While the problem of describing and classifying fibrations of roofs over
a smooth projective variety has been addressed in (Kan18; ORS20), we
focus on a special class of such objects, which we call *homogeneous*
*roof bundles: they provide a natural relativization of homogeneous*
roofs, while keeping many of the properties of the latter objects in a
relative setting.

**Definition 4.4.1.** *Let* 𝐺 *be a semisimple Lie group and* 𝑃 *a parabolic*
*subgroup such that* 𝐺/𝑃 *is a homogeneous roof. Let* V *be a principal*
𝐺*-bundle over a smooth projective variety* 𝐵*. We define a*homogeneous
roof bundle*over*𝐵*the variety*V ×^{𝐺} 𝐺/𝑃*.*

We get this diagram:

V ×^{𝐺} 𝐺/𝑃

V ×^{𝐺} 𝐺/𝑃_{1} V ×^{𝐺} 𝐺/𝑃_{2}

𝑝_{1} 𝑝_{2}

(4.4.1)

*Remark*4.4.2. Note that V ×^{𝐺} 𝐺/𝑃 is a locally trivial fibration over 𝐵
with fiber 𝐹_{𝑏} '𝐺/𝑃.

**Lemma 4.4.3.** *Let*𝐺*be a semisimple Lie group and*𝑃 ⊂ 𝐺 *a parabolic*
*subgroup such that* 𝐺/𝑃 *is a homogeneous roof with projective bundle*
*structures* ℎ_{𝑖} : 𝐺/𝑃 ' P(𝐸_{𝑖}) −→ 𝐺/𝑃_{𝑖} *for*𝑖 ∈ {1; 2}*. Let*V −→ 𝐵 *be*
*a principal*𝐺*-bundle over a smooth projective variety*𝐵*. Then there are*
*vector bundles*E𝑖*such that the homogeneous roof bundle*Z=V ×^{𝐺}𝐺/𝑃

*admits projective bundle structures* 𝑝_{𝑖}:Z 'P(E𝑖) −→ Z𝑖*for*𝑖 ∈ {1; 2}
*such that the following diagram is commutative:*

Z

Z_{1} Z_{2}

𝐵

𝑝_{1} 𝑝_{2}

𝑟_{1} 𝑟_{2}

(4.4.2)

*where*𝑟_{1}*and*𝑟_{2}*are smooth extremal contractions and*Z𝑖 :=V ×^{𝐺}𝐺/𝑃_{𝑖}*.*
*Moreover, there exists a line bundle*L*on*Z*such that*L*restricts to*O (1)
*on the fibers of both*𝑝_{1}*and* 𝑝_{2}*, and such that*𝑝_{𝑖}_{∗}L ' E𝑖*.*

*Proof.* Let us call 𝜋 :Z −→ 𝐵 the map induced by the structure map
V −→ 𝐵. Then, for every 𝑏 ∈ 𝐵 we have 𝜋^{−1}(𝑏) ' 𝐺/𝑃. We obtain
the following diagram:

Z

Z_{1} Z_{2}

𝐵

𝑝_{1} 𝑝_{2}

𝜋

𝑟_{1} 𝑟_{2}

(4.4.3)

where 𝑝_{1} and 𝑝_{2}, restricted to the preimage of a point 𝑏 ∈ 𝐵, are
the P^{𝑟−1}-bundle structures of the roof 𝐺/𝑃, therefore they are P^{𝑟−1}
-fibrations over Z_{1} andZ_{2}.

For each homogeneous roof of the list (Kan18, Section 5.2.1), there exist

homogeneous vector bundles 𝐸_{1} and 𝐸_{2} such that P(𝐸_{1}) ' P(𝐸_{2}) '
𝐺/𝑃. Hence, for 𝑖 =1,2, they have the form:

𝐸_{𝑖} =𝐺×^{𝑃}^{𝑖} 𝑉_{𝑖} (4.4.4)

for a given representation space𝑉_{𝑖}. From the data of 𝐸_{𝑖} we can define
vector bundles on Z𝑖 with the following construction:

E𝑖 :=V ×^{𝐺} 𝐺×^{𝑃}^{𝑖}𝑉_{𝑖} (4.4.5)
Note that for every 𝑏 ∈ 𝐵, we have 𝑟^{−}^{1}

𝑖 (𝑏) ' 𝐺/𝑃_{𝑖} and E𝑖|_{𝑟}−1

𝑖 (𝑏) ' 𝐸_{𝑖}.
Since 𝐺/𝑃 is a roof, this implies that (𝑟^{−}^{1}

𝑖 (𝑏),E𝑖|_{𝑟}−1

𝑖 (𝑏)) is a simple Mukai pair.

The line bundle L can be constructed in the following way: let us consider the line bundle 𝐿 on 𝐺/𝑃 introduced in Proposition 4.1.4.

Since 𝐿 is homogeneous, there exists a one-dimensional representation
space𝑊 such that 𝐿 =𝐺×^{𝑃}𝑊. Let us fix:

L :=V ×^{𝐺} 𝐺×^{𝑃}𝑊 . (4.4.6)

Such bundle restricts to 𝐿 on every fiber of𝜋, hence it restricts toO (1)
on the fibers of both 𝑝_{1} and 𝑝_{2}.

In order to prove the isomorphisms Z 'P(E_{1}) 'P(E_{2}) we proceed in
the following way. Recall that, given a principal 𝐻-bundle W over a
variety 𝑋 there is the following exact functor (see (Nor82, Section 2.2),
or the survey (Bal06, Page 8)):

𝐻−Mod ^{W×} Vect(𝑋)

𝐻(−)

(4.4.7)

which sends the 𝐻-module 𝑅 to the vector bundle W ×^{𝐻} 𝑅 over 𝑋.
Observe now that V ×^{𝐺} 𝐺 is a principal 𝑃-bundle over Z, because
V −→ V/𝑃 is a principal 𝑃-bundle and V/𝑃 ' V ×^{𝐺} 𝐺/𝑃' Z (see
for example (Mit01, Proposition 3.5)). This allows to construct an exact
functor:

𝑃−Mod Vect(Z).

V×^{𝐺}𝐺×^{𝑃}(−)

(4.4.8) Let us now recall that there exists an equivalence of categories

Rep(𝑃) ^{𝐹} Vect^{𝑃}(𝐺/𝑃) (4.4.9)
which sends the𝑃-module𝐻 to the homogeneous vector bundle𝐺×^{𝑃}𝐻,
hence 𝐹^{−1} is an exact functor. Summing all up we can construct an
exact functor sending homogeneous vector bundles over 𝐺/𝑃 to vector
bundles over Z:

Vect^{𝑃}(𝐺/𝑃) Vect(Z)

𝑃−Mod

V×^{𝐺}𝐺×^{𝑃}(−)◦𝐹^{−1}

𝐹^{−1} V×^{𝐺}𝐺×^{𝑃}(−)

(4.4.10)

By applying this functor to the following surjection (given by the
rela-tive Euler sequence of the projecrela-tive bundle structure ℎ_{𝑖})

ℎ^{∗}

𝑖𝐸_{𝑖} −→ 𝐿 −→0 (4.4.11)

we get the following surjective map (compatibility with pullback follows from (Mit01, Proposition 3.6):

𝑝^{∗}

𝑖E −→ L −→ 0 (4.4.12)

which determines an isomorphism Z −→ P(E) by (Har77, Ch. II.7,
Proposition 7.12), and thus 𝑝_{𝑖}_{∗}L ' E by (Har77, Ch. II.7, Proposition

7.11).

Let us now prove that 𝑟_{1} and 𝑟_{2} are contractions of extremal rays.

Observe that 𝑟_{𝑖} is locally projective (hence proper) and every fiber is
isomorphic to a Fano variety of Picard number one𝐺/𝑃_{𝑖}, which means
that −𝐾_{Z}

𝑖 is 𝑟_{𝑖}-ample because −𝐾_{Z}

𝑖|_{𝑟}−1

𝑖 (𝑏) is ample for every 𝑏 ∈ 𝐵
(Laz04a, Theorem 1.7.8). This proves that 𝑟_{𝑖} is a Fano–Mori
con-traction (Occ99, Definition I.2.2). To prove that 𝑟_{𝑖} is a contraction of
extremal ray (or elementary Fano–Mori contraction) we just need to
show that Pic(Z𝑖)/Pic(𝐵) ' Z (Occ99, Definition I.2.2). By (FI73,
Proposition 2.3), since Z𝑖 is a locally trivial𝐺/𝑃_{𝑖}-fibration we have an
exact sequence:

𝐻^{0}(𝐺/𝑃_{𝑖},O^{∗})/C^{∗} −→Pic(𝐵) −→Pic(Z𝑖) −→Pic(𝐺/𝑃_{𝑖}) −→0
(4.4.13)
and the first term vanishes because of the long cohomology sequence
associated to the exponential sequence for 𝐺/𝑃_{𝑖}:

0−→ 𝐻^{0}(𝐺/𝑃_{𝑖},Z) −→ 𝐻^{0}(𝐺/𝑃_{𝑖},O) −→

−→ 𝐻^{0}(𝐺/𝑃_{𝑖},O^{∗}) −→𝐻^{1}(𝐺/𝑃_{𝑖},Z) −→ · · ·

(4.4.14)

where we observe that 𝐻^{1}(𝐺/𝑃_{𝑖},Z) vanishes since the integral
coho-mology of rational homogeneous varieties is generated by their Bruhat
decompositon, and it is nonzero only in even degree. Our claim is
proven once we recall that by Definition 4.1.2 one has Pic(𝐺/𝑃_{𝑖}) ' Z.
*Remark*4.4.4. Note that, for every 𝑏 ∈ 𝐵, we have 𝑟^{−}^{1}

𝑖 (𝑏) '𝐺/𝑃_{𝑖} and

E𝑖|_{𝑟}−1

𝑖 (𝑏) ' 𝐸_{𝑖}. Since 𝐺/𝑃 is a roof, this implies that (𝑟^{−}^{1}

𝑖 (𝑏),E𝑖|_{𝑟}−1
𝑖 (𝑏))
is a simple Mukai pair.

**4.4.1** **Calabi–Yau fibrations**

One of our main interests is to investigate the zero loci of pairs of
sec-tions of E_{1} andE_{2}which are pushforwards of a section Σ ∈𝐻^{0}(Z,L),
hence relativizing the setting of Definition 4.1.8. Let us make this
clearer by the following lemma, the notation is established in Diagram
4.4.3.

**Lemma 4.4.5.** *Let*Z*be a homogeneous roof bundle of type*𝐺/𝑃 ; P^{𝑛}×
P^{𝑛}*over a smooth projective variety*𝐵*and fix*ℎ_{𝑖} :𝐺/𝑃' P(𝐸_{𝑖}) −→𝐺/𝑃_{𝑖}
*for*𝑖 ∈ {1; 2}*. Suppose there exists a basepoint-free vector bundle* L *on*
Z *such that for every* 𝑏 ∈ 𝐵 *one has* L |_{𝜋}−1(𝑏) ' 𝐿 *and the restriction*
*map*𝐻^{0}(Z,L) −→ 𝐻^{0}(𝜋^{−1}(𝑏), 𝐿)*is surjective. Given a general section*
Σ ∈𝐻^{0}(Z,L), let us call 𝑋_{𝑖} :=𝑍(𝑝_{𝑖}_{∗}Σ). Then there exist fibrations:

𝑋_{1} 𝑋_{2}

𝐵

𝑓_{1} 𝑓_{2}

(4.4.15)

*such that for a general*𝑏 ∈ 𝐵*the varieties*𝑌_{1}:= 𝑓^{−1}

1 (𝑏)*and*𝑌_{2} := 𝑓^{−1}

2 (𝑏)
*are a Calabi–Yau pair associated to the roof*𝐺/𝑃*in the sense of Definition*
4.1.8.

*Proof.* Since 𝑝_{𝑖}_{∗}L = E𝑖, 𝑋_{𝑖} ⊂ Z𝑖 is the zero locus of a section 𝑝_{𝑖}_{∗}Σ
of E𝑖. Let us call 𝑓_{𝑖} := 𝑟_{𝑖}|𝑋𝑖. By the condition E𝑖|_{𝑟}−1

𝑖 (𝑏) ' 𝐸_{𝑖} and

𝑟^{−}^{1}

𝑖 (𝑏) ' 𝐺/𝑃_{𝑖} it follows that (𝑟^{−}^{1}

𝑖 (𝑏),E𝑖|_{𝑟}−1

𝑖 (𝑏)) is a Mukai pair. If 𝑏
andΣ are general the varieties𝑌_{𝑖} = 𝑍(𝑝_{𝑖}_{∗}Σ|_{𝑟}−1

𝑖 (𝑏)) ⊂𝑟^{−}^{1}

𝑖 (𝑏) are Calabi–

Yau by Lemma 2.5.3 and the fact that the general Σ has smooth zero
locus. Moreover, 𝐸_{𝑖} ' ℎ_{𝑖}_{∗}𝐿 and the varieties 𝑌_{1} and 𝑌_{2} are the zero
loci of the pushforwards of the same section Σ_{𝜋}−1(𝑏), therefore they are
a Calabi–Yau pair associated to the roof of type 𝐺/𝑃 as in Definition
4.1.8.