In the following we will mostly focus on left mutations, since given
an admissible subcategory 𝔄 ⊂ ℭ of a triangulated category ℭ, the
functors L𝔄 :^{⊥} 𝔄 −→ 𝔄^{⊥} and R𝔄 : 𝔄^{⊥} −→^{⊥} 𝔄 are mutually inverses
(see, for example, (Kuz10, Lemma 2.7) and the source therein).

**Definition 10.3.1.** *Let*𝜋 :Z −→𝐵*be a flat proper morphism of smooth*
*projective varieties and let* M *be the smooth zero locus of a section*
*of a line bundle* L, embedded in Z *by* 𝜄 : M ↩−→ Z. Consider two
*relatively exceptional objects* E,F ∈ 𝐷^{𝑏}(Z) *and suppose there exist*
*strong semiorthogonal decompositions*

𝐷^{𝑏}(Z)= hC,E ⊗𝜋^{∗}𝐷^{𝑏}(𝐵),F ⊗𝜋^{∗}𝐷^{𝑏}(𝐵)i

𝐷^{𝑏}(M) = hD, 𝜄^{∗}E ⊗𝜄^{∗}𝜋^{∗}𝐷^{𝑏}(𝐵), 𝜄^{∗}F ⊗𝜄^{∗}𝜋^{∗}𝐷^{𝑏}(𝐵)i

(10.3.1)
*for some admissible subcategories* C,D*. We say that the left mutation*
LhE ⊗𝜋^{∗}𝐷^{𝑏}(𝐵)i(F ⊗𝜋^{∗}𝐷^{𝑏}(𝐵)) commutes *with* 𝜄^{∗} *if the following *
*equiva-lence holds:*

𝐷^{𝑏}(M) =hD,Lh𝜄^{∗}E ⊗𝜄^{∗}𝜋^{∗}𝐷^{𝑏}(𝐵)i(𝜄^{∗}F ⊗𝜄^{∗}𝜋^{∗}𝐷^{𝑏}(𝐵)), 𝜄^{∗}E ⊗𝜄^{∗}𝜋^{∗}𝐷^{𝑏}(𝐵)i
' hD, 𝜄^{∗}LhE ⊗𝜋^{∗}𝐷^{𝑏}(𝐵)i(F ⊗𝜋^{∗}𝐷^{𝑏}(𝐵)), 𝜄^{∗}E ⊗𝜄^{∗}𝜋^{∗}𝐷^{𝑏}(𝐵)i

(10.3.2)
**Definition 10.3.2.** *Let*Z −→𝐵*be a flat and proper morphism of smooth*
*projective varieties, let* L *be a line bundle on*Z. Consider two objects
E,F ∈ 𝐷^{𝑏}(Z). We say thatE*is*L-semiorthogonal*to*F *if the following*
*condition is fulfilled:*

𝜋_{∗}𝑅H𝑜𝑚_{Z}(E,F ⊗ L^{∨}) =0 (10.3.3)

**Lemma 10.3.3.** *In the setting of Definition 10.3.2, let*𝐵*be a point. Then*
E *is*L*-semiorthogonal to*F *if and only if*Ext^{•}_{Z}(E,F ⊗ L^{∨})=0.

*Proof.* Fix 𝐵' {𝑝𝑡}. One has:

𝜋_{∗}𝑅H𝑜𝑚_{𝐺/𝑃}(E,F ⊗ L^{∨}) ' 𝐻^{•}(𝐺/𝑃, 𝑅H𝑜𝑚

𝐺/𝑃(E,F ⊗ L^{∨}))
' Ext^{•}_{𝐺}_{/}_{𝑃}(E,F ⊗ L^{∨})

(10.3.4) where the first isomorphism is a consequence of (Mum12, Page 50,

Corollary 2).

**Lemma 10.3.4.** *Let* 𝜋 : Z −→ 𝐵 *be a flat and proper morphism of*
*smooth projective varieties, call* M ⊂ Z *the smooth zero locus of a*
*section of a line bundle*L*, embedded in*Z*by the morphism* 𝜄*. Suppose*
*there exist admissible subcategories* C ⊂ 𝐷^{𝑏}(M), D ⊂ 𝐷^{𝑏}(Z) *and*
*vector bundles* E,F ∈ 𝐷^{𝑏}(Z) *relatively exceptional over* 𝐵 *such that*
*one has the following strong,*𝐵*-linear semiorthogonal decompositions:*

𝐷^{𝑏}(Z)= hD,E ⊗𝜋^{∗}𝐷^{𝑏}(𝐵),F ⊗𝜋^{∗}𝐷^{𝑏}(𝐵)i
𝐷^{𝑏}(M) = hC, 𝜄^{∗}E ⊗𝜄^{∗}𝜋^{∗}𝐷^{𝑏}(𝐵), 𝜄^{∗}F ⊗𝜄^{∗}𝜋^{∗}𝐷^{𝑏}(𝐵)i

(10.3.5)
*Then, if* E *is* L*-semiorthogonal to* F*,* LhE ⊗𝜋^{∗}𝐷^{𝑏}(𝐵)iF ⊗𝜋^{∗}𝐷^{𝑏}(𝐵)
*com-mutes with*𝜄^{∗}*.*

*Proof.* In order to describe the left mutation of 𝜄^{∗}F ⊗ 𝜋^{∗}G through
h𝜄^{∗}E ⊗𝜄^{∗}𝜋^{∗}𝐷^{𝑏}(𝐵)i inside 𝐷^{𝑏}(M), we introduce the following functors
(and their right adjoints):

Ψ_{hE ⊗𝜋}∗𝐷^{𝑏}(𝐵)i :𝐷^{𝑏}(𝐵) 𝐷^{𝑏}(Z)

G 𝜋^{∗}G ⊗ E

(10.3.6)

Ψ^{!}

hE ⊗𝜋^{∗}𝐷^{𝑏}(𝐵)i :𝐷^{𝑏}(Z) 𝐷^{𝑏}(𝐵)

R 𝜋_{∗}𝑅H𝑜𝑚_{Z}(E,R)

(10.3.7)

We can apply Lemma 10.2.3 to the data 𝐿_{1} = 𝜄^{∗}, 𝑅_{1} = 𝜄_{∗}, 𝐿_{2} =
Ψ_{hE ⊗}𝜋^{∗}𝐷^{𝑏}(𝐵)i, 𝑅_{2} = Ψ^{!}

hE ⊗𝜋^{∗}𝐷^{𝑏}(𝐵)i, 𝔛 = 𝐷^{𝑏}(Z), 𝔅 = 𝐷^{𝑏}(𝐵), 𝔐 =
𝐷^{𝑏}(M). As a result, the following diagram commutes:

𝐿_{1}𝐿_{2}𝑅_{2}𝐴 𝐿_{1}𝐴

𝐿_{1}𝐿_{2}𝑅_{2}𝑅_{1}𝐿_{1}𝐴 𝐿_{1}𝐴

𝐿_{1}(𝜖_{2}, 𝐴)

𝐿_{1}𝐿_{2}𝑅_{2}(𝜂_{1}_{, 𝐴})

𝜖_{12}_{, 𝐿}

1𝐴

(10.3.8)

for any object 𝐴 of 𝐷^{𝑏}(Z). Let us now introduce the following
func-tors:

Θ_{h𝜄}∗E ⊗𝜄^{∗}𝜋^{∗}𝐷^{𝑏}(𝐵)i :𝐷^{𝑏}(𝐵) 𝐷^{𝑏}(M)
G 𝜄^{∗}𝜋^{∗}G ⊗𝜄^{∗}E

(10.3.9)

Θ^{!}

h𝜄^{∗}E ⊗𝜄^{∗}𝜋^{∗}𝐷^{𝑏}(𝐵)i : 𝐷^{𝑏}(M) 𝐷^{𝑏}(𝐵)

W 𝜋_{∗}𝜄_{∗}𝑅H𝑜𝑚_{M}(𝜄^{∗}E,W)
(10.3.10)
Note that, by definition, Θ_{h𝜄}∗E ⊗𝜄^{∗}𝜋^{∗}𝐷^{𝑏}(𝐵)i = 𝜄^{∗}Ψ_{hE ⊗𝜋}∗𝐷^{𝑏}(𝐵)i. Then,

Dia-gram 10.3.8 can be written in the following way:

𝜄^{∗}Ψ_{hE ⊗𝜋}∗𝐷^{𝑏}(𝐵)iΨ^{!}

hE ⊗𝜋^{∗}𝐷^{𝑏}(𝐵)i(F ⊗𝜋^{∗}G) 𝜄^{∗}(F ⊗𝜋^{∗}G)

𝜄^{∗}Ψ_{hE ⊗}𝜋^{∗}𝐷^{𝑏}(𝐵)iΨ^{!}

hE ⊗𝜋^{∗}𝐷^{𝑏}(𝐵)i𝜄_{∗}𝜄^{∗}(F ⊗𝜋^{∗}G) 𝜄^{∗}(F ⊗𝜋^{∗}G)

𝛼

𝜄^{∗}𝜖_{Z}

𝜖_{M}

(10.3.11)
where𝛼=𝜄^{∗}Ψ_{hE ⊗𝜋}∗𝐷^{𝑏}(𝐵)iΨ^{!}

hE ⊗𝜋^{∗}𝐷^{𝑏}(𝐵)i(𝜂_{1}_{,F ⊗𝜋}∗G),𝜖_{Z} =𝜖_{2}_{,F ⊗𝜋}∗G, 𝜖_{M} =
𝜖_{12}_{,𝜄}∗(F ⊗𝜋^{∗}G).

Let us now prove the following claim:

**Claim.** *The map* 𝛼 *is an isomorphism if* E *is* L-semiorthogonal to F.
To this purpose, let us focus on the following term:

Ψ^{!}_{hE ⊗𝜋}_{∗}

𝐷^{𝑏}(𝐵)i𝜄_{∗}𝜄^{∗}(F ⊗𝜋^{∗}G) =𝜋_{∗}𝑅H𝑜𝑚_{Z}(E, 𝜄_{∗}𝜄^{∗}(F ⊗𝜋^{∗}G))
(10.3.12)
Since E is a vector bundle we have:

𝜋_{∗}𝑅H𝑜𝑚_{Z}(E, 𝜄_{∗}𝜄^{∗}(F ⊗𝜋^{∗}G)) '𝜋_{∗}(E^{∨}⊗𝜄_{∗}𝜄^{∗}(F ⊗𝜋^{∗}G)). (10.3.13)
Observe that 𝜄_{∗}𝜄^{∗}(F ⊗𝜋^{∗}G) has a resolution given by the tensor product
of F with the Koszul resolution of 𝜄_{∗}𝜄^{∗}O:

0−→ F ⊗𝜋^{∗}G ⊗ L^{∨}−→ F ⊗𝜋^{∗}G −→𝜄_{∗}𝜄^{∗}(F ⊗𝜋^{∗}G) −→0 (10.3.14)
By left-exactness of the derived pushforward we get the following long

exact sequence:

0−→ 𝑅^{0}𝜋_{∗}(E^{∨}⊗ F ⊗𝜋^{∗}G ⊗ L^{∨}) −→ 𝑅^{0}𝜋_{∗}(E^{∨}⊗ F ⊗𝜋^{∗}G) −→

−→ 𝑅^{0}𝜋_{∗}(E^{∨}⊗𝜄_{∗}𝜄^{∗}(F ⊗𝜋^{∗}G)) −→ 𝑅^{1}𝜋_{∗}(E^{∨}⊗ F ⊗𝜋^{∗}G ⊗ L^{∨}) −→

−→ 𝑅^{1}𝜋_{∗}(E^{∨}⊗ F ⊗𝜋^{∗}G) −→𝑅^{1}𝜋_{∗}(E^{∨}⊗𝜄_{∗}𝜄^{∗}(F ⊗𝜋^{∗}G)) −→ · · ·
(10.3.15)
Hence, proving the claim reduces to show that

𝑅^{𝑘}𝜋_{∗}(E^{∨}⊗ F ⊗𝜋^{∗}G ⊗ L^{∨}) =0 (10.3.16)
for every 𝑘. By the (derived) projection formula one has:

𝑅^{𝑘}𝜋_{∗}(E^{∨}⊗ F ⊗𝜋^{∗}G ⊗ L^{∨}) ' 𝑅^{𝑘}𝜋_{∗}(E^{∨}⊗ F ⊗ L^{∨}) ⊗ G

' 𝑅^{𝑘}𝜋_{∗}𝑅H𝑜𝑚_{Z}(E,F ⊗ L^{∨}) ⊗ G =0
(10.3.17)
where the last isomorphism follows by the fact thatE isL-semiorthogonal
to F. This proves that 𝛼 is an isomorphism for every G ∈ 𝐷^{𝑏}(𝐵).
In order to prove that𝜄^{∗}LhE ⊗𝜋^{∗}𝐷^{𝑏}(𝐵)i(F ⊗𝜋^{∗}G)andLh𝜄^{∗}E ⊗𝜄^{∗}𝜋^{∗}𝐷^{𝑏}(𝐵)i𝜄^{∗}(F ⊗
𝜋^{∗}G) are isomorphic for every G ∈ 𝐷^{𝑏}(𝐵), note that such objects are
defined as the cones of respectively 𝜖_{M} and 𝜄^{∗}𝜖_{Z}, in the following
distinguished triangles:

Θ_{h𝜄}∗E ⊗𝜄^{∗}𝜋^{∗}𝐷^{𝑏}(𝐵)iΘ^{!}_{h𝜄}_{∗}_{E ⊗𝜄}_{∗}

𝜋^{∗}𝐷^{𝑏}(𝐵)i𝜄^{∗}F ⊗𝜄^{∗}𝜋^{∗}G−−→^{𝜖}^{M}

−→𝜄^{∗}F ⊗𝜄^{∗}𝜋^{∗}G −→Lh𝜄^{∗}E ⊗𝜄^{∗}𝜋^{∗}𝐷^{𝑏}(𝐵)i𝜄^{∗}F ⊗𝜄^{∗}𝜋^{∗}G;

𝜄^{∗}Ψ_{hE ⊗𝜋}∗𝐷^{𝑏}(𝐵)iΨ^{!}_{hE ⊗𝜋}_{∗}

𝐷^{𝑏}(𝐵)iF ⊗𝜋^{∗}G

𝜄^{∗}𝜖_{Z}

−−−→

−→𝜄^{∗}(F ⊗𝜋^{∗}G) −→𝜄^{∗}LhE ⊗𝜋^{∗}𝐷^{𝑏}(𝐵)iF ⊗𝜋^{∗}G.

(10.3.18)

Since 𝛼 is an isomorphism, the proof is concluded by (GM03, page 232, Corollary 4) applied to Diagram 10.3.11.

With the following lemma, in the setting of a roof bundle 𝜋 :Z −→𝐵, we show that L-semiorthogonality of vector bundles constructed by a representation of 𝑃 can be checked on the fibers of 𝜋.

**Lemma 10.3.5.** *Let*𝜋 :Z −→𝐵*be a roof bundle of type*𝐺/𝑃*where*𝐵*is*
*a smooth projective variety and*V*a principal*𝐺*-bundle on*𝐵*. Let*Γ*be a*
*representation of*𝑃*acting on the vector space*𝑉_{Γ}*such that*𝐿 =𝐺×^{𝑃}𝑉_{Γ}*.*
*Consider a line bundle* L *on* Z*such that* L = 𝜋^{∗}𝑇 ⊗ (V ×^{𝐺} 𝐺×^{𝑃}𝑉_{Γ})
*where*𝑇*is a line bundle on*𝐵*. Let*E,F *be vector bundles on*Z*such that*
*for every*𝑏 ∈ 𝐵 *one has* E |_{𝜋}−1(𝑏) ' 𝐸 *and*F |_{𝜋}−1(𝑏) ' 𝐹*, where* 𝐸 *is* 𝐿
*-semiorthogonal to*𝐹*. Then, for every*G ∈ 𝐷^{𝑏}(𝐵),E*is*L-semiorthogonal
*to*F ⊗𝜋^{∗}G.

*Proof.* Since E is a vector bundle, the following holds:

𝜋_{∗}𝑅H𝑜𝑚_{Z}(E,F ⊗𝜋^{∗}G ⊗ L^{∨}) ' 𝜋_{∗}(E^{∨}⊗ F ⊗𝜋^{∗}G ⊗ L^{∨}) (10.3.19)
thus our claim follows by proving that 𝑅^{𝑘}𝜋_{∗}(E^{∨}⊗ F ⊗𝜋^{∗}G ⊗ L^{∨}) =0
for every 𝑘. By the derived projection formula one has:

𝑅^{𝑘}𝜋_{∗}(E^{∨}⊗ F ⊗𝜋^{∗}G ⊗ L^{∨})= 𝑅^{𝑘}𝜋_{∗}(E^{∨}⊗ F ⊗ L^{∨}) ⊗ G. (10.3.20)
We will prove that the stalk 𝑅^{𝑘}𝜋_{∗}(E^{∨}⊗ F ⊗ L^{∨})𝑏 vanishes for every
𝑏∈ 𝐵. Once we fix𝑏, we observe that by our assumptions the following
holds:

◦ E^{∨}⊗ F ⊗ L^{∨} is flat over 𝐵 (Har77, Proposition III.9.2)

◦ The map 𝑏 ↦−→dim𝐻^{𝑘}(𝜋^{−1}(𝑏),E^{∨}⊗ F ⊗ L^{∨}|_{𝜋}−1(𝑏)) =
dim𝐻^{𝑘}(𝐺/𝑃, 𝐸^{∨}⊗𝐹 ⊗ 𝐿^{∨}) is constant for every 𝑘
Then, we apply (Mum12, Page 50, Corollary 2) and we find:

𝑅^{𝑘}𝜋_{∗}(E^{∨}⊗ F ⊗ L^{∨})𝑏 ' 𝐻^{𝑘}(𝐺/𝑃, 𝐸^{∨}⊗ 𝐹⊗ 𝐿^{∨})
'Ext_{𝐺}^{𝑘}_{/𝑃}(𝐸 , 𝐹⊗ 𝐿^{∨}) =0

(10.3.21)

where the last equality holds because 𝐸 is 𝐿-semiorthogonal to 𝐹 by
hypothesis. This proves that 𝑅^{𝑘}𝜋_{∗}(E^{∨}⊗ F ⊗ L^{∨}) =0, hence concluding

the proof.

**Notation 10.3.6.** *We establish the following data:*

◦ *Consider a homogeneous roof bundle*Z −−→^{𝜋} 𝐵*of type*𝐺/𝑃 *on a*
*smooth projective base* 𝐵*, such that* Z *admits two different *
*pro-jective bundle structures* 𝑝_{𝑖} : Z −→ Z𝑖 *for*𝑖 ∈ {1; 2}. CallV *a*
*principal*𝐺*-bundle over* 𝐵*such that*Z=V ×^{𝐺} 𝐺/𝑃*.*

◦ *Let* 𝐿 *be the line bundle on* 𝐺/𝑃 *which restricts to* O (1) *on the*
*fibers of both the projective bundle structures of*𝐺/𝑃*(which exists*
*by Proposition 4.1.4). Given a representation*Γ*of*𝑃*acting on the*
*vector space*𝑉_{Γ}*such that*𝐿=𝐺×^{𝑃}𝑉_{Γ}*, consider a line bundle*L*on*
Z*such that*L =𝜋^{∗}𝑇 ⊗ (V ×^{𝐺} 𝐺×^{𝑃}𝑉_{Γ})*where*𝑇 *is a line bundle*
*on* 𝐵*, and such that* L *restricts to* O (1) *on each fiber of both the*
*projective bundle structures of*Z.

◦ *A general section* Σ ∈ 𝐻^{0}(Z,L) *with smooth zero locus* M*, a*
*general section* 𝜎 ∈ 𝐻^{0}(𝐺/𝑃, 𝐿), a point 𝑏 ∈ 𝐵 *such that* 𝑀 :=

𝑍(𝜎) =𝑍(Σ|_{𝜋}−1(𝑏))*is smooth, and the following diagram:*

𝑀 =M ×𝐵 {𝑏} M

{𝑏} 𝐵

(10.3.22)

*Call*𝜄:M ↩−→ Z*and*𝑙 :𝑀 ↩−→𝐺/𝑃*the respective embeddings.*

◦ *A Calabi–Yau pair* (𝑌_{1}, 𝑌_{2}) *associated to the roof*𝐺/𝑃*, defined as*
𝑌_{𝑖} = 𝑍(ℎ_{𝑖∗}𝜎_{𝑖}) *where* ℎ_{𝑖} : 𝐺/𝑃 −→ 𝐺/𝑃_{𝑖} *are the two projective*
*bundle structures of*𝐺/𝑃*. Similarly, we consider a pair* (𝑋_{1}, 𝑋_{2})
*of Calabi–Yau fibrations defined as* 𝑋_{𝑖}= 𝑍(𝑝_{𝑖∗}Σ)*.*

◦ *Two full exceptional collections:*

𝐷^{𝑏}(𝐺/𝑃_{1})= h𝐽_{1}, . . . , 𝐽_{𝑚}i, 𝐷^{𝑏}(𝐺/𝑃_{2}) = h𝐾_{1}, . . . , 𝐾_{𝑚}i
(10.3.23)
*which by (Orl92, Corollary 2.7) induce the following *
*semiorthogo-nal decompositions for*𝐷^{𝑏}(𝐺/𝑃)*(recall that*𝐺/𝑃*has*P^{𝑟}^{−1}*-bundle*
*structures on both*𝐺/𝑃_{1}*and*𝐺/𝑃_{2}*):*

𝐷^{𝑏}(𝐺/𝑃) =h𝐸_{1}, . . . , 𝐸_{𝑁}i =h𝐹_{1}, . . . , 𝐹_{𝑁}i (10.3.24)
*where the bundles*𝐸_{𝑖}*have the form*𝐸_{𝑖} =𝐽_{𝑗}⊗𝐿^{⊗}^{𝑘}*and*𝐹_{𝑖}=𝐾_{𝑗}⊗𝐿^{⊗𝑘}
*for some integers* 𝑗 , 𝑘*. Moreover, by Theorem (Orl03 Proposition*
*2.10), one gets the following semiorthogonal decompositions for*
𝐷^{𝑏}(𝑀):

𝐷^{𝑏}(𝑀) = h𝜃_{1}𝐷^{𝑏}(𝑌_{1}), 𝑙^{∗}𝐸_{1}, . . . , 𝑙^{∗}𝐸_{𝑛}i

= h𝜃_{2}𝐷^{𝑏}(𝑌_{2}), 𝑙^{∗}𝐹_{1}, . . . , 𝑙^{∗}𝐹_{𝑛}i

(10.3.25)

*where* 𝜃_{1} *and* 𝜃_{2} *are defined in Equation 10.1.9. Note that the*
*numbers of exceptional objects in Equations 10.3.24 and 10.3.25*
*are different, but all exceptional objects of 10.3.25 are pullbacks of*
*objects from 10.3.24. We introduce the following ordered lists of*
*exceptional objects:*

𝑬 =(𝐸_{1}, . . . , 𝐸_{𝑛})
𝑭 =(𝐹_{1}, . . . , 𝐹_{𝑛})

(10.3.26)

*Observe that* 𝑬 *and* 𝑭 *are subsets of the generators of* 𝐷^{𝑏}(𝐺/𝑃)
*appearing in the exceptional collections 10.3.24.*

◦ *For every* 𝐽_{𝑖}*, which can be described as* 𝐽_{𝑖} = 𝐺 ×^{𝑃}^{1}𝑉_{Γ}

𝑖 *for some*
*vector space*𝑉_{Γ}

𝑖 *associated to a representation*Γ𝑖 *of* 𝑃_{1}*, consider*
*the vector bundle* J𝑖 :=V ×^{𝐺} 𝐺×^{𝑃}^{𝑖}𝑉_{Γ}

𝑖 *such as defined in Section*
*10.1.1, with the property that* J𝑖|_{𝜋}−1(𝑏) ' 𝐽_{𝑖} *for every*𝑖*. Then, by*
*(Sam06, Theorem 3.1), the subcategory*J𝑖⊗𝑟^{∗}

1𝐷^{𝑏}(𝐵)*is admissible*
*in*𝐷^{𝑏}(Z_{1})*. In the same way we can define admissible subcategories*
K𝑖 ⊗ 𝑟^{∗}

2𝐷^{𝑏}(𝐵) ⊂ 𝐷^{𝑏}(Z_{2})*. By applying (Sam06, Theorem 3.1)*
*and (Orl92, Corollary 2.7) to the collections 10.3.25 we find the*
*following semiorthogonal decompositions:*

*and by definition of* L*, one has* [𝐸_{𝑖}] ' [ℎ^{∗}
*such that each*𝜉^{(}^{𝜆}^{)} *falls in one of the following classes (type O1,*
*O2 or O3):*

𝑖+1 *is defined by the following distinguished *
*tri-angle in*𝐷^{𝑏}(𝐺/𝑃)*:*

*Similarly, we define the operation of exchanging the order*

𝑖 *is defined by the following distinguished *
*tri-angle in*𝐷^{𝑏}(𝐺/𝑃)*:*

1 *to the end, twisting it by*𝐿^{⊗(𝑟−1)} *and substituting*
𝜓^{(𝜆)} *with*R^{𝑙}^{∗}^{𝐸}1𝜓^{(𝜆)}*:*

**O3** *For any*𝑘 ∈ Z*, substituting* 𝐸^{(𝜆)}

◦ *An autoequivalence of* 𝐷^{𝑏}(𝑀) *given by a sequence of mutations*
*and twists on the semiorthogonal decompositions 10.3.25 which*
*acts in the following way:*
*Proof.* By Theorem 3.3.2 one has a semiorthogonal decomposition
𝐷^{𝑏}(Z) = h[𝑊_{1}], . . . ,[𝑊_{𝑁}]i, then applying (Kuz10, Corollary 2.9) one
finds:

𝐷^{𝑏}(Z) =h[𝑊_{1}], . . . ,[𝑊_{𝑖−}_{1}],L[𝑊_{𝑖}][𝑊_{𝑖+}_{1}],[𝑊_{𝑖}],[𝑊_{𝑖+}_{2}], . . . ,[𝑊_{𝑁}]i
(10.3.40)

On the other hand, for 1 ≤𝑖 ≤ 𝑁−1 one has:

𝐷^{𝑏}(𝐺/𝑃) = h𝑊_{1}, . . . , 𝑊_{𝑖}_{−1},L^{𝑊}𝑖𝑊_{𝑖}_{+1}, 𝑊_{𝑖}, 𝑊_{𝑖}_{+2}, . . . , 𝑊_{𝑚}i (10.3.41)
and by applying again Theorem 3.3.2 to the collection 10.3.41 one
finds:

𝐷^{𝑏}(Z)=h[𝑊_{1}], . . . ,[𝑊_{𝑖}_{−1}],[L^{𝑊}^{𝑖}𝑊_{𝑖}_{+1}],[𝑊_{𝑖}],[𝑊_{𝑖}_{+2}], . . . ,[𝑊_{𝑁}]i (10.3.42)
By comparison with Equation 10.3.40 we see that both [L^{𝑊}^{𝑖}𝑊_{𝑖+1}] and
L[𝑊_{𝑖}][𝑊_{𝑖+}_{1}] are equivalent to the subcategory ^{⊥}h[𝑊_{1}], . . . ,[𝑊_{𝑖−}_{1}]i ∩
h[𝑊_{𝑖}],[𝑊_{𝑖+}_{2}], . . . ,[𝑊_{𝑁}]i^{⊥}, hence they are equivalent.

**Lemma 10.3.8.** *In the language of Notation 10.3.6, consider a *
*semiorthog-onal decomposition* 𝐷^{𝑏}(𝑀) = h𝜙_{1}𝐷^{𝑏}(𝑌_{1}), 𝑙^{∗}𝑊_{1}, . . . , 𝑙^{∗}𝑊_{𝑛}i *where, for*
1 ≤ 𝑗 ≤ 𝑛*,*𝑊_{𝑗} *is a homogeneous vector bundle on*𝐺/𝑃*. Assume that*𝑊_{𝑖}
*is* 𝐿*-semiorthogonal to*𝑊_{𝑖}_{+1}*for some positive*𝑖 < 𝑛*. Then the following*
*holds:*

◦ L^{𝑊}𝑖𝑊_{𝑖}_{+1}*commutes with*𝑙^{∗}

◦ L[𝑊𝑖][𝑊_{𝑖}_{+1}] *commutes with*𝜄^{∗}

◦ *one has a semiorthogonal decomposition:*

𝐷^{𝑏}(M) =h𝜙_{1}𝐷^{𝑏}(𝑋_{1}), 𝜄^{∗}[𝑊_{1}], . . . , 𝜄^{∗}[𝑊_{𝑖−1}],
𝜄^{∗}[L^{𝑊}𝑖𝑊_{𝑖}_{+1}], 𝜄^{∗}[𝑊_{𝑖}], 𝜄^{∗}[𝑊_{𝑖}_{+2}], . . . , 𝜄^{∗}[𝑊_{𝑛}]i

(10.3.43)

*Proof.* Let us first recall that by Theorem 3.3.2 and Theorem 3.3.5
one has 𝐷^{𝑏}(M) = h𝜙_{1}(𝑌_{1}), 𝜄^{∗}[𝑊_{1}], . . . , 𝜄^{∗}[𝑊_{𝑛}]i. We now prove
com-mutativity. By Lemma 10.3.7, one has [L^{𝑊}^{𝑖}𝑊_{𝑖}_{+1}] = L[𝑊_{𝑖}][𝑊_{𝑖}_{+1}]. By
Lemma 10.3.5, since 𝑊_{𝑖} is 𝐿-semiorthogonal to 𝑊_{𝑖}_{+1} it follows that

W𝑖 is L-semiorthogonal to W_{𝑖+}_{1}. Then, by Lemma 10.3.4 applied
to 𝐺/𝑃 −→ {𝑝𝑡} one finds that L^{𝑊}^{𝑖}𝑊_{𝑖}_{+1} commutes with 𝑙^{∗}, while
by applying the same lemma to Z −→ 𝐵 it follows that L[𝑊_{𝑖}][𝑊_{𝑖}_{+1}]
commutes with 𝜄^{∗}. By the latter we get:

L^{𝜄}^{∗}[𝑊_{𝑖}]𝜄^{∗}[𝑊_{𝑖+}_{1}] ' 𝜄^{∗}L[𝑊_{𝑖}][𝑊_{𝑖+}_{1}] =𝜄^{∗}[L^{𝑊}^{𝑖}𝑊_{𝑖+}_{1}] (10.3.44)
By (Kuz10, Corollary 2.9) one has:

𝐷^{𝑏}(M) =h𝜙_{1}𝐷^{𝑏}(𝑋_{1}), 𝜄^{∗}[𝑊_{1}], . . . , 𝜄^{∗}[𝑊_{𝑖}_{−1}],
L^{𝜄}^{∗}[𝑊_{𝑖}]𝜄^{∗}[𝑊_{𝑖+}_{1}], 𝜄^{∗}[𝑊_{𝑖}], 𝜄^{∗}[𝑊_{𝑖+}_{2}], . . . , 𝜄^{∗}[𝑊_{𝑛}]i

(10.3.45) and substituting 10.3.44 in the decomposition 10.3.43 completes the proof.

**Proposition 10.3.9.** *In the language of Notation 10.3.6, assume there is*
*a semiorthogonal decomposition*𝐷^{𝑏}(𝑀) = h𝜙_{1}𝐷^{𝑏}(𝑌_{1}), 𝑙^{∗}𝑊_{1}, . . . , 𝑙^{∗}𝑊_{𝑛}i
*where every*𝑊_{𝑗} *is a homogeneous vector bundle on*𝐺/𝑃*. Then one has:*

𝐷^{𝑏}(M)=hL^{𝜄}^{∗}[𝑊_{𝑛}⊗𝐿⊗ (−𝑟+1)]𝜙_{1}𝐷^{𝑏}(𝑋_{1}), 𝜄^{∗}[𝑊_{𝑛}⊗𝐿^{⊗ (−𝑟}^{+}^{1}^{)}], 𝜄^{∗}[𝑊_{1}], . . . , 𝜄^{∗}[𝑊_{𝑛}_{−1}]i.
(10.3.46)
*Proof.* Since M is a smooth projective variety, by the Serre functor,
there is the following semiorthogonal decomposition:

𝐷^{𝑏}(M)=hL^{𝜄}^{∗}[𝑊_{𝑛}] ⊗𝜔_{M}𝜙_{1}𝐷^{𝑏}(𝑋_{1}), 𝜄^{∗}[𝑊_{𝑛}] ⊗𝜔_{M}, 𝜄^{∗}[𝑊_{1}], . . . , 𝜄^{∗}[𝑊_{𝑛−}_{1}]i.
(10.3.47)
One has (Ful98, Example 3.2.11):

𝜔_{Z} ' 𝑝^{∗}

𝑖𝜔_{Z}

𝑖 ⊗ 𝑝^{∗}

𝑖 detE𝑖⊗ L^{⊗(−}^{𝑟}^{)} (10.3.48)

but since Z is a roof bundle, (Z|_{𝑟}−1

𝑖 (𝑏),E𝑖|_{𝑟}−1

𝑖 (𝑏)) is a Mukai pair for
every 𝑏 ∈ 𝐵, which implies that 𝜔_{Z}

𝑖 ⊗ detE𝑖 ' 𝑟^{∗}

𝑖𝑇, where 𝑇 is a
line bundle on 𝐵. Then, by plugging this into 10.3.48 we get 𝜔_{Z} '
L^{⊗(−}^{𝑟}^{)} ⊗𝜋^{∗}𝑇. Due to the following normal bundle sequence:

0−→𝑇_{M} −→ 𝜄^{∗}𝑇_{Z} −→ L −→0 (10.3.49)
one has 𝜔_{M} ' 𝜄^{∗}𝜔_{Z} ⊗ 𝜄^{∗}L^{∨} ' 𝜄^{∗}L^{⊗(−𝑟+}^{1}^{)} ⊗𝜄^{∗}𝜋^{∗}𝑇. Then, the proof is
completed by the following computation:

𝜄^{∗}[𝑊_{𝑛}] ⊗𝜔_{M} =𝜄^{∗}[𝑊_{𝑛}] ⊗𝜄^{∗}L^{⊗(−𝑟+1)} =𝜄^{∗}[𝑊_{𝑛}⊗ 𝐿^{⊗(−𝑟+1)}]. (10.3.50)
and by plugging it in the decomposition 10.3.47.

Let us gather here the assumptions for the main theorem of this chap-ter.

**Assumption 10.3.10.** *The data of Notation 10.3.6 fulfill the following*
*requirements:*

**A1** L *is basepoint-free and the restriction map*

𝐻^{0}(Z,L) −→ 𝐻^{0}(𝜋^{−}^{1}(𝑏),L |_{𝜋}−1(𝑏)) (10.3.51)
*is surjective for every*𝑏 ∈𝐵

**A2** *The autoequivalence of*𝐷^{𝑏}(𝑀)*described in Notation 10.3.6 acts by*
*a composition of the following operations on the first collection of*
*Equation 10.3.25:*

◦ *Mutations of pairs of exceptional objects*L^{𝑙}^{∗}^{𝐸}𝑙^{∗}𝐹 *where* 𝐸*,*
𝐹 *satisfy the semiorthogonality condition*Ext^{•}_{𝐺/𝑃}(𝐹 , 𝐸) = 0

*and* 𝐸 *is* 𝐿*-semiorthogonal to* 𝐹 *(in short,* L^{𝑙}^{∗}^{𝐸}𝑙^{∗}𝐹 *satisfies*
*Condition (*†*) as defined in Chapter 9)*

◦ *Overall twists by a power of*𝐿*.*

◦ *Applying the Serre functor of*S𝑀*sending the last exceptional*
*object to the beginning of the semiorthogonal decompositions,*
*or applying the inverse functor*S^{−1}

𝑀 *.*

◦ *Applying the mutation*L^{𝑙}^{∗}^{𝐸}*or*R^{𝑙}^{∗}^{𝐸}*to the subcategory*𝜃_{𝑖}𝐷^{𝑏}(𝑌_{𝑖}),
*where*𝑙^{∗}𝐸 *is an exceptional object in the right (respectively*
*left) semiorthogonal complement of*𝜃_{𝑖}𝐷^{𝑏}(𝑌_{𝑖}).

**A3** *The sequence of operations* 𝜉 = 𝜉^{(}^{𝑅}^{)}. . . 𝜉^{(1)} *acts on* (𝑬, 𝜃_{1}) *in the*
*following way:*

𝜉 : (𝑬, 𝜃_{1}) ↦−→ (𝑭, 𝜓) (10.3.52)
*where* (𝑬^{(1)}, 𝜓^{(}^{1}^{)}) =(𝑬, 𝜃_{1}) *and*(𝑬^{(}^{𝑅+1)}, 𝜓^{(}^{𝑅+}^{1}^{)}) =(𝑭, 𝜓).

The condition A1 is needed to ensure smoothness of the general sections
of L, and the fact that zero loci of pushforwards of such sections
have the property of being Calabi–Yau fibrations. On the other hand,
assumption A2 is needed to construct the mutations in 𝐷^{𝑏}(M). The
last assumption A3 is needed to ensure that such mutations really yield
an equivalence 𝐷^{𝑏}(𝑋_{1}) ' 𝐷^{𝑏}(𝑋_{2}).

**Definition 10.3.11.** *In the language of Notation 10.3.6, we say that a pair*
(Z,L) *of a roof bundle* Z *of type* 𝐺/𝑃 *together with the line bundle*
L *satisfies Assumption 10.3.10 if* L *satisfies Assumption A1 and there*
*exist two full exceptional collections* 𝐺/𝑃 = h𝐸_{1}, . . . 𝐸_{𝑁}i = h𝐹_{1}. . . 𝐹_{𝑁}i
*and a section*𝜎 ∈𝐻^{0}(𝐺/𝑃, 𝐿) *as required in Notation 10.3.6, which are*

*compatible with Assumptions A2 and A3.*

**Theorem 10.3.12.** *Let*(Z,L) *satisfy Assumption 10.3.10. Then a *
*gen-eral section of*L *induces a derived equivalence of Calabi–Yau fibrations*
Φ:𝐷^{𝑏}(𝑋_{1}) −→ 𝐷^{𝑏}(𝑋_{2}).

*Proof.* Consider two full exceptional collections 𝐺/𝑃 =h𝐸_{1}, . . . 𝐸_{𝑁}i=
h𝐹_{1}. . . 𝐹_{𝑁}i and a general section 𝜎 ∈ 𝐻^{0}(𝐺/𝑃, 𝐿) with zero locus 𝑀,
compatible with Assumptions A2 and A3. Then, by Assumption A2
one has a derived equivalence:

𝐷^{𝑏}(𝑀)= h𝜃_{1}𝐷^{𝑏}(𝑌_{1}), 𝑙^{∗}𝐸_{1}, . . . , 𝑙^{∗}𝐸_{𝑛}i (10.3.53)

−→h𝜓 𝐷^{𝑏}(𝑌_{1}), 𝑙^{∗}𝐹_{1}, . . . , 𝑙^{∗}𝐹_{𝑛}i (10.3.54)
which consists in applying a sequence of operations to the first
semiorthog-onal decomposition, which only include mutations of 𝐿-semiorthogonal
exceptional pairs, the Serre functor of 𝑀, overall twists by a line bundle
and mutations of 𝜃_{1}𝐷^{𝑏}(𝑌_{1}) through the admissible subcategory
gener-ated by an exceptional object.

Consider the sequence of operations 𝜉 = 𝜉^{(}^{𝑅}^{)}· · ·𝜉^{(1)} defined in
No-tation 10.3.6. By Assumption A3 one has:

𝜉 : (𝑬, 𝜃_{1}) ↦−→ (𝑭, 𝜓) (10.3.55)
These operations, by A2, are in one-to-one correspondence with the
mutations used to transform the decomposition 10.3.53 into 10.3.54
(the case of mutations of pairs is treated in Lemma 10.3.8). In fact, for
1 ≤𝜆 ≤ 𝑅+1 one has a semiorthogonal decomposition:

𝐷^{𝑏}(𝑀) =h𝜓^{(}^{𝜆}^{)}𝐷^{𝑏}(𝑌_{1}), 𝑙^{∗}𝐸^{(𝜆)}

1 , . . . , 𝑙^{∗}𝐸_{𝑛}^{(𝜆)}i. (10.3.56)

Every operation 𝜉^{(𝜆)} commutes also with the mapping𝑊 ↦−→ [𝑊]: for
the cases of Operations O2 and O3 this follows by the fact that the
product of homogeneous vector bundles associated to representations
Γ,Γ^{0}is the homogeneous vector bundle associated to the representation
Γ⊗Γ^{0}, while the case of type O1 follows from Lemma 10.3.7.
Here-after we show that we can associate to the sequence of mutations
on the collection 10.3.53 a sequence of mutations on the
decompo-sition 𝐷^{𝑏}(M) = h𝜙_{1}𝐷^{𝑏}(𝑋_{1}), 𝜄^{∗}[𝐸_{1}], . . . , 𝜄^{∗}[𝐸_{𝑛}]i defined through the
operations [𝜉^{(𝜆)}], thus obtaining for every 𝜆:

𝐷^{𝑏}(M) =hΦ^{(}^{𝜆}^{)}𝐷^{𝑏}(𝑋_{1}), 𝜄^{∗}[𝐸^{(𝜆)}

1 ], . . . , 𝜄^{∗}[𝐸_{𝑛}^{(𝜆)}]i. (10.3.58)
To prove our claim, let us consider each of the allowed kinds of
muta-tions on 10.3.53, and describe the associated mutation on 10.3.58.

◦ Every time a left mutation of pairs
h. . . , 𝑙^{∗}𝐸^{(}

is performed in 10.3.56, we do the operation

We obtain a semiorthogonal decomposition because of the
fol-lowing argument: by Assumption A2, 𝐸^{(}

𝜆)

𝑖+1] by Lemma 10.3.7, we see that the operation described
in Equation 10.3.60 is simply the left mutation of𝜄^{∗}[𝐸^{(}

𝜆)

𝑖+1]through
𝜄^{∗}[𝐸^{(}

𝜆)

𝑖 ]. An analogous argument works for right mutations.

◦ Every time the Serre functor is applied to Equation 10.3.56:

h𝜓^{(𝜆)}𝐷^{𝑏}(𝑌_{1}), 𝑙^{∗}𝐸^{(}
perform the following operation on Equation 10.3.58:

hΦ^{(𝜆)}𝐷^{𝑏}(𝑋_{1}), 𝜄^{∗}[𝐸^{(}

In fact, by Proposition 10.3.9 the resulting collection above is the
one obtained by applying the Serre functor ofM to 𝜄^{∗}[𝐸^{(}

𝜆)
1 ] and
sending the subcategory equivalent to 𝐷^{𝑏}(𝑋_{1}) to the beginning
of the collection. The same holds for the inverse Serre functor.