Derived Equivalences for the Flops of Type C 2 and A G4 via Mutation of Semiorthogonal Decomposition

We give a new proof of the derived equivalence of a pair of varieties connected by the flop of type C 2 in the list of Kanemitsu (2018), which is originally due to Segal (Bull. Lond. Math. Soc., 48 (3) 533–538, 2016). We also prove the derived equivalence of a pair of varieties connected by the flop of type A G 4 in the same list. The latter proof follows that of the derived equivalence of Calabi–Yau 3-folds in Grassmannians Gr ( 2 , 5 ) and Gr ( 3 , 5 ) by Kapustka and Rampazzo (Commun. Num. Theor. Phys., 13 (4) 725–761 2019) closely.


Introduction
Let G be a semisimple Lie group and B a Borel subgroup of G. For distinct maximal parabolic subgroups P and Q of G containing B, three homogeneous spaces G/P , G/Q, and G/(P ∩ Q) form the following diagram: v v n n n n n n n n n n n n + ( ( P P P P P P P P P P P P The morphisms − and + are projective morphisms whose relative O (1) are O(H ) and O(h) respectively. We consider the diagram • ι − , ι + , and ι are the zero-sections, • ϕ − and ϕ + are blow-ups of the zero sections, and • φ − and φ + are the affinizations which contract the zero sections.
If V − and V + have the trivial canonical bundles, then one expects from [4,Conjecture 4.4] or [16,Conjecture 1.2] that V − and V + are derived-equivalent.
When G is the simple Lie group of type G 2 , Ueda [24] used sequence of mutations of semiorthogonal decompositions of D b (V) obtained by applying Orlov's theorem [20] to the diagram Eq. 1.1 to prove the derived equivalence of V − and V + . This sequence of mutations in turn follows that of Kuznetsov [18] closely.
In this paper, by using the same method, we give a new proof to the following theorem, which is originally due to Segal [22], where the flop was attributed to Abuaf: Theorem 1.1 Varieties connected by the flop of type C 2 are derived-equivalent.
The term the flop of type C 2 was introduced in [13], where simple K-equivalent maps in dimension at most 8 were classified. There are several ways to prove Theorem 1.1. In [22], Segal showed the derived equivalence by using tilting vector bundles. Hara [8] constructed alternative tilting vector bundles and studied the relation between functors defined by him and Segal.
The flop of type A G 2r−2 is also in the list of Kanemitsu [13]. It connects V − and V + for P = Gr(r − 1, 2r − 1) and Q = Gr(r, 2r − 1). Similarly, we prove the following theorem: Although the proof of Theorem 1.2 is parallel to that of the derived equivalence of Calabi-Yau complete intersections in P = Gr(2, 5) and Q = Gr (3,5) defined by global sections of the equivariant vector bundles dual to V − and V + in [15,Theorem 5.7], we write down a full detail for clarity. As explained in [24], the derived equivalence obtained in [15] in turn follows from Theorem 1.2 using matrix factorizations.
We also give a similar proof of derived equivalences for a Mukai flop and a standard flop. For a Mukai flop, Kawamata [16] and Namikawa [19] independently showed the derived equivalence by using the pull-back and the push-forward along the fiber product V − × V 0 V + . Addington, Donovan, and Meachan [1] introduced a generalization of the functor of Kawamata and Namikawa parametrized by an integer, and discovered that certain compositions of these functors give the P-twist in the sense of Huybrechts and Thomas [11]. They also considered the case of a standard flop, where the derived equivalence is originally proved by Bondal and Orlov [5]. Our proof is obtained by proceeding the mutation performed in [5] and [1] a little further in a straightforward way. Hara [7] also studied a Mukai flop in terms of non-commutative crepant resolutions.
For a standard flop, Segal [21] showed the derived equivalence by using the grade restriction rule for variation of geometric invariant theory quotients (VGIT) originally introduced by Hori, Herbst, and Page [10]. VGIT method was subsequently developed by Halpern-Leistner [6] and Ballard, Favero, and Katzarkov [2]. It is an interesting problem to develop this method further to prove the derived equivalence for the flop of type C 2 and A G 4 , and a Mukai flop.

Notations and conventions
We work over an algebraically closed field k of characteristic 0 throughout this paper. All pull-back and push-forward are derived unless otherwise specified. The complexes underlying Ext • (−, −) and H • (−) will be denoted by hom(−, −) and h(−) respectively.

Flop of Type C 2
Let P and Q be the parabolic subgroups of the simple Lie group G of type C 2 associated with the crossed Dynkin diagrams and . The corresponding homogeneous spaces are the projective space P = P(V ), the Lagrangian Grassmannian Q = LGr(V ), and the isotropic flag variety F = P P L ⊥ P /L P = P Q S Q . Here V is a 4-dimensional symplectic vector space, L ⊥ P is the rank 3 vector bundle given as the symplectic orthogonal to the tautological line bundle L P ∼ = O P (−h) on P, and S Q is the tautological rank 2 bundle on Q. Note that Q is also a quadric hypersurface in P 4 . Tautological sequences on Q = LGr(V ) and F ∼ = P Q S Q give where S F := * + S Q . We have whose determinants are given by O P (2h) and O Q (H ) respectively. Since ω P ∼ = O P (−4h), and from [17] (cf. also [14]) that . Since ϕ ± are blow-ups along the zero-sections, it follows from [20] that and . By abuse of notation, we use the same symbol for an object of D b (F) and its image in D b (V) by the push-forward ι * . Equations 2.3 and 2.4 give , by mutating the first term to the far right, and then − (D b (V − )) one step to the right, we obtain In the sequel, we will use the following fact.

Lemma 2.1 Given two vector bundles
is given by the short exact sequence Eq. 2.2. By mutating O F (−h + H ) one step to the left, O F (−h) to the far right, and then 1 (D b (V − )) one step to the right, we obtain By mutating 2 (D b (V − )) one step to the left, and then O F (h) to the far left, we obtain By comparing Eq. 2.7 with Eq. 2.5, we obtain a derived equivalence

4
Let P and Q be the parabolic subgroups of the simple Lie group G of type A 4 associated with the crossed Dynkin diagrams and . The corresponding homogeneous spaces are the Grassmannians P = Gr(2, V ), Q = Gr(3, V ), and the partial flag Here V is a 5-dimensional vector space, Q ∨ P is the dual of the universal quotient bundle on P, and S Q is the tautological rank 3 bundle on Q. We have First, we adapt several lemmas in [15] to our situation. To distinguish vector bundles which are obtained as a pull-back to F from P or Q, we put tilde on the pull-back from Q. By abuse of notation, we use the same symbol for an object of D b (F) and its image in D b (V) by the push-forward ι * .

Lemma 3.1 hom
Proof We have where the first and the second isomorphisms follow from Lemma 2.
The parallel result to the following lemma was tacitly used in [15].

Lemma 3.4 As an object of
, and Q F respectively.

Lemma 3.5 hom
Again, both Lemmas 3.4 and 3.5 follow from Lemma 2.1 and Borel-Bott-Weil theorem. Lemma 3.6 below and the exact sequences Proof We have where the isomorphisms follow from Lemma 2.1 and Borel-Bott-Weil theorem. Similarly, we have Recall from [17] (cf. also [14]) Since ϕ ± are blow-ups along the zero-sections, it follows from [20] that and

ih+jH ). Equations 3.1 and 3.3 give a semiorthogonal decomposition of the form
Since , by mutating the first five terms to the far right, and then + (D b (V + )) five steps to the right, we obtain One can easily see that O 1,1 is orthogonal to O 0,3 , Q 0,3 , O 0,4 , and Q 0,4 by Lemmas 3.1 and 3.2, so that By mutating Q 0,2 , Q 1,3 , Q 1,1 , and Q 2,2 one step to the right, we obtain by Q 1,1 ∼ = Q ∨ 1,2 , Lemmas 3.5, and 3.6 ) . By mutating O 1,2 and O 2,3 four steps to the left, we obtain by Lemmas 3.1, 3.2, and 3.6 One can easily see that S ∨ 1,2 is orthogonal to O 0,4 and Q 0,4 by Lemmas 3.4, so that . By mutating O 0,3 and O 1,4 two steps to the right, O 1,3 and O 2,4 three steps to the left, and then O 0,4 and O 1,5 two steps to the right, we obtain by Lemmas 3.5 and 3.6 ) . By mutating O 1,1 to the far right, and then 1 (D b (V + )) one step to the right, we obtain By Lemmas 3.2, 3.3, and 3.4, we obtain . By mutating 2 (D b (V + )) ten steps to the left, and then last ten terms to the far left, we obtain . By mutating Q 0,2 and Q 1,3 two steps to the left, the first two terms to the far right, and then 3 (D b (V + )) two steps to the right, we obtain by S ∨ 0,0 S 1,0 , Lemmas 3.4, and 3.6 (3.4) where 4 := R S ∨ 2,3 ,O 3,3 • 3 . By comparing Eq. 3.4 with Eq. 3.2, we obtain a derived equivalence is the left adjoint functor of − .

Mukai Flop
For n ≥ 2, let P and Q be the maximal parabolic subgroups of the simple Lie group of type A n associated with the crossed Dynkin diagrams and . The corresponding homogeneous spaces are the projective spaces P = PV , Q = PV ∨ , and the partial flag variety F = F (1, n; V ), where V is an (n + 1)-dimensional vector space. Since Proof We have which vanishes by Lemma 4.1.
Recall from [3] that and Since ϕ ± are blow-ups along the zero-sections, it follows from [20] that and Derived Equivalences for the Flops of Type C 2 and A G where A 0 is given by (4.5) Note from Lemma 4.2 that there are no morphisms from right to left in Eq. 4.5. Since to the far right, and then − (D b (V − )) to the far right, we obtain By mutating 1 (D b (V − )) one step to the left, and then O 2n−2,n−2 to the far left, we obtain

Standard Flop
For n ≥ 1, let P and Q be the maximal parabolic subgroups of the semisimple Lie group G = SL(V ) × SL(V ∨ ) associated with the crossed Dynkin diagram ⊕ and ⊕ . The corresponding homogeneous spaces are the projective spaces P = PV , Q = PV ∨ , and their product Proof We have which vanishes for 1 ≤ i ≤ n − j ≤ n − 1.
It follows from [20] that

Remark 1
The way of presenting our proof in Section 4 and 5 is called chess game by some authors [12,23].