In this section, we prove that a general section 𝑠 ∈ 𝐻^{0}(𝐹 ,O (1,1))
gives rise to two non-isomorphic Calabi–Yau threefolds 𝑋 = 𝑍(𝑝_{∗}𝑠)
and 𝑌 = 𝑍(𝑞_{∗}𝑠), this result will be stated in Theorem 7.2.6.
Be-fore proving the theorem, we will discuss some auxiliary results. In
(BCP20), an argument to show that every 𝑋e⊂ X_{25} is contained in just
one pair of Grassmannians has been explained. Using similar ideas,
we will prove an analogous result for the boundary ¯X_{25} of the family,
namely that every Calabi–Yau threefold in ¯X_{25} is contained in just one
Grassmannian.

**Lemma 7.2.1.** *Let* 𝑋 *be a Calabi–Yau threefold described as the zero*
*locus of a section of*Q^{∨}

2(2)*. Then the following equalities hold for every*
𝑡 ≥0:

𝐻^{0}(𝐺(2, 𝑉_{5}),Q_{2}(−𝑡))=𝐻^{0}(𝑋 ,Q_{2}|𝑋(−𝑡)); (7.2.1)

𝐻^{0}(𝐺(2, 𝑉_{5}),∧^{2}Q_{2}(−𝑡))=𝐻^{0}(𝑋 ,∧^{2}Q_{2}|𝑋(−𝑡)). (7.2.2)

*In particular,*𝐻^{0}(𝑋 ,Q_{2}|𝑋) '𝑉_{5}*and*

𝐻^{0}(𝑋 ,Q_{2}|𝑋(−𝑡)) =𝐻^{0}(𝑋 ,∧^{2}Q_{2}|𝑋(−𝑡)) =0
*for*𝑡*strictly positive.*

*Proof.* Let us consider the following short exact sequence which comes
from tensoring the ideal sheaf sequence of 𝑋 with Q_{2}:

0−→ I_{𝑋/𝐺}_{(2},𝑉_{5}) ⊗ Q_{2}(−𝑡) −→ Q_{2}(−𝑡) −→ Q_{2}|𝑋(−𝑡) −→0 (7.2.3)
Given this sequence, we need to show the vanishing of the first two
degrees of cohomology for I_{𝑋/𝐺}_{(}_{2},𝑉_{5})⊗ Q_{2}. To do this, we consider the
sequence obtained tensoring with Q_{2} the Koszul resolution of the ideal
sheaf of 𝑋:

0−→ Q_{2}(−5−𝑡)−→ Q−^{𝜃} _{2}⊗ Q_{2}^{∨}(−3−𝑡) −→

−→ Q_{2}⊗ Q_{2}(−2−𝑡)−→ I−^{𝜙} _{𝑋/𝐺}_{(}_{2},𝑉_{5}) ⊗ Q_{2}(−𝑡) −→0

(7.2.4)

The bundles Q^{∨}

2(−5−𝑡) and Q_{2}⊗ Q^{∨}

2(−3−𝑡) have no cohomology in degree smaller than six: this follows from the isomorphisms

Q^{∨}_{2}(−5−𝑡) ' ∧^{2}Q_{2}^{∨}⊗ (∧^{3}Q_{2}^{∨})^{⊗(4+}^{𝑡}^{)}
Q_{2}⊗ Q^{∨}_{2}(−3−𝑡) ' (∧^{3}Q_{2}^{∨})^{⊗(}^{2}^{+𝑡)} ⊗ ∧^{2}Q^{∨}_{2} ⊗ Q^{∨}_{2}

(7.2.5)
and by a Borel–Weil–Bott computation. This, in turn, proves that
𝐻^{<}^{6}(𝐺(2, 𝑉_{5}),ker(𝜙)) = 0 in (7.2.4). Similarly, one finds that Q_{2} ⊗
Q_{2}(−2−𝑡) has no cohomology in degree smaller than four, due to Q_{2}⊗
Q_{2}(−2−𝑡) ' (∧^{2}Q^{∨}

2)^{⊗(2+𝑡)}. Therefore𝐻^{0}(𝐺(2, 𝑉_{5}),I𝑋/𝐺(2,𝑉_{5}) ⊗ Q_{2}) =0
and 𝐻^{1}(𝐺(2, 𝑉_{5}),I𝑋/𝐺(2,𝑉_{5}) ⊗ Q_{2}) = 0. This, together with (7.2.3),
proves our claim (7.2.1). The second equality follows from a totally

analogous computation, namely it involves the tensor product of the
ideal sheaf sequence with the wedge square of Q_{2}.
**Lemma 7.2.2.** *Let* 𝑋 *be a Calabi–Yau threefold described as the zero*
*locus of a section of*Q^{∨}

2(2). Then the restrictionQ^{∨}

2(2) |𝑋 *is slope-stable.*

*Proof.* Consider a subobject F ⊂ Q^{∨}

2|𝑋(2). Then, since 𝐺(2, 𝑉_{5}) has
Picard number one, we have 𝑐_{1}(F ) =O (𝑡) for some𝑡 and this leads to
the injection

0−→ O −→ ∧^{𝑟}Q^{∨}_{2}|𝑋(2𝑟−𝑡) (7.2.6)
where 𝑟 is the rank of F, which can be either one or two. To have
F as a destabilizing object for Q^{∨}

2|𝑋(2), 𝑡 must satisfy the following

On the other hand, for the injection in (7.2.6) to exist it means that

∧^{𝑟}Q^{∨}

2|𝑋(2𝑟 − 𝑡) has global sections. Let us now consider the case
𝑟 =1. ThenQ^{∨}

2|𝑋(2−𝑡) has sections only for𝑡 ≤1, but for such values
the inequality 7.2.7 cannot be satisfied. We can prove that the same
happens for 𝑟 =2: in fact,∧^{2}Q^{∨}

2|𝑋(4−𝑡) ' Q_{2}(3−𝑡) has sections only
for 𝑡 ≤ 3, but the inequality 7.2.7 cannot be fulfilled with these values

of 𝑡.

Let us suppose 𝑋 is contained in two Grassmannians𝐺_{1}and𝐺_{2}, where
the latter is the image of the former under an isomorphism of P^{9}. Since
both the restrictions of the normal bundlesN𝑖|𝑋 =N_{𝐺}

𝑖/P^{9}|𝑋 =Q^{∨}

2𝑖(2) |𝑋

are stable with the same slope, every morphism between them must

be either zero or an isomorphism. Below we furthermore prove that the isomorphism class of the normal bundle determines the Grassman-nian. Combining these two facts will give us the uniqueness of the Grassmannian containing 𝑋.

**Lemma 7.2.3.** *Let* 𝑋 *be a Calabi–Yau threefold described as the zero*
*locus of a section of*Q^{∨}

2(2). Then the following isomorphism holds:

𝐻^{0}(P^{9},O (1)) ' 𝐻^{0}(𝑋 ,O𝑋(1)) (7.2.8)
*Proof.* The claim follows by proving separately the following claims:

𝐻^{0}(P^{9},O

P^{9}(1)) ' 𝐻^{0}(𝐺(2, 𝑉_{5}),O𝐺(2,𝑉_{5})(1)) (7.2.9)
𝐻^{0}(G(2, 𝑉_{5}),O𝐺(2,𝑉_{5})(1)) ' 𝐻^{0}(𝑋 ,O𝑋(1)) (7.2.10)
Let us begin by verifying Equation 7.2.10. A twist of the Koszul
resolution of 𝑋 ⊂𝐺(2, 𝑉_{5}) yields:

0−→ O (−4) −→ ∧^{2}Q_{2}(−3) −→ Q_{2}(−1) −→ O (1) −→𝑖_{∗}O𝑋(1) −→0
(7.2.11)
where𝑖 is the embedding of 𝑋 in𝐺(2, 𝑉_{5}). The desired isomorphism is
a consequence of the vanishing of cohomology of the first three bundles.

To prove the validity of Equation 7.2.9 we follow basically the same
argument applied to the Pfaffian resolution of 𝐺(2, 𝑉_{5}), yielding the
following exact sequence on P^{9}:

0−→ O (−4) −→𝑉_{5}⊗ O (−2) −→𝑉_{5}⊗ O (−1) −→

−→ OP^{9}(1) −→ 𝑗_{∗}O𝐺(2,𝑉_{5})(1) −→0

(7.2.12)
where 𝑗 is the embedding of 𝐺(2, 𝑉_{5}) in P^{9}. Again the first three
bundles have no cohomology. For both computations, the vanishings

can be computed by the Borel–Weil–Bott theorem (or, in the first case,

by Lemma 7.2.1).

**Lemma 7.2.4.** *Consider a Calabi–Yau threefold* 𝑋 ∈ X_{25}*such that* 𝑋 *is*
*contained in two translates*𝐺_{1}, 𝐺_{2} *of*𝐺(2, 𝑉_{5}) *in*P^{9}*. If*N_{𝐺}

1|P(∧^{2}𝑉_{5})|𝑋 '
N_{𝐺}

2|P(∧^{2}𝑉_{5})|𝑋*, then*𝐺_{1} =𝐺_{2}*.*

*Proof.* Let E be a globally generated rank three vector bundle on 𝑋
such that𝐻^{0}(𝑋 ,E) =𝑉_{5}. Then it defines a unique morphism 𝑓 : 𝑋 −→

The proof is concluded by observing that sinceN_{𝐺}

1|P(∧^{2}𝑉_{5})|𝑋 ' N_{𝐺}

2|P(∧^{2}𝑉_{5})|𝑋,

one has 𝑓_{1}= 𝑓_{2}, and hence 𝐺_{1}=𝐺_{2}.

**Corollary 7.2.5.** *If* 𝑋 ⊂ P^{9} *is a Calabi–Yau threefold from the family*

X¯_{25}*, then* 𝑋 *is contained as a zero locus of a vector bundle in a unique*
*Grassmannian*𝐺(2,5)*in its Plücker embedding.*

*Proof.* Suppose that 𝑋 is contained in two Grassmannians𝐺_{1}, 𝐺_{2}such
that for each of them we have an exact sequence:

0→ N𝑋|𝐺_{𝑖} → N_{𝑋}_{|}
isomorphism it induces an isomorphism N_{𝐺}

1|P^{9}|𝑋 ' N_{𝐺}

2|P^{9}|𝑋 and we
conclude by Lemma 7.2.4. If it is trivial it lifts to an isomorphism
N_{𝑋|𝐺}_{1} ' N𝑋|𝐺_{2} which again gives an isomorphism N_{𝐺}

1|P^{9}|𝑋 ' N_{𝐺}

2|P^{9}|𝑋

and permits us to conclude again by Lemma 7.2.4.

Now we are ready to prove the main theorem of this chapter.

**Theorem 7.2.6.** *Let*𝐹 *be the partial flag manifold*𝐹(2,3, 𝑉_{5}), let 𝑝*and*

4. Because of Lemma 7.1.1, we deduce that if there
ex-ists an isomorphism mapping 𝑋 to 𝑌, then it is given by a map
𝑓 : 𝐺(2, 𝑉_{5}) → 𝐺(3, 𝑉_{5}). Recall that such a map is determined by
a linear isomorphism from𝑇_{𝑓} :𝑉_{5}→𝑉^{∨}

5.

Thus, because of Corollary 7.2.5, 𝑋 and 𝑌 are isomorphic only if
there exists 𝑓 :𝐺(2, 𝑉_{5}) →𝐺(3, 𝑉_{5}) such that 𝑋 is 𝑓-dual to 𝑋. This,
by Corollary 7.1.8 translates to the fact that a section 𝑠_{𝑋} ∈ H𝐹 from
Lemma 7.1.5 defining 𝑋 on 𝐹 satisfies 𝑀^{−1}

𝑓 𝑆^{𝑇}𝑀_{𝑓} = 𝜆𝑆 for 𝑆 being
the matrix associated to the section 𝑠_{𝑋} and some constant𝜆. But since
𝑆 and 𝑆^{𝑇} are similar matrices then multiplication by 𝜆 must then
pre-serve the spectrum of 𝑆. The proof amounts now to find a matrix 𝑆
corresponding to an element of H𝐹 with spectrum that is not fixed by
multiplication with 𝜆≠ 1 and such that the equation

𝑆^{𝑇}𝑀− 𝑀 𝑆=0

has no solutions among matrices 𝑀 of the form 𝑀 = ∧^{2}𝑇, and then
expand by openness to the general element ofH𝐹. This is done via the
following script in Macaulay2 (GS19):

R=QQ[a_1..a_25]

S=matrix{

{ 1 ,0,0,0,0,0,0,0,0,0}, {0, 2 ,0,0,0,0,0,0,0,0}, {0,0, 0 ,0,0,0,0,0,0,0}, {0,0,0, 0 ,0,0,0,0,0,0}, {0,1,0,0, 0 ,0,0,0,0,0}, {0,0,0,0,0, 1 ,0,0,0,0}, {0,0,0,0,0,0,-1 ,0,0,0}, {0,0,0,0,0,0,0,-1 ,0,0}, {0,0,0,0,0,0,0,0,-1 ,0},

{0,0,0,0,0,0,0,0,0,-1 }}

T=genericMatrix(R,5,5) M=exteriorPower(2,T)

Sol=ideal flatten(transpose(S)*M-M*S) saturate(Sol, ideal det T)

Here we chose a matrix 𝑆 satisfying the equations defining H𝐹 =
𝐻^{0}(𝐼_{𝐹}∨|𝑃^{∨})^{⊥} as in Remark 7.1.6.

This implies that a general hyperplane section 𝑠 of the flag variety 𝐹 yields two Calabi–Yau threefolds 𝑋 and 𝑌 which are dual, but not projectively isomorphic. By the fact that the studied manifolds have Picard number one we conclude that they are not birational (OR17,

proof of Theorem 4.1).

The proof above being very explicit has the advantage that it permits to construct concrete examples of pairs of Calabi–Yau varieties in our family which are dual but not birational. We can however perform a more conceptual proof, which is more suitable to generalization and allows to estimate the expected codimension of the fixed locus of our duality.