In the following section we will describe the only non homogeneous roof present in the list of (Kan18) , and we will describe the 𝐾3 pairs arising by such construction.

**4.2.1** **Homogeneous vector bundles on the five ** **dimen-sional quadric**

Let 𝑄 ⊂ P^{6} be a quadric hypersurface of dimension five, let S be its
rank 4 spinor bundle. Ottaviani constructed a 7-dimensional moduli
space of rank 3 bundles G such that

0−→ O −→ S^{∨} −→ G −→0 (4.2.1)

More precisely, there exists a moduli space isomorphic to P^{7}\𝑄_{6} of
rank 3 vector bundles G with Chern class 𝑐(G) = (2,2,2), and those
bundles are the ones satisfying Equation 4.2.1 (Ott88, Theorem 3.2).

By the Borel–Weil–Bott theorem and the sequence 4.2.1, one proves
that dim𝐻^{0}(𝑄 ,G (1)) = 41 and by the above we have 𝑐(G (1)) =
(5,9,12). Hence, a section 𝑠 in such 41-dimensional vector space
defines a 𝐾3 surface of degree 12 in P^{6}.

LetOdenote the complexified Cayley octonions (for details see (Kan19, Definition 2.4) and the source therein). It is known by (Kan19, Theorem 2.6) that the projectivization of the Ottaviani bundle can be described in the following way:

P(G^{∨}) ={(𝑥 , 𝑦) ∈P(ImO^{∨}) ×P(ImO^{∨}) |𝑥·𝑥 =𝑦·𝑦 =𝑥·𝑦 =0}:= 𝑋
(4.2.2)
This variety has two natural projections to the quadrics 𝑄 = {𝑥 ∈
P(ImO^{∨}) |𝑥·𝑥 = 0} and 𝑄e = {𝑦 ∈ P(ImO^{∨}) |𝑦· 𝑦 = 0} leading to the

following diagram:

𝑋

𝑌 𝑄 𝑄e 𝑌e

𝜋 e𝜋

(4.2.3)

where both𝑌 and𝑌eare 𝐾3 surfaces described as zero loci of (twisted)
Ottaviani bundles G (1) and G (1), ande P(G^{∨}(−1)) ' P(Ge^{∨}(−1)) '
𝑋.

*Remark*4.2.1. Diagram 4.2.3 appears as the roof of type 𝐺^{†}

2 in the list
of (Kan18, Section 5.2.1), and it is the only non homogeneous example
of such construction. In fact,the Fano 7-fold 𝑋, in contrast with the
other examples, is not a generalized flag. However, if we consider
the surjection from the 𝐺_{2} flag to the five dimensional quadric, we
obtain the projectivization of a rank two vector bundle, which admits a
second projective bundle structure alongside with a surjection to the𝐺_{2}
-Grassmannian. This construction yields the roof of type 𝐺_{2} studied in
(IMOU19; Kuz18), which gives derived equivalent but non isomorphic
Calabi–Yau threefolds.

**4.2.2** **The roof of type** 𝐷

_{4}

**and its degeneration**

Recall the homogeneous roof of type 𝐷_{4}.
𝑋_{𝐷}

4

𝑌 𝑄_{6} 𝑄e_{6} 𝑌e

𝜋 e𝜋

(4.2.4)

Here 𝑄_{6} and 𝑄e_{6} are six-dimensional quadrics representing spinor
va-rieties 𝑂 𝐺(4,8)_{±} and 𝑋_{𝐷}

4 = 𝑂 𝐺(3,8) = P^{𝑄}6(S^{∨}(1)) = P𝑄_{e}_{6}(Se^{∨}(1))
where S and Se are the spinor bundles respectively on 𝑄_{6} and 𝑄e_{6}.
Note that 𝑂 𝐺(3,8) admits two different projective bundle structures
given by the maps 𝜋 and e𝜋, which can be interpreted as the projections
determined by the embeddings of the parabolic subgroup of 𝑂 𝐺(3,8)
inside the parabolic subgroups defining 𝑂 𝐺(4,8)±.

There exists the following short exact sequence on 𝑄_{6} (Ott88, Section
3), whose restriction to 𝑄_{5} is Equation 4.2.1:

0−→ O (1) −→ S^{∨}(1) −→ G (1) −→0. (4.2.5)
Note that when the Mukai pair moves in a moduli we get a family of
roofs. In this way one can also obtain degenerations of roofs which
involve bundles which are not necessarily stable. This is the case for
instance in the following context.

**Degeneration of roofs**

Considering the family of extensions between O (1) and G (1) we see
the trivial extension O (1) ⊕ G (1) as a degeneration of S^{∨}(1). It
follows that 𝑋_{𝐷}

4 admits a degeneration to ˆ𝑋_{𝐷}

4 = P^{𝑄}6(O (1) ⊕ G (1)).

The latter variety is not a roof, but it admits a natural surjection to𝑄_{6}.
A general hyperplane section of ˆ𝑋_{𝐷}

4 now gives rise to a 𝐾3 surface obtained as a zero locus of a section of O (1) ⊕ G (1) on such quadric.

In consequence the 𝐾3 is given as the zero locus of the restriction
of the corresponding section of G (1) to a five-dimensional quadric
𝑄_{5} obtained as a hyperplane section of 𝑄_{6}. If we now consider the

restriction of G (1) to the zero locus of a section of O (1) we obtain
the roof 𝐺^{†}

2. The latter roof is a subvariety of some degeneration of
the roof of type 𝐷_{4}. Moreover, the 𝐾3 surfaces associated to this roof
are degenerations of 𝐾3 surfaces associated to 𝑋_{𝐷}

4. We however see in Subsection 4.2.3 that a general𝐾3 surface of degree 12 appears also in the degenerate description.

*Remark* 4.2.2. Note that we can further degenerate the 𝐺^{†}

2 roof using the exact sequence:

0−→ C (2) −→ G (1) −→ O (2) −→0. (4.2.6)
where C is the Cayley bundle on 𝑄_{5}. The zero locus of C (2) ⊕ O (2)
is the intersection of a del Pezzo threefold of degree 6 with a quadric.

We can then consider the restriction of C (2) to the zero locus of a section of O (2) which is just a complete intersection of two quadrics.

This is however not a roof as it does not appear in the classification of (Kan18, Theorem 5.12). The 𝐾3 surfaces obtained in this way are also not general 𝐾3 surfaces of degree 12 as their Picard number is ≥ 2.

**4.2.3** **Completeness of the family of K3 varieties of type** 𝐺

^{†}

2

In the remainder of this section we prove that the family of 𝐾3 surfaces described as sections of an Ottaviani bundle G (1) represents a dense open subset of the family of polarized 𝐾3 surfaces of degree 12. In particular the general element of this family has Picard number one.

We then prove that pairs of 𝐾3 surfaces associated to the roof 𝐺^{†}

2 are in general not isomorphic.

It is well-known (Muk87, Corollary 0.3) that a polarized 𝐾3 surfaces
of degree 2𝑔−2 has an embedding (defined by its polarization) in the
projective space P^{𝑔}. If we can prove that our degree 12 𝐾3 surfaces in
P^{6} form a 26-dimensional family up to automorphisms of P^{6}, then our
family can be recovered by the complete 19-dimensional family in P^{7}
by means of a projection from one point. Since the general element of
a complete family of 𝐾3 surfaces has Picard number one, we conclude
that the same holds for our family.

**Lemma 4.2.3.** *Let*𝑄 ⊂ P^{6} *be a five dimensional quadric hypersurface*
*and*G*an Ottaviani bundle on*𝑄*. If*𝑌 = 𝑍(𝑠)*for* 𝑠∈ 𝐻^{0}(𝑄 ,G (1))*, then*
𝑌 *determines the bundle*G*and the section* 𝑠*up to scalar multiplication.*

*Proof.* Let us consider a 𝐾3 surface 𝑌 ⊂ 𝑄 and let G and Ge be
two Ottaviani bundles on 𝑄, such that there exist two sections 𝑠 ∈
𝐻^{0}(𝑄 ,G (1)) and e𝑠 ∈ 𝐻^{0}(𝑄 ,G (1))e with 𝑌 = 𝑍(𝑠) = 𝑍(e𝑠). Then we
have the following diagram:

· · · G^{∨}(−1) I𝑌|𝑄 0

· · · Ge^{∨}(−1) I𝑌|𝑄 0

𝛼_{𝑠}
𝛽

𝛼

e 𝑠

(4.2.7)

where I𝑌|𝑄 is the ideal sheaf of𝑌 ⊂ 𝑄 and the rows are given by the Koszul resolutions ofI𝑌 with respect to the two sections. The existence of 𝛽 is a consequence of the following claim.

**Claim.** *The following map is surjective:*

Hom(G^{∨}(−1),Ge^{∨}(−1)) −→Hom(G^{∨}(−1),I𝑌|𝑄)

Proving the claim is equivalent to show that

𝑓 : 𝐻^{0}(𝑄 ,G ⊗ G^{∨}) −→𝐻^{0}(𝑄 ,G (1) ⊗ I𝑌|𝑄) (4.2.8)
is surjective. To this purpose, we compute the tensor product of G (1)
with the Koszul resolution of I𝑌 with respect to e𝑠. Using detGe^{∨} =
O (−2) and ∧^{2}Ge^{∨}' G (−2)e we find:

0−→ G (−4) −→ G ⊗G (−3) −→ G ⊗e Ge^{∨}−→ G (1) ⊗ I𝑌|𝑄 −→ 0
(4.2.9)
Cohomology can be computed with the Borel–Weil–Bott theorem: in
fact, using the sequence 4.2.1, we can resolve all the bundles of 4.2.9
in terms of twists of tensor products of S and its dual. In particular,
the leftmost term of 4.2.9 is acyclic beacuse both O (−4) and S^{∨}(−4)
are, while the cohomology of the second one follows from the diagram:

O (−3) S^{∨}(−3)

S^{∨}(−3) S^{∨}⊗ S^{∨}(−3)

G (−3) S^{∨}⊗ G (−3) G ⊗G (−e 3)

(4.2.10)

In the first two rows, the only term which is not acyclic isS^{∨}⊗S^{∨}(−3) =

∧^{2}S^{∨}(−3) ⊕ Sym^{2}S^{∨}(−3): the first summand has no cohomology,
while the second one has cohomology C[−2]. Let us call K the
cokernel of G (−4) −→ G ⊗G (−e 3). Then one has 𝐻^{•}(𝑄 ,K) =C[−2],
and by the sequence

0−→ K −→ G ⊗Ge^{∨}−→ G (1) ⊗ I𝑌|𝑄 −→ 0 (4.2.11)

we conclude that 𝑓 is surjective, thus proving the claim.

Since Ottaviani bundles are stable (Ott88, Theorem 3.2), the map 𝛽
can be either zero or an isomorphism, so we deduce that 𝑠 and e𝑠
must be sections of isomorphic Ottaviani bundles. Hence, the proof is
completed by observing that Hom(G,G) =C.
**Lemma 4.2.4.** *Let*𝑌 ⊂ 𝑄 *be a* 𝐾3*surface satisfying the hypotheses of*
*Lemma*4.2.3. Then𝑌 *is contained in a unique quadric in*P^{6}*.*

*Proof.* The proof follows from observing that ℎ^{0}(𝑄 ,I𝑌|𝑄(2)) = 0. By
the Koszul resolution of I𝑌|𝑄 and the relation G^{∨} ' ∧^{2}G (−2) we find
the following exact sequence:

0−→ O (−3) −→ G (−2) −→ G^{∨}(1) −→ I𝑌|𝑄(2) −→0

and the desired result is obtained by an application of the Borel–Weil–

Bott theorem. In fact, as in the proof of Lemma 4.2.3, one can resolve the first three bundles in terms of twists of O, S and its dual.

**Proposition 4.2.5.** *The general* 𝐾3*surface described as a zero locus of*
*a section of* G (1), whereG *is an Ottaviani bundle, has Picard number*
*one.*

*Proof.* The space of sections of an Ottaviani bundle has dimension
41, and the moduli space of Ottaviani bundles on 𝑄 is 7-dimensional.

Since the action of Aut𝑄 =Spin(7) is transitive on the moduli space of Ottaviani bundles, and a 𝐾3 surface 𝑌 ⊂ 𝑄 determines the section,

the (projective) dimension of the family is given by:

40−21+7=26

where 21−7 is the dimension of the space of automorphisms of 𝑄
fixing an Ottaviani bundle. Hence, the family we are describing is
a 26 dimensional family (of classes up to automorphisms of P^{6}) of
embedded𝐾3 surfaces of degree 12 inP^{6}. Since each 𝐾3 of degree 12
has a projective embedding inP^{7} a complete family of𝐾3 of degree 12
inP^{6} can be described by a 19+7=26-parameter space, via projection
from a point in P^{7}. This proves that our family is complete, therefore

the general element has Picard number one.