Let us recall here the geometry of the roof of typre 𝐴^{𝐺}

4:

𝑀

𝐹

𝑋 𝐺(2, 𝑉_{5}) 𝐺(3, 𝑉_{5}) 𝑌

𝑓_{1} 𝑓_{2}

𝑝 𝑞

(7.1.1)

The notation is the following:

◦ 𝑉_{5} is a five-dimensional vector space and 𝐹 =𝐹(2,3, 𝑉_{5}).

◦ 𝑝 and𝑞 are the natural projections from𝐹 to the two Grassman-nians.

◦ The flag variety 𝐹 has Picard group generated by the pullbacks of
the hyperplane bundles of the two Grassmannians 𝐺(2, 𝑉_{5}) and
𝐺(3, 𝑉_{5}). We denote the pullbacks ofO𝐺(2,𝑉_{5})(1) andO𝐺(3,𝑉_{5})(1)
by O (1,0) and O (0,1) respectively. In this notation 𝑀 is the
zero locus of a section 𝑠 ∈𝐻^{0}(𝐹 ,O (1,1)).

◦ One has that 𝑝_{∗}O (1,1) = Q^{∨}

2(2) and 𝑞_{∗}O (1,1) = U_{3}(2), where
we call U𝑖 the universal bundle of a Grassmannian 𝐺(𝑖, 𝑉_{5}) and
Q𝑖 its universal quotient bundle. The varieties 𝑋 and 𝑌 are,
respectively, the zero loci of the sections 𝑝_{∗}𝑠 and 𝑞_{∗}𝑠 of Q^{∨}

2(2)
and U_{3}(2),

◦ 𝑓_{1} is a fibration over 𝐺(2, 𝑉_{5}) with fiber isomorphic to P^{1}, for
points outside the subvariety 𝑋 whereas the fibers are isomorphic
toP^{2} for points on 𝑋. Similarly 𝑓_{2} is a map onto𝐺(3, 𝑉_{5}) whose
fibers are P^{1} outside𝑌 and P^{2} over𝑌.

The rest of this chapter is focused on proving that the general Calabi–

Yau pair associated to the roof of type 𝐴_{4}, in the sense of Definition
4.1.8 is non birationally equivalent.

**Lemma 7.1.1.** *Let*𝑋 *be the zero locus of a regular section*
𝑠_{2}∈ 𝐻^{0}(𝐺(2, 𝑉_{5}),Q^{∨}_{2}(2)).

*Then* 𝑠_{2} *is uniquely determined by* 𝑋 *up to scalar multiplication. *
*Simi-larly, if*𝑌 *is the zero locus of a regular section*𝑠_{3}*of*U_{3}(2)*on*𝐺(3, 𝑉_{5})*,*
𝑠_{3}*is uniquely determined by*𝑌*.*

*Proof.* We will prove the result for 𝐺(2, 𝑉_{5}), the proof for the case
of 𝐺(3, 𝑉_{5}) is identical. Let us suppose 𝑋 is the zero locus of two
sections 𝑠_{2} ande𝑠_{2}. Then, the Koszul resolutions with respect to these
two sections can be extended to the diagram:

· · · Q_{2}(−2) I𝑋 0

where the existence of the arrow 𝛽 is given by the following claim:

**Claim.** The map

Hom(Q_{2}(−2),Q_{2}(−2)) −→Hom(Q_{2}(−2),I𝑋) (7.1.3)
is surjective.

This can be verified by proving surjectivity of the following map:

𝐻^{0}(Q_{2}^{∨}⊗ Q_{2}) −→ 𝐻^{0}(Q^{∨}_{2}(2) ⊗ I𝑋) (7.1.4)
This can be achieved by tensoring the Koszul resolution ofI𝑋 byQ^{∨}

2(2).

In fact, by the identities detQ_{2}=O (1) and Q_{2}' ∧^{2}Q^{∨}

2(1) one has the exact sequence

0−→ Q_{2}^{∨}(−3) −→ Q^{∨}_{2} ⊗ Q_{2}^{∨}(−1) −→ Q^{∨}_{2} ⊗ Q_{2} −→ Q (2) ⊗ I𝑋 −→0
(7.1.5)
where by the Borel-Weil-Bott theorem one finds:

𝐻^{•}(𝐺(2, 𝑉_{5}),Q^{∨}_{2}(−3))=0,
𝐻^{•}(𝐺(2, 𝑉_{5}),Q_{2}^{∨}⊗ Q^{∨}_{2}(−1))=C[−2],

𝐻^{•}(𝐺(2, 𝑉_{5}),Q_{2}^{∨}⊗ Q_{2}) =C[0].

(7.1.6)

which proves our claim.

In particular, if two sections define the same 𝑋, then the identity of
the ideal sheaf lifts to an automorphism of Q_{2}(−2). However, since
Ext^{•}(Q_{2},Q_{2}) = C[0], the only possible automorphisms of Q_{2}(−2) are
scalar multiples of the identity. That implies that the sections differ by

multiplication with a nonzero constant.

**Corollary 7.1.2.** *Let* 𝑋 = 𝑍(𝑠_{2}) ⊂ 𝐺(2, 𝑉_{5})*. Then there exists a unique*
*hyperplane section* 𝑀 *of* 𝐹 *such that the fiber* 𝑝|^{−}^{1}

𝑀(𝑥) *is isomorphic to*
P^{2}*for*𝑥 ∈ 𝑋 *and is isomorphic to*P^{1}*for*𝑥 ∈ 𝐺(2, 𝑉_{5}) \𝑋*. Similarly for*
𝑌 = 𝑍(𝑠_{3}) ⊂ 𝐺(3, 𝑉_{5}) *there exists a unique hyperplane section*𝑀 *of* 𝐹
*such that the fiber*𝑞|^{−1}

𝑀(𝑥)*is isomorphic to*P^{2}*for*𝑥 ∈𝑌*and is isomorphic*
*to*P^{1}*for*𝑥 ∈𝐺(3, 𝑉_{5}) \𝑌*.*

*Proof.* We consider only the case 𝑋 =𝑍(𝑠_{2}) ⊂𝐺(2,5) the other being
completely analogous. Since𝐹 is the projectivization of a vector bundle
over 𝐺(2, 𝑉_{5}), then the pushforward 𝑝_{∗} defines a natural isomorphism

𝐻^{0}(𝐹 ,O (1,1))= 𝐻^{0}(𝐺(2, 𝑉_{5}),Q^{∨}_{2}(2)).

Hence𝑠_{2}= 𝑝_{∗}(𝑠)for a unique𝑠 ∈ 𝐻^{0}(𝐹 ,O (1,1)). We define𝑀 =𝑍(𝑠)
which satisfies the assertion by the discussion above. The uniqueness
of 𝑀 follows from Lemma 7.1.1. Indeed, for any hyperplane section

˜

𝑀 = 𝑍(𝑠˜), the fibers 𝑝|^{−}_{˜}^{1}

𝑀(𝑥) are isomorphic to P^{2} exactly for 𝑥 ∈
𝑍(𝑝_{∗}𝑠˜), but 𝑍(𝑝_{∗}𝑠˜) = 𝑋 only if 𝑝_{∗}𝑠˜ is proportional to 𝑠_{2} which means
that ˜𝑠 is proportional to 𝑠 and this proves uniqueness.

Let us consider an isomorphism

𝑓 :𝐺(2, 𝑉_{5}) −→𝐺(3, 𝑉_{5}). (7.1.7)
Every such isomorphism is induced by a linear isomorphism𝑇_{𝑓} :𝑉_{5} −→

𝑉^{∨}

5 in the following way:

𝑓 =𝐷◦𝜙_{2}:𝐺(2, 𝑉_{5}) −→𝐺(3, 𝑉_{5}). (7.1.8)

where 𝐷 is the canonical isomorphism

𝐷 :𝐺(𝑖, 𝑉_{5}) −→ 𝐺(5−𝑖, 𝑉^{∨}

5) (7.1.9)

and 𝜙_{𝑖} is the induced action of 𝑇_{𝑓} on the Grassmannian:

𝜙_{𝑖} :𝐺(𝑖, 𝑉_{5}) −→𝐺(𝑖, 𝑉^{∨}

5) (7.1.10)

Similarly, we consider dual maps 𝑓^{∨} : 𝐺(3, 𝑉^{∨}

5) −→ 𝐺(2, 𝑉^{∨}

5),
ex-pressed as 𝑓^{∨}=𝜙^{∨}

2 ◦𝐷^{∨}.

Note that above maps 𝑓, 𝐷, 𝜙_{2}, 𝜙_{3} are restrictions of linear maps
between the Plücker spaces of the corresponding Grassmannians. By
abuse of notation we shall use the same name for their linear extensions.

We can now introduce the following notion of duality.

**Definition 7.1.3.** *Given an isomorphism* 𝑓 : 𝐺(2, 𝑉_{5}) −→ 𝐺(3, 𝑉_{5})*, we*
*say*𝑋 ⊂ 𝐺(2, 𝑉_{5})*is* 𝑓*-dualto*𝑌 ⊂ 𝐺(2, 𝑉_{5})*if*(𝑋 , 𝑓(𝑌))*is a Calabi–Yau*
*pair associated to the roof of type* 𝐴^{𝐺}

4*, in the sense of Definition 4.1.8.*

Let us start by defining 𝑃=P(∧^{2}𝑉_{5}) ×P(∧^{2}𝑉^{∨}

5), where ∧^{2}𝑉_{5} is
identi-fied with ∧^{3}𝑉^{∨}

5 by means of 𝐷. In that case 𝐹 is a linear section of 𝑃 (in its Segre embedding) by a codimension 25 linear space.

*Remark* 7.1.4. Recall that (Wey03, Proposition 3.1.9) the equations of
𝐹 in 𝑃 are described by the following sections 𝑠_{𝑥}∗⊗𝑦 ∈𝐻^{0}(𝑃,O (1,1))

𝑠_{𝑥}∗⊗𝑦(𝛼, 𝜔) =𝜔(𝑥^{∗}) ∧𝛼∧𝑦 (7.1.11)
for 𝜔 ∈Λ^{2}𝑉^{∨}

5 = Λ^{3}𝑉_{5}, 𝛼∈Λ^{2}𝑉_{5} and for every 𝑥^{∗}⊗ 𝑦 ∈𝑉^{∨}

5 ⊗𝑉_{5}.

In other words, we have

𝑠_{𝑥}∗⊗𝑦(𝛼, 𝜔) =0 for ( [𝛼],[𝜔]) ∈𝐹(2,3, 𝑉_{5}) ⊂ P(Λ^{2}𝑉_{5}) ×P(Λ^{3}𝑉_{5}).
This defines a 25 dimensional subspace𝐻^{0}(𝑃,I𝐹(1,1)) ⊂𝐻^{0}(𝑃,O (1,1))
spanned by linearly independent sections corresponding to 𝑥^{∗} = 𝑒^{∗}

𝑖,
𝑦=𝑒_{𝑗} for 𝑖, 𝑗 ∈ {1. . .5} and a chosen basis {𝑒_{𝑖}} for 𝑉_{5}.

Now, for every 𝑓 as in (7.1.7) we define the following function:

𝑃 𝑃

(𝑥 , 𝑦) ( (𝑓^{∨})^{−1}(𝑦), 𝑓(𝑥))

𝜄𝑓

(7.1.12)

which induces the following map at the level of sections:

𝐻^{0}(𝑃,O𝑃(1,1)) 𝐻^{0}(𝑃,O𝑃(1,1))

𝑠 𝑠◦𝜄_{𝑓}

e𝜄_{𝑓}

(7.1.13)

Note that 𝜄_{𝑓} is a linear extension of an automorphism of the flag variety
𝐹 ⊂ 𝑃. It is constructed in such a way that we have that 𝑋 is defined by
a section 𝑝_{∗}(𝑠) ∈ 𝐻^{0}(𝐺(2, 𝑉_{5}),Q^{∨}

2(2)) if and only if 𝑓(𝑋) is defined
by 𝑞_{∗}(

e𝜄_{𝑓}(𝑠)) ∈𝐻^{0}(𝐺(3, 𝑉_{5}),U^{∨}

3(2)).

Our aim is to interpret 𝑓-duality in the setting above as explicitly as
possible. For that we will identify 𝐻^{0}(𝐹 ,O (1,1)) with a subspaceH𝐹

of sections in 𝐻^{0}(𝑃,O (1,1)) invariant under our transformations. The
following lemmas will be useful in the proof of non-birationality of
general Calabi–Yau pairs.

**Lemma 7.1.5.** *The space*𝐻^{0}(𝑃,O (1,1))*decomposes as*𝐻^{0}(I𝐹|𝑃(1,1))⊕

3, which is the representation space of the product of
represen-tations of weights𝜔_{2}and𝜔_{3}of𝐺 𝐿(𝑉_{5}). By the Littlewood-Richardson
rule, this space decomposes in the following way, and the
decomposi-tion is 𝐺 𝐿(𝑉_{5})-invariant:
Moreover, again by the Borel–Weil–Bott theorem, one has 𝑉_{𝜔}

2+𝜔_{3} =
𝐻^{0}(𝐹 ,O (1,1)) from which we get a surjection:

𝐻^{0}(𝑃,O (1,1)) −→𝐻^{0}(𝐹 ,O (1,1)) (7.1.15)
from which the claim follows once we set H𝐹 :=𝑉_{𝜔}

2+𝜔_{3}.

Alternatively, one can proceed in the following way: it is well known
that Aut(𝐹) ' 𝐺 𝐿(𝑉_{5})o Z/2. Moreover, the action of Aut(𝐹) on 𝐹
is linear and extends to an action of Aut(𝐹) on 𝑃 compatible withe𝜄_{𝑓}.
It follows that 𝐻^{0}(I𝐹|𝑃(1,1)) is invariant under e𝜄_{𝑓} since it is clearly
invariant under Aut(𝐹). Furthermore the dual action of Aut(𝐹) on
𝑃^{∨} preserves the dual flag variety, hence 𝐻^{0}(I𝐹^{∨}|𝑃^{∨}(1,1)) is invariant
under the dual action ofe𝜄_{𝑓}. We can define H𝐹 = 𝐻^{0}(I𝐹^{∨}|𝑃^{∨}(1,1))^{⊥}.
The latter space is invariant under Aut(𝐹), so it is also invariant under
e𝜄_{𝑓} and the map H𝐹 → 𝐻^{0}(𝐹 ,O (1,1)) defined by restriction is an

isomorphism.

Note that, by construction, the action of e𝜄_{𝑓} on 𝐻^{0}(𝐹 ,O (1,1))
cor-responds to the action e𝜄_{𝑓} on H𝐹. It means that we can think of
𝐻^{0}(𝐹 ,O (1,1)) equipped with the action induced by e𝜄_{𝑓} as a subset
of 𝐻^{0}(𝑃,O (1,1)) invariant under the action of e𝜄_{𝑓} on 𝐻^{0}(𝑃,O (1,1))
.

*Remark* 7.1.6. Note that, by applying the procedure of Remark 7.1.4
to describe the equations of the dual flag 𝐹^{∨} with respect to the dual
basis of 𝑉_{5}, one can find explicit equations defining H𝐹 in terms of
matrices in 𝐻^{0}(𝑃,O (1,1)) ' 𝑀_{10}_{×}_{10}. In particular, in our choice of
basis, Equation 7.1.11 provides explicit linear conditions on the entries
of 10×10 matrices to be elements of H𝐹. This will be useful in the
proof of Theorem 7.2.6.

**Lemma 7.1.7.** *The variety* 𝑋 *is* 𝑓*-dual to*𝑌 *if and only if there exists a*
*constant*𝜆 ∈ C^{∗}*such that sections*𝑠_{𝑋} ∈ H𝐹*,* 𝑠_{𝑌} ∈ H𝐹 *defining* 𝑋 *and*𝑌
*respectively satisfy*e𝜄_{𝑓}(𝑠_{𝑌}) =𝜆 𝑠_{𝑋}*.*

*Proof.* By definition, 𝑋 is 𝑓-dual to 𝑌 if there exists a section ˆ𝑠 ∈
𝐻^{0}(𝐹 ,O (1,1)) such that 𝑝_{∗}𝑠ˆ defines 𝑋 while 𝑞_{∗}𝑠ˆ defines 𝑓(𝑌). By
Lemma 7.1.5 there then exists a unique section 𝑠 ∈ H𝐹 such that

ˆ

𝑠 = 𝑠|𝐹. Now, by definition of e𝜄_{𝑓}, since 𝑞_{∗}𝑠 defines 𝑓(𝑌) we have
𝑝_{∗}(

e𝜄_{𝑓})^{−1}(𝑠) defines 𝑌. Furthermore by Lemma 7.1.5 we know that
(e𝜄_{𝑓})^{−1}(𝑠) ∈ H𝑓. We conclude from Lemmas 7.1.1 and 7.1.5 that up to
multiplication by constants 𝑠=𝑠_{𝑋} and (

e𝜄_{𝑓})^{−1}(𝑠) =𝑠_{𝑌}.
From now on, let us fix a basis of𝑉_{5} inducing a dual basis on 𝑉^{∨}

5, and
natural bases on ∧^{2}𝑉_{5} and ∧^{2}𝑉^{∨}

5 which are dual to each other.

A section 𝑠 ∈𝐻^{0}(𝑃,O𝑃(1,1)) is represented by a 10×10 matrix 𝑆 in
the following way

𝑠: (𝑥 , 𝑦) 𝑦^{𝑇}𝑆𝑥 (7.1.16)

where 𝑥 and 𝑦 are expansions of 𝑥 and 𝑦 in the chosen bases of ∧^{2}𝑉_{5}
and ∧^{2}𝑉^{∨}

5. Once fixed our bases, 𝜙_{2}is represented by a 10×10
invert-ible matrix 𝑀_{𝑓}, which is the second exterior power of the invertible
matrix associated to𝑇_{𝑓}.

We can now describe very explicitly the 𝑓-duality in terms of matrices using the following.

**Lemma 7.1.8.** *If*𝑆*is the matrix associated to*𝑠 ∈ 𝐻^{0}(𝑃,O𝑃(1,1))*then*
*the matrix associated to*e𝜄_{𝑓}(𝑠)*is*𝑀^{−1}

𝑓 𝑆^{𝑇}𝑀_{𝑓}*.*

*Proof.* On a pair(𝑥 , 𝑦), the map𝜄_{𝑓} acts via𝜄_{𝑓}(𝑥 , 𝑦) = ( (𝜙^{∨}

2)^{−1}(𝑦), 𝜙_{2}(𝑥)).

Furthermore, in our choice of basis 𝜙_{2}(𝑥) = 𝑀_{𝑓}𝑥 and (𝜙^{∨}

2)^{−1}(𝑦) =
(𝑀^{𝑇}

𝑓)^{−1}𝑦.
This yields:

e𝜄_{𝑓}(𝑠) (𝑥 , 𝑦)=𝑠◦𝜄_{𝑓}(𝑥 , 𝑦) =(𝑀_{𝑓}𝑥)^{𝑇}𝑆(𝑀^{𝑇}

𝑓)^{−1}𝑦 =𝑦^{𝑇}𝑀^{−1}

𝑓 𝑆^{𝑇}𝑀_{𝑓}𝑥
(7.1.17)

.

*Remark*7.1.9. In (OR17, sec. 5), it is proven that[𝑣] ∈P(𝔤𝔩(𝑉))defines
a section𝑠_{𝑣}of∧^{2}𝑉(1), whose projection to𝐻^{0}(𝐺(2, 𝑉_{5}),∧^{2}Q_{2}(2))cuts
out the threefold 𝑋_{[}_{𝑣}_{]}. Then 𝑠_{𝑣} corresponds to a 10×10 matrix 𝑆 that
we defined in (7.1.16). Hence, from Lemmas 7.1.7 and 7.1.8 follows

that 𝑋_{[}_{𝑣}_{]} and 𝑋_{[}

𝑣^{𝑇}] are 𝐷-dual. This means that our duality relation on
X_{25} between 𝑋 and𝑌 given by the condition of (𝑋 , 𝑌) being a Calabi–

Yau pair associated to the roof of type 𝐴^{𝐺}

4 is equivalent to the duality
notion defined in (OR17, Section 5), extending the duality defined on
X_{25}.