetry unless W is a Calabi-Yau polynomial (defined below). A more likely candidate, when it exists, is
given in Clarke [4].
48 | Patrick Clarke
Theorem 3.5. If quotient Landau-Ginzburg models ([Cn+1 /G], W G ) and ([Cn+1 /G], W G0 ) are quotients by the
0
0
q
same group G, then the weight vectors of ([Cm+1 /G T ], W G T ) and ([Cm +1 /G T ], W G0 T ) coincide, and Z W
T ,G T is
q
birational to Z 0T T0 for each q.
W ,G
Proof. As before, A = A0 . The set of weight vectors for the duals are those q which satisfy qA τ (e j − e k ) = 0 for
all j, k, and we know that W G T = W G0 T0 on F(M τ ) by Proposition 3.1.
Definition 3.6. W is called Calabi-Yau if P is square and invertible over Q,
(1, . . . , 1) ag(P) > 0 or (1, . . . , 1) ag(P) < 0
and
(1, . . . , 1) ag(P)(1, . . . , 1)τ = det(P).
Where ag(P) is the adjugate matrix of P.
We recover the the main theorems of [11] and [9] here.
Corollary 3.7. Consider a group G ≤ (C× )n+1 and Calabi-Yau polynomials W and W 0 which are invariant under
q0
the action of G. Then weight vectors of the duals coincide and form cyclic subgroup of Hom(M τ , Z), and Z W
T ,G T
q0
and Z 0T T0 are birational for a generator q0 .
W ,G
Proof. Again A = A0 , so the set of weight vectors is cut out by the n-equations qA τ (e0 − e j ) = 0 for j = 1, . . . , n.
These equations are linearly independent because A is invertible, so these equations are linearly independent
and the weight vectors form a group isomorphic to Z. Now apply the above results.
Theorem 3.8. Consider a finite group G ≤ (C× )n+1 and invertible polynomials W and W 0 which are invariant
q0
q0
under the action of G. Then weight vectors of the duals are cyclic, and Z W
0 are birational for a
T ,G T and Z 0T
W ,G T
generator q0 .
Proof. Write P and P0 for the matrices corresponding to W and W 0 . Because we are using a fixed G, we know
that there is a free abelian group M and homomorphisms B τ , B0τ : Zm+1 → M and A : M → Zn+1 such that
we have factorizations P = A ◦ B τ and P0 = A ◦ B0τ . This leads to the diagram
0
B
B
/ (Zm0 +1 )τ
(Zm+1 )dJτ o
MO τ
9
JJ
ttt
JJ
t
t
JJ A τ
tt 0
JJ
P
ttt P
(Zn+1 )τ
0
0
Applying F we see (C× )m+1 /G T = (C× )m +1 /G T = F(M τ ), and W T = W 0T (+) ◦ F(A0 ). So we need only show
that that the weight vector for the transposed matrices agree when pulled back to M are colinear. Consider
q T B = (1, . . . , 1) ag(P T )B and q0T B0 = (1, . . . , 1) ag(P0T )B0 . Writing this out we get
qT B
=
(1, . . . , 1) ag(A τ ) ag(B)B
=
(1, . . . , 1) ag(A τ ) det(B).
Similarly, q0T B0 = (1, . . . , 1) ag(A τ ) det(B0 ).
A BHK as Duality for toric LG models
Berglund-Hübsch [2] introduced a mirror construction for certain hypersurfaces in weighted projective spaces
given by Calabi-Yau polynomials [2]. This was generalized by Krawitz [10] where, to a group G ≤ (C× )N and
Calabi-Yau polynomial W , a dual group G† is introduced and
([(C× )n+1 /G], W G ) and ([(C× )m+1 /G† ], W GT† )
A proof of the birationality of certain BHK-mirrors
| 49
are deemed dual. The construction of [2] occurs when G = {1}. We show that when the BHK-dual is defined
(i.e. W is Calabi-Yau), it is the same as the one above.
A.1 Krawitz Duality.
The notation in this paper is a departure from that of Krawitz [10], so before proving the equivalence of the
constructions, we point out some of the main objects under consideration.
In our construction, given a subgroup H of a torus T, we applied the functor F to obtain an abelian group
F(H). Since H ≤ T, we can associate another abelian group to H; namely, those elements of the Lie algebra t
which map to H under the exponential map t → T. We will denote this subgroup by Λ H .
Definition A.1. The assignment H 7→ Λ H defines a covariant functor Λ− from the category of subgroups of
tori to the category of abelian groups.
To put ourselves in a position that is amenable to the conventions of [10], in the case T = (C× )N we denote Λ T
by CN , the exponential map will be exp(2πi−), and so Λ{1} = ZN .
In [10] the notation A W is used for our P τ , the matrix P square and invertible, and N is written for our
n + 1 and m + 1. Additionally:
– ρ1 , . . . , ρ N = the columns of (P τ )−1 ,
– ρ1 , . . . , ρ N = the rows of (P τ )−1 ,
– ρ k is also denote exp(2πiρ k ) ∈ (C× )N , where exp(2πi−) is applied entry-wise to ρ k , and
×
– similarly, ρ k is also used to denote exp(2πiρ k ) ∈ (C )N where we have put the to distinguish between
the N-tori.
For clarity, we will write g k = exp(2πiρ k ) and g k = exp(2πiρ k ) ∈ (C× )N rather than overloading the notations
ρ k and ρ k .
Krawitz expresses a subgroup G of (C× )N under whose action W is invariant in terms of the elements
{g1 , . . . , g N }. This subgroup is specified by which exponents [a1 , . . . , a N ] ∈ ZN to produce an element
g1a1 · · · g Na N ∈ G. Observe that this set of a-vectors is exactly Λ G .
The group hg1 , . . . , g N i was denoted Gmax in [10], and (tautologically) we see that the ZN in which the
a-vectors live is Λ Gmax . The other ZN is Λ{1} , and the vectors [a1 , . . . , a N ] which have g1a1 · · · g Na N = 1 ∈ (C× )N
can be thought of as the columns of P τ (because (P τ )−1 P τ = the identity matrix). This can be packaged into
the inclusions
N
ZN ∼
= Λ{1} ,→ Λ G ,→ Λ Gmax ∼
=Z
whose composition is given by multiplication on the left by P τ .
×
The dual group G† is defined to be the elements g 1r1 · · · g rNN ∈ (C )N satisfying the condition (18) of [10]:


a1
h
i
 . 

r1 , . . . , r N (P τ )−1 
 ..  ∈ Z
aN
for all a1 , · · · , a N such that g1a1 · · · g Na N ∈ G. This means
h
r1 , . . . , r N
i
is in the Z-dual space to Λ G when this
dual space is thought of as a subspace of Λ{τ 1} . So if we write Gmax for the group hg 1 , . . . , g N i, and 1 for the
×
identity element in (C )N we have identifications
Λ τGmax
,→
Λ τG
,→
Λ G†
k
Λ{1}
and the composition is given by P.
,→
Λ{τ 1}
,→
Λ Gmax .
k
k
50 | Patrick Clarke
A.2 G† = G T .
The basic observation relating the two constructions is the precise relationship between the functors F, −τ ,
and Λ.
Lemma A.2. Considered as functors on algebraic tori, there is a natural isomorphism
λ : F(−)τ ⇒ Λ{1∈(−)} .
Proof. Consider an algebraic torus T. Λ{1} is the kernel of its exponential map. In particular, we can write
T = Λ{1} ⊗Z (C/2πiZ). Given ϕ : Λ{1} → Z considered as an element of Λ{τ 1} , there is a homomorphism
χ : T → C× by the rule (k ⊗ z) 7→ ϕ(k) · z ∈ C/2πiZ ∼
= C× . This defines a isomorphism Λ{τ 1} → F(T).
τ
Transposing this gives us λ : F(−) ⇒ Λ{1∈(−)} .
Corollary A.3. Given a finite group G ≤ T of a torus, there is natural surjection
q : T → F(Λ τG )
with kernel G.
Proof. Consider the quotient map T → T/G. Since G is finite, this induces an isomorphism Λ G≤T → Λ1∈T/G .
Now transpose and apply F to Lemma A.2.
Theorem A.4. Consider a morphism W : CN → C and a group G ≤ (C× )N for which W is invariant. Assume the
matrix P is invertible, then the Krawitz dual G† equals G T .
Proof. From Corollary A.3, we have an exact sequence
×
1 → G† → (C )N → F(Λ G ) → 1
Applying F yields
Λ G → ZN → F(G† ).
Thus if we can verify that the map F(q) : Λ G → ZN equals the map B : M τ → (Zm+1 )τ from figure 2 we are
done. Since both F(q) and B factor the invertible matrix P τ , if suffices to check that A τ : (Zn+1 )τ → M τ equals
the map ZN → Λ G . By definition, A τ is obtained by applying F to the map
(C× )N → (C× )N /G.
and then transposing. On the other hand, Corollary A.3 guarantees that ZN = Λ{1} → Λ G is also obtained
this way.
Acknowledgement: The author is grateful for the encouragement of Y. Ruan and for conversations with M.
Krawitz, T. Kelly, and M. Shoemaker.
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