# On the Abuaf-Ueda flop
via non-commutative crepant resolutions

wahei hara  The Mathematics and Statistics Building, University of Glasgow,
University Place, Glasgow, G12 8QQ, UK<EMAIL_ADDRESS>
###### Abstract.

The Abuaf-Ueda flop is a $7$-dimensional flop related to $G_{2}$ homogeneous
spaces. The derived equivalence for this flop is first proved by Ueda using
mutations of semi-orthogonal decompositions. In this article, we give an
alternative proof for the derived equivalence in which we use tilting bundles.
Our proof also show the existence of non-commutative crepant resolutions of
the singularity appearing in the flopping contraction. We also give some
results on moduli spaces of finite-length modules over our non-commutative
crepant resolution.

###### Key words and phrases:

Derived category, Non-commutative crepant resolution, Flop, Tilting bundle

###### 2010 Mathematics Subject Classification:

14F05

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Tilting bundles and derived equivalences
  4. 4 Moduli problem

##  1\. Introduction

When there is a flop $Y_{+}\dashrightarrow Y_{-}$ between two smooth
varieties, it is important to compare their derived categories. According to a
famous conjecture due to Bondal and Orlov [BO02], we expect that we have a
derived equivalence
$\operatorname{D^{b}}(Y_{+})\simeq\operatorname{D^{b}}(Y_{-})$. The aim of
this article is to give a proof of this conjecture for a $7$-dimensional flop
that has an interesting geometric feature by using tilting bundles.

###  1.1. The Abuaf-Ueda flop

First of all we give the construction of the flop we study in this article.
Let us consider the $G_{2}$ Dynkin diagram
$\bigcirc\equiv\rangle\equiv\bigcirc$. Then by the classification theory of
homogeneous varieties, projective homogeneous varieties of the semi-simple
algebraic group of type $G_{2}$ corresponds to a marked Dynkin diagram. The
one $\lower
1.29167pt\hbox{\scalebox{1.5}{$\times$}}\equiv\rangle\equiv\bigcirc$
corresponds to the $G_{2}$-Grassmannian
$\operatorname{\mathbf{G}}=\operatorname{Gr}_{G_{2}}$. Another one
$\bigcirc\equiv\rangle\equiv\lower 1.29167pt\hbox{\scalebox{1.5}{$\times$}}$
corresponds to the $5$-dimensional quadric
$\operatorname{\mathbf{Q}}\subset\operatorname{\mathbb{P}}^{6}$. The last one
$\lower 1.29167pt\hbox{\scalebox{1.5}{$\times$}}\equiv\rangle\equiv\lower
1.29167pt\hbox{\scalebox{1.5}{$\times$}}$ corresponds to the (full) flag
variety $\operatorname{\mathbf{F}}$ of type $G_{2}$. There are projections
$\operatorname{\mathbf{F}}\to\operatorname{\mathbf{G}}$ and
$\operatorname{\mathbf{F}}\to\operatorname{\mathbf{Q}}$, and both of them give
$\operatorname{\mathbb{P}}^{1}$-bundle structures of
$\operatorname{\mathbf{F}}$.

Let us consider the Cox ring of $\operatorname{\mathbf{F}}$

$C:=\bigoplus_{a,b=0}^{\infty}H^{0}(\operatorname{\mathbf{F}},\mathcal{O}_{\operatorname{\mathbf{F}}}(a,b))\simeq\bigoplus_{a,b=0}^{\infty}V_{(a,b)}^{\vee},$

where $\mathcal{O}_{\operatorname{\mathbf{F}}}(a,b)$ (resp.
$V_{(a,b)}^{\vee}$) is a line bundle on $\operatorname{\mathbf{F}}$ (resp. the
dual of an irreducible representation of $G_{2}$) that corresponds to the
dominant weight $(a,b)$. Put
$C_{a,b}:=H^{0}(\operatorname{\mathbf{F}},\mathcal{O}_{\operatorname{\mathbf{F}}}(a,b))$
and

$C_{n}:=\bigoplus_{a\in\operatorname{\mathbb{Z}}}C_{n+a,a}.$

Using them we can define a $\operatorname{\mathbb{Z}}$-grading on $C$ by

$C=\bigoplus_{n\in\operatorname{\mathbb{Z}}}C_{n}.$

This grading corresponds to a $\operatorname{\mathbb{G}_{m}}$-action on
$\operatorname{Spec}C$ that is obtained by a map
$\operatorname{\mathbb{G}_{m}}\to(\operatorname{\mathbb{G}_{m}})^{2},\alpha\mapsto(\alpha,\alpha^{-1})$
and the natural $(\operatorname{\mathbb{G}_{m}})^{2}$-action on
$\operatorname{Spec}C$ coming from the original bi-grading.

Then we can take the geometric invariant theory quotients

$Y_{+}:=\operatorname{Proj}(C_{+})$, $Y_{-}:=\operatorname{Proj}(C_{-})$, and
$X:=\operatorname{Proj}C_{0}$,

where

$\text{$C_{+}:=\bigoplus_{n\geq 0}C_{n}$ and $C_{-}:=\bigoplus_{n\leq
0}C_{n}$}.$

The projective quotients $Y_{+}$ and $Y_{-}$ are the total spaces of rank two
vector bundles on $\operatorname{\mathbf{G}}$ and $\operatorname{\mathbf{Q}}$
respectively. The affinization morphism $\phi_{+}:Y_{+}\to X$ and
$\phi_{-}:Y_{-}\to X$ are small resolution of the singular affine variety $X$
and they contract the zero-sections. Furthermore we can show that the
birational map $Y_{+}\dashrightarrow Y_{-}$ is a $7$-dimensional simple flop
with an interesting feature that the contraction loci are not isomorphic to
each other.

The author first learned this interesting flop from Abuaf. Later the author
noticed that the same flop was found by Ueda independently [Ued16]. Thus the
author would like to attribute this new flop to both of them, and would like
to call this flop the Abuaf-Ueda flop.

Ueda confirmed that Bondal-Orlov conjecture is true for the Abuaf-Ueda flop,
using the theory of semi-orthogonal decompositions and their mutations.
However, since there are many other methods to construct an equivalence
between derived categories, it is still an interesting problem to prove the
derived equivalence using other methods.

###  1.2. Results in this article

The main purpose of this article to construct tilting bundles on both sides of
the flop $Y_{+}\dashrightarrow Y_{-}$, and construct equivalences between the
derived categories of $Y_{+}$ and $Y_{-}$ using those tilting bundles. A
tilting bundle $T_{\ast}$ on $Y_{\ast}$ ($\ast\in\\{+,-\\}$) is a vector
bundle on $Y_{\ast}$ that gives an equivalence

$\operatorname{RHom}_{Y_{\ast}}(T_{\ast},-):\operatorname{D^{b}}(Y_{\ast})\to\operatorname{D^{b}}(\operatorname{End}_{Y_{\ast}}(T_{\ast}))$

between two derived categories. In particular, if we find tilting bundles
$T_{+}$ and $T_{-}$ with the same endomorphism ring, then we have an
equivalence $\operatorname{D^{b}}(Y_{+})\simeq\operatorname{D^{b}}(Y_{-})$ as
desired.

The advantage of this method is that it enables us to study a flop from the
point of view of the theory of non-commutative crepant resolutions (= NCCRs)
that is first introduced by Van den Bergh [VdB04]. In our case, an NCCR
appears as the endomorphism algebra $\operatorname{End}_{Y_{\ast}}(T_{\ast})$
of a tilting bundle $T_{\ast}$. Via the theory of NCCRs, we also study the
Abuaf-Ueda flop from the moduli-theoretic point of view.

Recall that $Y_{+}$ and $Y_{-}$ are the total spaces of rank two vector
bundles on $\operatorname{\mathbf{G}}$ and $\operatorname{\mathbf{Q}}$
respectively. If there is a variety $Z$ that gives a rational resolution of an
affine singular variety and that is the total space of a vector bundle on a
projective variety $W$ admitting a tilting bundle $T$, it is natural to expect
that the pull back of $T$ via the projection $Z\to W$ gives a tilting bundle
on $Z$. Indeed, in many known examples, we can produce tilting bundles in such
a way [BLV10, H17a, WZ12].

However, in our case, we cannot obtain tilting bundles on $Y_{+}$ or $Y_{-}$
as a pull back of known tilting bundles on $\operatorname{\mathbf{G}}$ or
$\operatorname{\mathbf{Q}}$. Thus the situation is different from previous
works. Nevertheless, by modifying bundles that are obtained from tilting
bundles on the base $\operatorname{\mathbf{G}}$ or
$\operatorname{\mathbf{Q}}$, we can find tilting bundles on $Y_{+}$ and
$Y_{-}$. Namely, tilting bundles we construct are the direct sum of
indecomposable bundles that are obtained by taking extensions of other bundles
obtained from $\operatorname{\mathbf{G}}$ or $\operatorname{\mathbf{Q}}$. We
can also check that they produce derived equivalences
$\operatorname{D^{b}}(Y_{+})\simeq\operatorname{D^{b}}(Y_{-})$.

###  1.3. Related works

If we apply a similar construction to Dynkin diagrams $A_{2}$ or $C_{2}$, then
we have the four-dimensional Mukai flop or the (five-dimensional) Abuaf flop
[Seg16] respectively. Therefore this article is a sequel of papers [H17a,
Seg16, H17b].

Recently, Kanemitsu [Kan18] classified simple flops of dimension up to eight,
which is a certain generalization of the theorem of Li [Li17]. It is
interesting to prove the derived equivalence for all simple flops that appear
in Kanemitsu’s list using tilting bundles, and we can regard this article as a
part of such a project.

This flop is also related to certain (compact) Calabi-Yau threefolds which are
studied in [IMOU16a, IMOU16b, Kuz18]. Let us consider the (geometric) vector
bundle $Y_{+}\to\operatorname{\mathbf{G}}$ over $\operatorname{\mathbf{G}}$.
Then as a zero-locus of a regular section of this bundle we have a smooth
Calabi-Yau threefold $V_{+}$ in $\operatorname{\mathbf{G}}$. Similarly, we can
construct another Calabi-Yau threefold $V_{-}$ in $\operatorname{\mathbf{Q}}$.
Papers [IMOU16b, Kuz18] show that Calabi-Yau threefolds $V_{+}$ and $V_{-}$
are L-equivalent, derived equivalent but NOT birationally equivalent to each
other. (L-equivalence and non-birationality is due to [IMOU16b], and derived
equivalence is due to [Kuz18].) As explained in [Ued16], we can construct a
derived equivalence
$\operatorname{D^{b}}(V_{+})\xrightarrow{\sim}\operatorname{D^{b}}(V_{-})$ for
Calabi-Yau threefolds from a derived equivalence
$\operatorname{D^{b}}(Y_{+})\xrightarrow{\sim}\operatorname{D^{b}}(Y_{-})$
with a certain nice property.

###  1.4. Open questions

It would be interesting to compare the equivalences in this article and the
one constructed by Ueda. It is also interesting to find Fourier-Mukai kernels
that give equivalences. In the case of the Mukai flop or the Abuaf flop, the
structure sheaf of the fiber product $Y_{+}\times_{X}Y_{-}$ over the
singularity $X$ gives a Fourier-Mukai kernel of an equivalence (see [Kaw02,
Nam03, H17b]). Thus it is interesting to ask whether this fact remains to hold
or not for the Abuaf-Ueda flop.

Another interesting topic is to study the autoequivalence group of the derived
category. Since we produce some derived equivalences that are different to
each other in this article, we can find some non-trivial autoequivalences by
combining them. It would be interesting to find an action of an interesting
group on the derived category of $Y_{+}$ (and $Y_{-}$) that contains our
autoequivalences.

Acknowledgements. The author would like to express his sincere gratitude to
his supervisor Professor Yasunari Nagai and Professor Michel Van den Bergh for
their advice and encouragement. The author is grateful to Professor Roland
Abuaf for letting him know about this interesting flop, for careful reading of
the first version of this article and for giving useful comments. The author
would like to thank Professor Shinnosuke Okawa for letting him know about the
paper [Ued16]. The author also would like to thank Professors Hajime Kaji and
Yuki Hirano for their interest and comments. A part of this work was done
during the author’s stay in Hasselt University. The author would like to thank
Hasselt University for the hospitality and excellent working condition. This
work is supported by Grant-in-Aid for JSPS Research Fellow 17J00857.

##  2\. Preliminaries

###  2.1. Tilting bundle and derived category

First we prepare some basic terminologies and facts about tilting bundles.

######  Definition 2.1.

Let $Y$ be a quasi-projective variety and $T$ a vector bundle (of finite rank)
on $Y$. Then we say that $T$ is partial tilting if
$\operatorname{Ext}_{Y}^{\geq 1}(T,T)=0$. We say that a partial tilting bundle
$T$ on $Y$ is tilting if $T$ is a generator of the unbounded derived category
$\mathrm{D}(\operatorname{Qcoh}(Y))$, i.e. if an object
$E\in\mathrm{D}(\operatorname{Qcoh}(Y))$ satisfies
$\operatorname{RHom}(T,E)\simeq 0$ then $E\simeq 0$.

If we find a tilting bundle on a projective scheme (over an affine variety),
we can construct a derived equivalence between the derived category of the
scheme and the derived category of a non-commutative algebra obtained as the
endomorphism ring of the bundle.

######  Proposition 2.2.

Let $Y$ be a projective scheme over an affine scheme $\operatorname{Spec}(R)$.
Assume that $Y$ admits a tilting bundle $T$. Then we have the following
derived equivalence

$\operatorname{RHom}_{Y}(T,-):\operatorname{D^{b}}(Y)\to\operatorname{D^{b}}(\operatorname{End}_{Y}(T)).$

These equivalences coming from tilting bundles are very useful to construct
equivalences between the derived categories of two crepant resolutions.

######  Lemma 2.3.

Let $X=\operatorname{Spec}R$ be a normal Gorenstein affine variety of
dimension greater than or equal to two, and let $\phi:Y\to X$ and
$\phi^{\prime}:Y^{\prime}\to X$ be two crepant resolutions of $X$. Put
$U:=X_{\mathrm{sm}}=Y\setminus\operatorname{exc}(\phi)=Y^{\prime}\setminus\operatorname{exc}(\phi^{\prime})$.
Assume that there are tilting bundles $T$ and $T^{\prime}$ on $Y$ and
$Y^{\prime}$, respectively, such that

$T|_{U}\simeq T^{\prime}|_{U}.$

Then there is a derived equivalence

$\operatorname{D^{b}}(Y)\simeq\operatorname{D^{b}}(\operatorname{End}_{Y}(T))\simeq\operatorname{D^{b}}(\operatorname{End}_{Y^{\prime}}(T^{\prime}))\simeq\operatorname{D^{b}}(Y^{\prime}).$

###### Proof.

See [H17c, Lemma 3.4]. ∎

The existence of a tilting bundle on a crepant resolution does not hold in
general. For this fact, see [IW14, Theorem 4.20]. In addition, even in the
case that a tilting bundle exists, it is still non-trivial to construct a
tilting bundle explicitly. The following lemma is very useful to find a
tilting bundle.

######  Lemma 2.4.

Let $\\{E_{i}\\}_{i=1}^{n}$ be a collection of vector bundles on a quasi-
projective scheme $Y$. Assume that

  1. (i)

The direct sum $\bigoplus_{i=1}^{n}E_{i}$ is a generator of
$\mathrm{D}(\operatorname{Qcoh}(Y))$.

  2. (ii)

There is no former $\operatorname{Ext}^{\geq 1}_{Y}$, i.e.
$\operatorname{Ext}_{Y}^{\geq 1}(E_{i},E_{j})=0$ for $i\leq j$. In particular,
this assumption implies that $E_{i}$ is a partial tilting bundle for any $i$.

  3. (iii)

There is no backward $\operatorname{Ext}^{\geq 2}_{Y}$, i.e.
$\operatorname{Ext}_{Y}^{\geq 2}(E_{i},E_{j})=0$ for $i>j$.

Then there exists a tilting bundle on $Y$.

###### Proof.

We use an induction on $n$. If $n=1$, the statement is trivial. Let $n>1$.
Choose generators of $\operatorname{Ext}_{Y}^{1}(E_{1},E_{2})$ as a right
$\operatorname{End}_{Y}(E_{1})$-module, and let $r$ be the number of the
generators. Then we can take the corresponding sequence

$0\to E_{2}\to F\to E_{1}^{\oplus r}\to 0.$

Then we have $\operatorname{Ext}_{Y}^{\geq 1}(E_{1},F)=0$. Indeed, if we apply
the functor $\operatorname{Ext}_{Y}^{i}(E_{1},-)$ to the sequence above, we
have the long exact sequence

$\cdots\to\operatorname{End}_{Y}(E_{1})^{\oplus
r}\xrightarrow{\delta}\operatorname{Ext}_{Y}^{1}(E_{1},E_{2})\to\operatorname{Ext}^{1}_{Y}(E_{1},F)\to\operatorname{Ext}_{Y}^{1}(E_{1},E_{1}^{\oplus
r})=0\to\cdots.$

Now $\delta$ is surjective by construction, and hence we have

$\operatorname{Ext}_{Y}^{\geq 1}(E_{1},F)=0.$

Applying the derived functor $\operatorname{Ext}_{Y}^{i}(E_{2},-)$ to the same
short exact sequence, we have $\operatorname{Ext}^{\geq 1}(E_{2},F)=0$ from
the assumption that there is no former $\operatorname{Ext}^{\geq 1}$. Thus we
have $\operatorname{Ext}_{Y}^{\geq 1}(F,F)=0$. One can also show that
$\operatorname{Ext}_{Y}^{\geq 1}(F,E_{1})=0$, and therefore $E_{1}\oplus F$ is
a partial tilting bundle.

Put $E_{1}^{\prime}=E_{1}\oplus F$ and $E^{\prime}_{i}=E_{i-1}$ for $1<i<n$.
Then it is easy to see that the new collection
$\\{E^{\prime}_{i}\\}_{i=1}^{n-1}$ satisfies the assumptions (i), (ii) and
(iii). Note that the condition (i) holds since the new collection
$\\{E^{\prime}_{i}\\}_{i=1}^{n-1}$ split-generates the original collection
$\\{E_{i}\\}_{i=1}^{n}$. Thus we have the result by the assumption of the
induction. ∎

###  2.2. Geometry and representation theory

Next we recall the representation theory and the geometry of homogeneous
varieties we need. We also explain the geometric aspect of the Abuaf-Ueda flop
in the present subsection.

####  2.2.1. Representation of $G_{2}$

In the present subsection, we recall the representation theory of the semi-
simple algebraic group of type $G_{2}$. We need the representation theory when
we compute cohomologies of homogeneous vector bundles using Borel-Bott-Weil
theorem in Section 3.

Let $V=\\{(x,y,z)\in\operatorname{\mathbb{R}}^{3}\mid x+y+z=0\\}$ be a
hyperplane in $\operatorname{\mathbb{R}}^{3}$. Then the $G_{2}$ root system in
$V$ is the following collection of twelve vectors in $V$.

$\Delta=\\{(0,\pm 1,\mp 1),(\pm 1,0,\mp 1),(\pm 1,\mp 1,0),(\pm 2,\mp 1,\mp
1),(\mp 1,\pm 2,\mp 1),(\mp 1,\mp 1,\pm 2)\\}.$

The vector in $\Delta$ is called root. Especially,

$\alpha_{1}=(1,-1,0)$ and $\alpha_{2}=(-2,1,1)$

are called simple roots, and we say that a root $\alpha\in\Delta$ is a
positive root if $\alpha=a\alpha_{1}+b\alpha_{2}$ for some $a\geq 0$ and
$b\geq 0$.

By definition, the fundamental weights $\\{\pi_{1},\pi_{2}\\}\subset V$ are
the set vectors in $V$ such that

$\langle\alpha_{i},\pi_{j}\rangle=\delta_{ij}.$

Here the parting $\langle-,-\rangle$ is a usual one

$\langle(a,b,c),(x,y,z)\rangle:=ax+by+cz.$

An easy computation shows that

$\pi_{1}=(0,-1,1)$ and $\pi_{2}=(-\frac{1}{3},-\frac{1}{3},\frac{2}{3})$.

The lattice
$L=\operatorname{\mathbb{Z}}\pi_{1}+\operatorname{\mathbb{Z}}\pi_{2}$ in $V$
generated by $\pi_{1}$ and $\pi_{2}$ is called the weight lattice of $G_{2}$,
and a vector in this lattice is called a weight. We call an weight of the form
$a\pi_{1}+b\pi_{2}$ for $a,b\in\operatorname{\mathbb{Z}}_{\geq 0}$ a dominant
weight. The set of dominant weights plays a central role in the representation
theory because they corresponds to irreducible representations.

Let $\alpha\in\Delta$ be a root. Then we can consider the reflection
$\mathrm{S}_{\alpha}$ defined by the root $\alpha$. That is a linear map
$\mathrm{S}_{\alpha}:V\to V$ defined as

$\mathrm{S}_{\alpha}(v):=v-\frac{2\langle\alpha,v\rangle}{\langle\alpha,\alpha\rangle}\alpha.$

The Weyl group $W$ is defined by a subgroup of the orthogonal group
$\mathrm{O}(V)$ generated by $\mathrm{S}_{\alpha}$ for $\alpha\in\Delta$:

$W:=\langle\mathrm{S}_{\alpha}\mid\alpha\in\Delta\rangle\subset\mathrm{O}(V).$

It is known that $W$ is generated by two reflections $\mathrm{S}_{\alpha_{1}}$
and $\mathrm{S}_{\alpha_{2}}$ defined by simple roots. Using this generator,
we define the length of an element in $W$ as follow. The length $l(w)$ of an
element $w\in W$ is the smallest number $n$ so that $w$ is a composition of
$n$ reflections by simple roots. In the case of $G_{2}$, the Weyl group $W$
has twelve elements. The Table 1 shows all elements in $W$ and their length.
In that table, we denote
$\mathrm{S}_{\alpha_{i_{k}}}\cdots\mathrm{S}_{\alpha_{i_{2}}}\mathrm{S}_{\alpha_{i_{1}}}$
by $\mathrm{S}_{i_{k}\cdots i_{2}i_{1}}$ for short.

Table 1. Elements of Weyl group and their length element | length
---|---
$1$ | $0$
$\mathrm{S}_{1}$ | $1$
$\mathrm{S}_{2}$ | $1$
$\mathrm{S}_{12}$ | $2$
$\mathrm{S}_{21}$ | $2$
$\mathrm{S}_{121}$ | $3$
$\mathrm{S}_{212}$ | $3$
$\mathrm{S}_{1212}$ | $4$
$\mathrm{S}_{2121}$ | $4$
$\mathrm{S}_{12121}$ | $5$
$\mathrm{S}_{21212}$ | $5$
$\mathrm{S}_{121212}=\mathrm{S}_{212121}$ | $6$

Put $\rho=\pi_{1}+\pi_{2}$. Using this weight, we can define another action of
the Weyl group $W$ on the weight lattice $L$ that is called dot-action. The
dot-action is defined by

$\mathrm{S}_{\alpha}\cdot v:=\mathrm{S}_{\alpha}(v+\rho)-\rho.$

In our $G_{2}$ case, the dot-action is the following affine transform.

$\displaystyle\mathrm{S}_{\alpha_{1}}\cdot(a\pi_{1}+b\pi_{2})$
$\displaystyle=(-a-2)\pi_{1}+(3a+b+3)\pi_{2},$
$\displaystyle\mathrm{S}_{\alpha_{2}}\cdot(a\pi_{1}+b\pi_{2})$
$\displaystyle=(a+b+1)\pi_{1}+(-b-2)\pi_{2}.$

####  2.2.2. Geometry of $G_{2}$-homogeneous varieties

Next we recall the geometry of $G_{2}$-homogeneous varieties.

The $G_{2}$-Grassmannian $\operatorname{\mathbf{G}}=\operatorname{Gr}_{G_{2}}$
is a $5$-dimensional closed subvariety of $\operatorname{Gr}(2,7)$, and has
Picard rank one. The Grassmannian $\operatorname{Gr}(2,7)$ admits the
universal quotient bundle $Q$ of rank $5$ and $G$ is the zero-locus of a
regular section of the bundle $Q^{\vee}(1)$. Since
$\det(Q^{\vee}(1))\simeq\mathcal{O}_{\operatorname{Gr}(2,7)}(4)$ and
$\omega_{\operatorname{Gr}(2,7)}\simeq\mathcal{O}_{\operatorname{Gr}(2,7)}(-7)$,
we have
$\omega_{\operatorname{\mathbf{G}}}\simeq\mathcal{O}_{\operatorname{\mathbf{G}}}(-3)$.
Thus $\operatorname{\mathbf{G}}$ is a five dimensional Fano variety of Picard
rank one and of Fano index three. We denote the restriction of the universal
subbundle on $\operatorname{Gr}(2,7)$ to $\operatorname{\mathbf{G}}$ by $R$.
The bundle $R$ has rank two and
$\det(R)\simeq\mathcal{O}_{\operatorname{\mathbf{G}}}(-1)$. It is known that
the derived category $\operatorname{D^{b}}(\operatorname{\mathbf{G}})$ of
$\operatorname{\mathbf{G}}$ admits a full strong exceptional collection

$\operatorname{D^{b}}(\operatorname{\mathbf{G}})=\langle
R(-1),\mathcal{O}_{\operatorname{\mathbf{G}}}(-1),R,\mathcal{O}_{\operatorname{\mathbf{G}}},R(1),\mathcal{O}_{\operatorname{\mathbf{G}}}(1)\rangle$

(see [Kuz06]). In particular, the variety $\operatorname{\mathbf{G}}$ admits a
tilting bundle

$R(-1)\oplus\mathcal{O}_{\operatorname{\mathbf{G}}}(-1)\oplus
R\oplus\mathcal{O}_{\operatorname{\mathbf{G}}}\oplus
R(1)\oplus\mathcal{O}_{\operatorname{\mathbf{G}}}(1).$

The other $G_{2}$-homogeneous variety of Picard rank one is the five
dimensional quadric variety $\operatorname{\mathbf{Q}}=Q_{5}$. On
$\operatorname{\mathbf{Q}}$ there are two important vector bundles of higher
rank. One is the spinor bundle $S$ on $\operatorname{\mathbf{Q}}$. The spinor
bundle $S$ has rank $4$ and appear in a full strong exceptional collection

$\operatorname{D^{b}}(\operatorname{\mathbf{Q}})=\langle\mathcal{O}_{\operatorname{\mathbf{Q}}}(-2),\mathcal{O}_{\operatorname{\mathbf{Q}}}(-1),S,\mathcal{O}_{\operatorname{\mathbf{Q}}},\mathcal{O}_{\operatorname{\mathbf{Q}}}(1),\mathcal{O}_{\operatorname{\mathbf{Q}}}(2)\rangle.$

The following are important properties of the spinor bundle.

######  Lemma 2.5 ([Ott88]).

For the spinor bundle $S$ on $\operatorname{\mathbf{Q}}$, we have

  1. (1)

$S^{\vee}\simeq S(1)$ and $\det
S\simeq\mathcal{O}_{\operatorname{\mathbf{Q}}}(-2)$.

  2. (2)

There exists an exact sequence

$0\to S\to\mathcal{O}_{\operatorname{\mathbf{Q}}}^{\oplus 8}\to S(1)\to 0.$

Another important vector bundle on $\operatorname{\mathbf{Q}}$ is the Cayley
bundle $C$. The Cayley bundle $C$ is a homogeneous vector bundle of rank two,
and $\det C\simeq\mathcal{O}_{\operatorname{\mathbf{Q}}}(-1)$. Historically,
this bundle was first studied by Ottaviani [Ott90]. Later we will see that the
variety $Y_{-}$ that gives one side of the Abuaf-Ueda flop is the total space
of $C(-2)$.

The $G_{2}$-flag variety $\operatorname{\mathbf{F}}$ is a $6$-dimensional
variety of Picard rank two. There is a projection
$p^{\prime}:\operatorname{\mathbf{F}}\to\operatorname{\mathbf{G}}$, and via
this projection, $\operatorname{\mathbf{F}}$ is isomorphic to the
projectivization of the universal subbundle $R(1)$ (with some line bundle
twist):

$\operatorname{\mathbf{F}}\simeq\operatorname{\mathbb{P}}_{\operatorname{\mathbf{G}}}(R(-1)):=\operatorname{Proj}_{\operatorname{\mathbf{G}}}\operatorname{Sym}^{\bullet}(R(-1))^{\vee}.$

Similarly, via a projection
$q^{\prime}:\operatorname{\mathbf{F}}\to\operatorname{\mathbf{Q}}$, we have

$\operatorname{\mathbf{F}}\simeq\operatorname{\mathbb{P}}_{\operatorname{\mathbf{Q}}}(C(-2)):=\operatorname{Proj}_{\operatorname{\mathbf{Q}}}\operatorname{Sym}^{\bullet}(C(-2))^{\vee}.$

Fix general members
$H\in\lvert(p^{\prime})^{*}\mathcal{O}_{\operatorname{\mathbf{G}}}(1)\rvert$
and
$h\in\lvert(q^{\prime})^{*}\mathcal{O}_{\operatorname{\mathbf{Q}}}(1)\rvert$.
Then we can write

$\mathcal{O}_{\operatorname{\mathbf{F}}}(aH+bh)\simeq\mathcal{O}_{\operatorname{\mathbf{G}}}(a)\boxtimes\mathcal{O}_{\operatorname{\mathbf{Q}}}(b).$

####  2.2.3. Borel-Bott-Weil theorem

For homogeneous vector bundles on homogeneous varieties, we can compute their
sheaf cohomologies using the Borel-Bott-Weil theorem.

######  Theorem 2.6 (Borel-Bott-Weil).

Let $E$ be a homogeneous vector bundle on a projective homogeneous variety $Z$
that corresponds to a weight $\pi$. Then one of the following can happen.

  1. (i)

There exists an element $w$ of the Weyl group $W$ such that $w\cdot\pi$ is a
dominant weight.

  2. (ii)

There exists $w\in W$ such that $w\cdot\pi=\pi$.

Furthermore,

  1. (I)

In the case of (i), we have

$\displaystyle H^{i}(Z,E)\simeq\begin{cases}(V_{S\cdot\pi})^{\vee}&\text{if
$i=l(w)$}\\\ 0&\text{otherwise}\end{cases}$

  2. (II)

In the case of (ii), we have

$\operatorname{R\Gamma}(Z,E)\simeq 0.$

Note that we use the dot-action in this theorem. We also note that the
condition (ii) is equivalent to the condition (ii’) in our case:

  1. (ii’)

$\pi+\rho\in\operatorname{\mathbb{R}}\cdot\alpha$ for some $\alpha\in\Delta$,
where $\operatorname{\mathbb{R}}\cdot\alpha$ is a line spanned by a root
$\alpha$.

On the $G_{2}$-Grassmannian $\operatorname{\mathbf{G}}$, a homogeneous vector
bundle corresponding to a weight $a\pi_{1}+b\pi_{2}$ exists if and only if
$b\geq 0$, and that bundle is $\operatorname{Sym}^{b}(R^{\vee})(a)$. On the
five dimensional quadric $\operatorname{\mathbf{Q}}$, a homogeneous vector
bundle corresponding to a weight $a\pi_{1}+b\pi_{2}$ exists if and only if
$a\geq 0$, and that bundle is $\operatorname{Sym}^{a}(C^{\vee})(a+b)$. On the
flag variety $\operatorname{\mathbf{F}}$, a line bundle
$\mathcal{O}_{\operatorname{\mathbf{F}}}(aH+bh)$ corresponds to a weight
$a\pi_{1}+b\pi_{2}$. Thus we can compute the cohomology of these bundles using
the Borel-Bott-Weil theorem.

####  2.2.4. Geometry of the Abuaf-Ueda flop

We explain the geometric description of the Abuaf-Ueda flop. First as
explained in [Ued16], $Y_{+}$ is the total space of a vector bundle $R(-1)$ on
$\operatorname{\mathbf{G}}$. Since
$\det(R(-1))\simeq\mathcal{O}_{\operatorname{\mathbf{G}}}(-3)\simeq\omega_{\operatorname{\mathbf{G}}}$,
the variety $Y_{+}$ is local Calabi-Yau of dimension seven.

The other side of the flop $Y_{-}$ is also a total space of a vector bundle of
rank two on $\operatorname{\mathbf{Q}}$. The bundle is $C(-2)$. Note that
$\det(C(-2))\simeq\mathcal{O}_{\operatorname{\mathbf{Q}}}(-5)\simeq\omega_{\operatorname{\mathbf{Q}}}$.

Let $\operatorname{\mathbf{G}}_{0}\subset Y_{+}$ and
$\operatorname{\mathbf{Q}}_{0}\subset Y_{-}$ be the zero-sections. Then the
blowing-ups of these zero-sections give the same variety

$\mathrm{Bl}_{\operatorname{\mathbf{G}}_{0}}(Y_{+})\simeq\mathrm{Bl}_{\operatorname{\mathbf{Q}}_{0}}(Y_{-})=:Y,$

and exceptional divisors of $p:Y\to Y_{+}$ and $q:Y\to Y_{-}$ are same, which
we denote by $E$. There is a morphism $Y\to\operatorname{\mathbf{F}}$, and via
this morphism, $Y$ is isomorphic to the total space of
$\mathcal{O}_{\operatorname{\mathbf{F}}}(-H-h)$. The zero-section
$\operatorname{\mathbf{F}}_{0}$ (via this description of $Y$) is the
exceptional divisor $E$.

Thus we have the following diagram

${E}$${\operatorname{\mathbf{F}}_{0}}$${Y}$${\operatorname{\mathbf{G}}_{0}}$${Y_{+}}$${Y_{-}}$${\operatorname{\mathbf{Q}}_{0}}$${\operatorname{\mathbf{G}}}$${X}$${\operatorname{\mathbf{Q}}.}$$\scriptstyle{p^{\prime}}$$\scriptstyle{q^{\prime}}$$\scriptstyle{p}$$\scriptstyle{q}$$\scriptstyle{\pi_{+}}$$\scriptstyle{\phi_{+}}$flop$\scriptstyle{\pi_{-}}$$\scriptstyle{\phi_{-}}$

Using projections $\pi_{+}:Y_{+}\to\operatorname{\mathbf{G}}$ and
$\pi_{-}:Y_{-}\to\operatorname{\mathbf{Q}}$, we define vector bundles

$\mathcal{O}_{Y_{+}}(a):=\pi_{+}^{*}\mathcal{O}_{\operatorname{\mathbf{G}}}(a)$
and $\mathcal{R}:=\pi_{+}^{*}R$

on $Y_{+}$ and

$\mathcal{O}_{Y_{-}}(a):=\pi_{-}^{*}\mathcal{O}_{\operatorname{\mathbf{Q}}}(a)$
and $\operatorname{\mathcal{S}}:=\pi_{-}^{*}S$

on $Y_{-}$. As for $\operatorname{\mathbf{F}}$, we define

$\mathcal{O}_{Y}(aH+bh):=\mathcal{O}_{Y_{+}}(a)\boxtimes\mathcal{O}_{Y_{-}}(b).$

By construction a line bundle $\mathcal{O}_{Y}(aH+bh)$ coincides with the
pull-back of $\mathcal{O}_{\operatorname{\mathbf{F}}}(aH+bh)$ by the
projection $Y\to\operatorname{\mathbf{F}}$ and thus we have

$\mathcal{O}_{Y}(E)\simeq\mathcal{O}_{Y}(-H-h).$

##  3\. Tilting bundles and derived equivalences

###  3.1. Tilting bundles on $Y_{+}$

First, we construct tilting bundles on $Y_{+}$. Recall that the derived
category $\operatorname{D^{b}}(\operatorname{\mathbf{G}})$ has an exceptional
collection

$R(-1),\mathcal{O}_{\operatorname{\mathbf{G}}}(-1),R,\mathcal{O}_{\operatorname{\mathbf{G}}},R(1),\mathcal{O}_{\operatorname{\mathbf{G}}}(1),$

where $R$ is the universal subbundle. Pulling back this collection, we have a
collection of vector bundles on $Y_{+}$ that is

$\mathcal{R}(-1),\mathcal{O}_{Y_{+}}(-1),\mathcal{R},\mathcal{O}_{Y_{+}},\mathcal{R}(1),\mathcal{O}_{Y_{+}}(1).$

The direct sum of these vector bundles gives a generator of
$\mathrm{D}(\operatorname{Qcoh}(Y_{+}))$ by the following Lemma 3.1. However,
the following Proposition 3.2 shows that the direct sum of these vector
bundles is NOT a tilting bundle on $Y_{+}$.

######  Lemma 3.1.

Let $\pi:Z\to W$ be an affine morphism and
$E\in\mathrm{D}(\operatorname{Qcoh}(W))$ is a generator. Then the derived pull
back $L\pi^{*}(E)$ is a generator of $\mathrm{D}(\operatorname{Qcoh}(Z))$.

###### Proof.

Let $F\in\mathrm{D}(\operatorname{Qcoh}(Z))$ be an object with
$\operatorname{RHom}_{Z}(L\pi^{*}(E),F)=0$. Then since
$\operatorname{RHom}_{Z}(L\pi^{*}(E),F)=\operatorname{RHom}_{W}(E,R\pi_{*}(F))$
and $E$ is a generator, we have $R\pi_{*}(F)=0$. The affineness of the
morphism $\pi$ implies $F=0$. ∎

######  Proposition 3.2.

We have

  1. (1)

$H^{\geq 1}(Y_{+},\mathcal{O}_{Y_{+}}(a))=0$ for all $a\geq-2$.

  2. (2)

$H^{\geq 1}(Y_{+},\mathcal{R}(a))=0$ for $a\geq-2$.

  3. (3)

$\operatorname{Ext}_{Y_{+}}^{\geq 1}(\mathcal{R},\mathcal{O}_{Y_{+}}(a))=0$
for $a\geq-3$.

  4. (4)

$\operatorname{Ext}_{Y_{+}}^{\geq 1}(\mathcal{R},\mathcal{R}(a))=0$ for
$a\geq-1$.

  5. (5)

$\operatorname{Ext}_{Y_{+}}^{\geq 2}(\mathcal{R},\mathcal{R}(-2))=0$ and
$\operatorname{Ext}_{Y_{+}}^{1}(\mathcal{R},\mathcal{R}(-2))\simeq\operatorname{\mathbb{C}}$.

###### Proof.

Here we prove (4) and (5) only. Other cases follow from similar (and easier)
computations.

Let $a\geq-2$ and $i\geq 1$. Since there are irreducible decompositions

$\displaystyle\mathcal{R}^{\vee}\otimes\mathcal{R}(a)\simeq(\operatorname{Sym}^{2}\mathcal{R}^{\vee})(a-1)\oplus\mathcal{O}_{Y_{-}}(a),$

we have

$\operatorname{Ext}_{Y_{+}}^{i}(\mathcal{R},\mathcal{R}(a))\simeq
H^{i}(Y_{+},\operatorname{Sym}^{2}\mathcal{R}^{\vee}(a-1))\oplus
H^{i}(Y_{+},\mathcal{O}_{Y_{+}}(a)).$

The second term of this decomposition is zero by (1), and hence we have

$\displaystyle\operatorname{Ext}_{Y_{+}}^{i}(\mathcal{R},\mathcal{R}(a))$
$\displaystyle\simeq
H^{i}(Y_{+},\operatorname{Sym}^{2}\mathcal{R}^{\vee}(a-1))$
$\displaystyle\simeq
H^{i}\left(\operatorname{\mathbf{G}},(\operatorname{Sym}^{2}R^{\vee})(a-1)\otimes\bigoplus_{k\geq
0}(\operatorname{Sym}^{k}R^{\vee})(k)\right)$

by adjunction. To compute this cohomology, we use the following decomposition

$\displaystyle(\operatorname{Sym}^{k}R^{\vee})(k)\otimes(\operatorname{Sym}^{2}R^{\vee})(a-1)\simeq$
$\displaystyle\begin{cases}(\operatorname{Sym}^{k+2}R^{\vee})(k+a-1)\oplus(\operatorname{Sym}^{k}R^{\vee})(k+a)\oplus(\operatorname{Sym}^{k-2}R^{\vee})(k+a+1)&\text{if
$k\geq 2$}\\\
(\operatorname{Sym}^{3}R^{\vee})(a)\oplus(\operatorname{Sym}^{1}R^{\vee})(a+1)&\text{if
$k=1$}\\\ (\operatorname{Sym}^{2}R^{\vee})(a-1)&\text{if $k=0$.}\\\
\end{cases}$

According to this irreducible decomposition, it is enough to compute the
cohomology of the following vector bundles.

  1. (i)

$(\operatorname{Sym}^{k+2}R^{\vee})(k+a-1)$ for $k\geq 0$ and $a\geq-2$.

  2. (ii)

$(\operatorname{Sym}^{k}R^{\vee})(k+a)$ for $k\geq 1$ and $a\geq-2$.

  3. (iii)

$(\operatorname{Sym}^{k-2}R^{\vee})(k+a+1)$ for $k\geq 2$ and $a\geq-2$.

To compute the cohomology of these bundles, we use the Borel-Bott-Weil
theorem. A bundle of type (i) corresponds to a weight
$(k+a-1)\pi_{1}+(k+2)\pi_{2}$. This weight is dominant if and only if $k+a\geq
1$, i.e.

$(k,a)\notin\\{(0,-2),(0,-1),(0,0),(1,-2),(1,-1),(2,-2)\\}.$

In this case the bundle has no higher cohomology. If $(k,a)=(0,-2)$, then we
have

$-3\pi_{1}+2\pi_{2}+\rho=-2\pi_{1}+3\pi_{2}=(0,2,-2)+(-1,-1,2)=(-1,1,0)$

and this vector is a root. Thus we have that the corresponding bundle is
acyclic, i.e.

$\operatorname{R\Gamma}(\operatorname{\mathbf{G}},\operatorname{Sym}^{2}R^{\vee}(-3))=0.$

One can show that the same things hold for $(k,a)=(0,-1),(0,0),(1,-1),(2,-2)$.
Let us compute the case if $(k,a)=(1,-2)$. In this case we have

$\mathrm{S}_{\alpha_{1}}\cdot(-2\pi_{1}+3\pi_{2})=0.$

Thus the Borel-Bott-Weil theorem implies

$\operatorname{R\Gamma}(\operatorname{\mathbf{G}},(\operatorname{Sym}^{3}R^{\vee})(-2))\simeq\operatorname{\mathbb{C}}[-1].$

Using the Borel-Bott-Weil theorem in the same way, we can show that bundles of
type (ii) and (iii) have no higher cohomology. This shows (4) and (5). ∎

######  Definition 3.3.

Let $\Sigma$ be a rank $4$ vector bundle on $Y_{+}$ that lies in the following
unique non-trivial extension

$0\to\mathcal{R}(-1)\to\Sigma\to\mathcal{R}(1)\to 0.$

Now we can show that the bundle $\Sigma$ is partial tilting and that a bunde

$\mathcal{O}_{Y_{+}}(-1)\oplus\mathcal{O}_{Y_{+}}\oplus\mathcal{O}_{Y_{+}}(1)\oplus\mathcal{R}\oplus\mathcal{R}(1)\oplus\Sigma$

is a tilting bundle on $Y_{+}$ as in the proof of Lemma 2.4. We also note that
the dual $\Sigma^{\vee}$ of $\Sigma$ is isomorphic to $\Sigma(1)$. Indeed, the
bundle $\Sigma^{\vee}$ lies in the sequence

$0\to\mathcal{R}^{\vee}(-1)\to\Sigma^{\vee}\to\mathcal{R}^{\vee}(1)\to 0.$

The isomorphism $\mathcal{R}^{\vee}\simeq\mathcal{R}(1)$ and the uniqueness of
such a non-trivial extension imply that $\Sigma^{\vee}\simeq\Sigma(1)$.

We can apply the same method to another collection

$\mathcal{O}_{Y_{+}}(-1),\mathcal{R},\mathcal{O}_{Y_{+}},\mathcal{R}(1),\mathcal{O}_{Y_{+}}(1),\mathcal{R}(2),$

and then we get another tilting bundle. As a consequence, we have the
following.

######  Theorem 3.4.

The following vector bundles on $Y_{+}$ are tilting bundles.

  1. (1)

$T_{+}^{\spadesuit}:=\mathcal{O}_{Y_{+}}(-1)\oplus\mathcal{O}_{Y_{+}}\oplus\mathcal{O}_{Y_{+}}(1)\oplus\mathcal{R}\oplus\mathcal{R}(1)\oplus\Sigma$

  2. (2)

$T_{+}^{\clubsuit}:=\mathcal{O}_{Y_{+}}(-1)\oplus\mathcal{O}_{Y_{+}}\oplus\mathcal{O}_{Y_{+}}(1)\oplus\mathcal{R}\oplus\mathcal{R}(1)\oplus\Sigma(1)$

  3. (3)

$T_{+}^{\heartsuit}:=\mathcal{O}_{Y_{+}}(-1)\oplus\mathcal{O}_{Y_{+}}\oplus\mathcal{O}_{Y_{+}}(1)\oplus\mathcal{R}(-1)\oplus\mathcal{R}\oplus\Sigma$

  4. (4)

$T_{+}^{\diamondsuit}:=\mathcal{O}_{Y_{+}}(-1)\oplus\mathcal{O}_{Y_{+}}\oplus\mathcal{O}_{Y_{+}}(1)\oplus\mathcal{R}(1)\oplus\mathcal{R}(2)\oplus\Sigma(1)$

Note that the pair $T_{+}^{\spadesuit}$ and $T_{+}^{\clubsuit}$ are dual to
each other, and the pair $T_{+}^{\heartsuit}$ and $T_{+}^{\diamondsuit}$ are
dual to each other.

###  3.2. Tilting bundles on $Y_{-}$

To find explicit tilting bundles on $Y_{-}$, we need to use not only the
Borel-Bott-Weil theorem but also some geometry of the flop. Recall that the
derived category $\operatorname{D^{b}}(\operatorname{\mathbf{Q}})$ has an
exceptional collection

$\mathcal{O}_{\operatorname{\mathbf{Q}}}(-2),\mathcal{O}_{\operatorname{\mathbf{Q}}}(-1),S,\mathcal{O}_{\operatorname{\mathbf{Q}}},\mathcal{O}_{\operatorname{\mathbf{Q}}}(1),\mathcal{O}_{\operatorname{\mathbf{Q}}}(2),$

where $S$ is the rank $4$ spinor bundle on the five dimensional quadric
$\operatorname{\mathbf{Q}}$. Pulling back this collection by the projection
$\pi_{-}:Y_{-}\to\operatorname{\mathbf{Q}}$, we have a collection of vector
bundles on $Y_{-}$

$\mathcal{O}_{Y_{-}}(-2),\mathcal{O}_{Y_{-}}(-1),\operatorname{\mathcal{S}},\mathcal{O}_{Y_{-}},\mathcal{O}_{Y_{-}}(1),\mathcal{O}_{Y_{-}}(2).$

The direct sum of these vector bundles is a generator of
$\mathrm{D}(\operatorname{Qcoh}(Y))$ by Lemma 3.1, but does NOT give a tilting
bundle on $Y_{-}$. First, we compute cohomologies of line bundles.

######  Proposition 3.5.

  1. (1)

$H^{\geq 1}(Y_{-},\mathcal{O}_{Y_{-}}(a))=0$ for all $a\geq-2$.

  2. (2)

$H^{\geq 2}(Y_{-},\mathcal{O}_{Y_{-}}(a))=0$ for all $a\geq-4$.

  3. (3)

$H^{1}(Y_{-},\mathcal{O}_{Y_{-}}(-3))=\operatorname{\mathbb{C}}$.

###### Proof.

Let $a\geq-4$. We have the following isomorphism by adjunction

$H^{i}(Y_{-},\mathcal{O}_{Y_{-}}(a))\simeq\bigoplus_{k\geq
0}H^{i}(\operatorname{\mathbf{Q}},(\operatorname{Sym}^{k}C^{\vee})(2k+a)).$

A bundle $(\operatorname{Sym}^{k}C^{\vee})(2k+a)$ corresponds to a weight
$k\pi_{1}+(k+a)\pi_{2}$. This weight is dominant if and only if $k+a\geq 0$,
i.e.

$(k,a)\notin\\{(0,-4),(0,-3),(0,-2),(0,-1),(1,-4),(1,-3),(1,-2),(2,-4),(2,-3),(3,-4)\\}.$

In this case, the corresponding vector bundle has no higher cohomologies. If
$k=0$ and $a\leq-1$, then the corresponding bundle is an acyclic line bundle
$\mathcal{O}_{\operatorname{\mathbf{Q}}}(a)$.

Let us consider remaining cases. If $(k,a)=(1,-4),(1,-2),(2,-3),(3,-4)$ then

$k\pi_{1}+(k+a)\pi_{2}+\rho=(k+1)\pi_{1}+(k+a+1)\pi_{2}=\begin{cases}2\pi_{1}-2\pi_{2}=(\frac{2}{3},-\frac{4}{3},\frac{2}{3})&\text{if
$(k,a)=(1,-4)$}\\\ 2\pi_{1}&\text{if $(k,a)=(1,-2)$}\\\ 3\pi_{1}&\text{if
$(k,a)=(2,-3)$}\\\ 4\pi_{1}&\text{if $(k,a)=(3,-4)$}\\\ \end{cases}$

and the weight lies in a line spanned by a root. Thus the corresponding bundle
is acyclic in those cases. If $(k,a)=(1,-3)$ then we have

$\mathrm{S}_{\alpha_{2}}\cdot(\pi_{1}-2\pi_{2})=0$

and hence we obtain

$\operatorname{R\Gamma}(\operatorname{\mathbf{Q}},C^{\vee}(-1))\simeq\operatorname{\mathbb{C}}[-1].$

If $(k,a)=(2,-4)$ then we have

$\mathrm{S}_{\alpha_{2}}\cdot(2\pi_{1}-2\pi_{2})=\pi_{1}$

and thus we get

$\operatorname{R\Gamma}(\operatorname{\mathbf{Q}},\operatorname{Sym}^{2}C^{\vee})\simeq
V_{\pi_{1}}^{\vee}[-1].$

This shows the result. ∎

######  Definition 3.6.

Let $\mathcal{P}$ be the rank $2$ vector bundle on $Y_{-}$ which lies in the
following unique non-trivial extension

$0\to\mathcal{O}_{Y_{-}}(-2)\to\mathcal{P}\to\mathcal{O}_{Y_{-}}(1)\to 0.$

One can show that the bundle $\mathcal{P}$ is partial tilting as in Lemma 2.4.
Note that, by the uniqueness of such a non-trivial sequence, we have
$\mathcal{P}^{\vee}\simeq\mathcal{P}(1)$.

######  Proposition 3.7.

We have $H^{\geq 1}(Y_{-},\mathcal{P}(a))=0$ for $a\geq-2$.

To prove this Proposition, we have to use the geometry of the flop. The
following two lemmas are important.

######  Lemma 3.8.

On the full flag variety $\operatorname{\mathbf{F}}$, there is an exact
sequence of vector bundles

$0\to\mathcal{O}_{\operatorname{\mathbf{F}}}(-h)\to
p^{\prime*}R\to\mathcal{O}_{\operatorname{\mathbf{F}}}(-H+h)\to 0$

###### Proof.

See [Kuz18]. ∎

######  Lemma 3.9.

There is an isomorphism $\mathcal{P}\simeq Rq_{*}(p^{*}\mathcal{R}(-h))$.

###### Proof.

From the lemma above, we have an exact sequence on $Y$

$0\to\mathcal{O}_{Y}(-h)\to p^{*}\mathcal{R}\to\mathcal{O}_{Y}(-H+h)\to 0.$

Since $\mathcal{O}_{Y}(E)\simeq\mathcal{O}_{Y}(-H-h)$, we have an exact
sequence

$0\to\mathcal{O}_{Y}(-2h)\to p^{*}\mathcal{R}(-h)\to\mathcal{O}_{Y}(h+E)\to
0.$

Using projection formula and
$Rq_{*}\mathcal{O}_{Y}(E)\simeq\mathcal{O}_{Y_{-}}$, we have
$Rq_{*}(p^{*}\mathcal{R}(-h))\simeq q_{*}(p^{*}\mathcal{R}(-h))$, and this
bundle lies in the exact sequence

$0\to\mathcal{O}_{Y_{-}}(-2)\to
q_{*}(p^{*}\mathcal{R}(-h))\to\mathcal{O}_{Y_{-}}(1)\to 0.$

This sequence is not split. Indeed if it is split, the bundle
$q_{*}(p^{*}\mathcal{R}(-h))|_{(Y_{-}\setminus\operatorname{\mathbf{Q}}_{0})}$
is also split. However, under the natural identification
$Y_{-}\setminus\operatorname{\mathbf{Q}}_{0}\simeq
Y_{+}\setminus\operatorname{\mathbf{G}}_{0}$, the bundle
$q_{*}(p^{*}\mathcal{R}(-h))|_{(Y_{-}\setminus\operatorname{\mathbf{Q}}_{0})}$
is identified with
$\mathcal{R}(1)|_{(Y_{+}\setminus\operatorname{\mathbf{G}}_{0})}$. Since the
zero-section $\operatorname{\mathbf{G}}_{0}$ has codimension two in $Y_{+}$,
if the bundle
$\mathcal{R}(1)|_{(Y_{+}\setminus\operatorname{\mathbf{G}}_{0})}$ is split,
the bundle $\mathcal{R}(1)$ is also split. This is contradiction.

Thus, by Proposition 3.5, we have $\mathcal{P}\simeq
Rq_{*}(p^{*}\mathcal{R}(-h))$. ∎

###### Proof of Proposition 3.7.

First, we have

$H^{\geq 1}(Y_{-},\mathcal{P}(a))=0$ for all $a\geq 0$,

and

$H^{\geq 2}(Y_{-},\mathcal{P}(a))=0$ for all $a\geq-2$,

by the definition of $\mathcal{P}$ and Proposition 3.5. Thus the non-trivial
parts are the vanishing of $H^{1}(Y_{-},\mathcal{P}(-1))$ and
$H^{1}(Y_{-},\mathcal{P}(-2))$. The first part also follows from the
definition of $\mathcal{P}$ using the same argument as in the proof of Lemma
2.4.

In the following, we show the vanishing of $H^{1}(Y_{-},\mathcal{P}(-2))$.
First by Lemma 3.9, we have $\mathcal{P}\simeq Rq_{*}(p^{*}\mathcal{R}(-h))$.

Therefore we can compute the cohomology as follows.

$\displaystyle H^{1}(Y_{-},\mathcal{P}(-2))$ $\displaystyle\simeq
H^{1}(Y_{-},Rq_{*}(p^{*}\mathcal{R}(-h))\otimes\mathcal{O}_{Y_{-}}(-2))$
$\displaystyle\simeq H^{1}(Y_{-},Rq_{*}(p^{*}\mathcal{R}(-3h)))$
$\displaystyle\simeq H^{1}(Y,p^{*}\mathcal{R}(-3h))$ $\displaystyle\simeq
H^{1}(Y,p^{*}\mathcal{R}(3H+3E))$ $\displaystyle\simeq
H^{1}(Y_{+},\mathcal{R}(3)\otimes Rp_{*}\mathcal{O}_{Y}(3E)).$

To compute this cohomology, we use a spectral sequence

$E^{k,l}_{2}=H^{k}(Y_{+},\mathcal{R}(3)\otimes
R^{l}p_{*}\mathcal{O}_{Y}(3E))\Rightarrow H^{k+l}(Y_{+},\mathcal{R}(3)\otimes
Rp_{*}\mathcal{O}_{Y}(3E)).$

Since $p_{*}\mathcal{O}_{Y}(3E)\simeq\mathcal{O}_{Y_{+}}$, we have

$E_{2}^{k,0}=H^{k}(Y_{+},\mathcal{R}(3))=0$ for $k\geq 1$.

This shows that there is an isomorphism of cohomologies

$H^{1}(Y_{+},\mathcal{R}(3)\otimes Rp_{*}\mathcal{O}_{Y}(3E))\simeq
H^{0}(Y_{+},\mathcal{R}(3)\otimes R^{1}p_{*}\mathcal{O}_{Y}(3E)).$

Let us consider the exact sequence

$0\to\mathcal{O}_{Y}(2E)\to\mathcal{O}_{Y}(3E)\to\mathcal{O}_{E}(3E)\to 0.$

Now we have

$\displaystyle p_{*}(\mathcal{O}_{E}(3E))=0,$ $\displaystyle
R^{1}p_{*}\mathcal{O}_{E}(3E)\simeq R(-1)\otimes\det(R(-1))\simeq
R(-4),~{}\text{and}$ $\displaystyle R^{1}p_{*}\mathcal{O}_{Y}(2E)\simeq
R^{1}p_{*}\mathcal{O}_{E}(2E)\simeq\det(R(-1))\simeq\mathcal{O}_{\operatorname{\mathbf{G}}_{0}}(-3).$

Hence there is an exact sequence

$0\to\mathcal{O}_{\operatorname{\mathbf{G}}_{0}}(-3)\to
R^{1}p_{*}(\mathcal{O}_{Y}(3E))\to R(-4)\to 0.$

Since

$\displaystyle
H^{0}(Y_{+},\mathcal{R}(3)\otimes\mathcal{O}_{\operatorname{\mathbf{G}}_{0}}(-3))\simeq
H^{0}(\operatorname{\mathbf{G}},R)=0$

and

$H^{0}(Y_{+},\mathcal{R}(3)\otimes R(-4))\simeq
H^{0}(\operatorname{\mathbf{G}},R\otimes
R(-1))\simeq\operatorname{Hom}_{\operatorname{\mathbf{G}}}(R^{\vee},R(-1))=0,$

we finally have the desired vanishing

$H^{1}(Y_{-},\mathcal{P}(-2))\simeq H^{0}(Y_{+},\mathcal{R}(3)\otimes
R^{1}p_{*}\mathcal{O}_{Y}(3E))=0.$

∎

######  Corollary 3.10.

We have

  1. (1)

$\operatorname{Ext}_{Y_{-}}^{\geq 1}(\mathcal{P}(1),\mathcal{P})=0$ and

  2. (2)

$\operatorname{Ext}_{Y_{-}}^{\geq 1}(\mathcal{P},\mathcal{P}(1))=0$.

###### Proof.

Let us consider the exact sequence

$0\to\mathcal{O}_{Y_{-}}(-2)\to\mathcal{P}\to\mathcal{O}_{Y_{-}}(1)\to 0$

that defines the bundle $\mathcal{P}$. Applying functors
$\operatorname{RHom}_{Y_{-}}(\mathcal{P}(1),-)$ and
$\operatorname{RHom}_{Y_{-}}(-,\mathcal{P}(1))$, we get exact triangles

$\displaystyle\operatorname{RHom}_{Y_{-}}(\mathcal{P}(1),\mathcal{O}_{Y_{-}}(-2))\to\operatorname{RHom}_{Y_{-}}(\mathcal{P}(1),\mathcal{P})\to\operatorname{RHom}_{Y_{-}}(\mathcal{P}(1),\mathcal{O}_{Y_{-}}(1)),$
$\displaystyle\operatorname{RHom}_{Y_{-}}(\mathcal{O}_{Y_{-}}(1),\mathcal{P}(1))\to\operatorname{RHom}_{Y_{-}}(\mathcal{P},\mathcal{P}(1))\to\operatorname{RHom}_{Y_{-}}(\mathcal{O}_{Y_{-}}(-2),\mathcal{P}(1)).$

Now the results follow from Proposition 3.7. ∎

Next we compute the cohomology of (the pull back of) the spinor bundle
$\mathcal{S}$. For this computation, we use the geometry of the flop again.

The following lemma is due to Kuznetsov.

######  Lemma 3.11.

There is an exact sequence on the flag variety $\operatorname{\mathbf{F}}$

$0\to p^{\prime*}R\to q^{\prime*}S\to p^{\prime*}R(H-h)\to 0.$

###### Proof.

See [Kuz18, Proposition 3 and Lemma 4]. ∎

######  Remark 3.12.

Interestingly, to prove this geometric lemma, Kuznetsov used derived
categories (namely, mutations of exceptional collections).

Using this lemma, we have the following.

######  Lemma 3.13.

An object $Rq_{*}(p^{*}\operatorname{\mathcal{R}}(H-h))$ is a sheaf on $Y_{-}$
and there exists an exact sequence on $Y_{-}$

$0\to\mathcal{P}(1)\to\operatorname{\mathcal{S}}\to
Rq_{*}(p^{*}\operatorname{\mathcal{R}}(H-h))\to 0.$

###### Proof.

By Kuznetsov’s lemma, there is an exact sequence on $Y$

$0\to p^{*}\operatorname{\mathcal{R}}\to q^{*}\operatorname{\mathcal{S}}\to
p^{*}\operatorname{\mathcal{R}}(H-h)\to 0.$

Since $Rq_{*}(p^{*}\operatorname{\mathcal{R}})\simeq\mathcal{P}(1)$, the
object $Rq_{*}(p^{*}\operatorname{\mathcal{R}}(H-h))$ is a sheaf on $Y_{-}$
and we have an exact sequence on $Y_{-}$

$0\to\mathcal{P}(1)\to\operatorname{\mathcal{S}}\to
Rq_{*}(p^{*}\operatorname{\mathcal{R}}(H-h))\to 0.$

∎

Using this exact sequence, we can have the following computations.

######  Lemma 3.14.

We have

  1. (1)

$\operatorname{Ext}_{Y_{-}}^{\geq
1}(\mathcal{O}_{Y_{-}}(a),\operatorname{\mathcal{S}})=0$ for $a\leq 1$, and

  2. (2)

$\operatorname{Ext}_{Y_{-}}^{\geq
1}(\operatorname{\mathcal{S}},\mathcal{O}_{Y_{-}}(b))=0$ for $b\geq-2$.

###### Proof.

Since $\operatorname{\mathcal{S}}^{\vee}\simeq\operatorname{\mathcal{S}}(1)$,
it is enough to show that $H^{\geq 1}(Y_{-},\operatorname{\mathcal{S}}(a))=0$
for $a\geq-1$. By Lemma 2.5, for any $a\in\operatorname{\mathbb{Z}}$, there is
an exact sequence

$0\to\operatorname{\mathcal{S}}(a)\to\mathcal{O}_{Y_{-}}(a)^{\oplus
8}\to\operatorname{\mathcal{S}}(a+1)\to 0.$

Since $H^{\geq 1}(Y_{-},\mathcal{O}_{Y_{-}}(a))=0$ if $a\geq-2$, it is enough
to show the case if $a=-1$. Let us consider the exact sequence

$0\to\mathcal{P}\to\operatorname{\mathcal{S}}(-1)\to
Rq_{*}(p^{*}\operatorname{\mathcal{R}}(H-2h))\to 0.$

Now we have $H^{\geq 1}(Y_{-},\mathcal{P})=0$, and therefore we can compute as

$\displaystyle H^{i}(Y_{-},\operatorname{\mathcal{S}}(-1))$
$\displaystyle\simeq
H^{i}(Y_{-},Rq_{*}(p^{*}\operatorname{\mathcal{R}}(H-2h)))$
$\displaystyle\simeq H^{i}(Y,p^{*}\operatorname{\mathcal{R}}(3H+2E))$
$\displaystyle\simeq H^{i}(Y_{+},\operatorname{\mathcal{R}}(3)\otimes
Rp_{*}\mathcal{O}_{Y}(2E))$

for any $i\geq 1$.

Let us consider a spectral sequence

$E_{2}^{k,l}=H^{k}(Y_{+},\operatorname{\mathcal{R}}(3)\otimes
R^{l}p_{*}\mathcal{O}_{Y}(2E))\Rightarrow
H^{k+l}(Y_{+},\operatorname{\mathcal{R}}(3)\otimes
Rp_{*}\mathcal{O}_{Y}(2E)).$

Now since $p_{*}\mathcal{O}_{Y}(2E)\simeq\mathcal{O}_{Y_{+}}$ we have

$E_{2}^{k,0}=H^{k}(Y_{+},\operatorname{\mathcal{R}}(3)\otimes
p_{*}\mathcal{O}_{Y}(2E))\simeq H^{k}(Y_{+}\operatorname{\mathcal{R}}(3))=0$

for $k\geq 1$ and hence we have

$H^{i}(Y_{+},\operatorname{\mathcal{R}}(3)\otimes
Rp_{*}\mathcal{O}_{Y}(2E))\simeq
H^{i-1}(Y_{+},\operatorname{\mathcal{R}}(3)\otimes
R^{1}p_{*}\mathcal{O}_{Y}(2E))$

for $i\geq 1$. Now we can compute as $R^{1}p_{*}\mathcal{O}_{Y}(2E)\simeq
Rp_{*}\mathcal{O}_{E}(2E)\simeq\mathcal{O}_{\operatorname{\mathbf{G}}_{0}}(-3)$
and thus

$H^{i-1}(Y_{+},\operatorname{\mathcal{R}}(3)\otimes
R^{1}p_{*}\mathcal{O}_{Y}(2E))\simeq H^{i-1}(\operatorname{\mathbf{G}},R)=0$

for all $i\geq 1$. This finishes the proof. ∎

From the lemma above, we have the following.

######  Theorem 3.15.

The following hold.

  1. (1)

$\operatorname{Ext}_{Y_{-}}^{\geq
1}(\operatorname{\mathcal{S}},\mathcal{P})=0$ and
$\operatorname{Ext}_{Y_{-}}^{\geq
1}(\mathcal{P},\operatorname{\mathcal{S}})=0$.

  2. (2)

$\operatorname{Ext}_{Y_{-}}^{\geq
1}(\operatorname{\mathcal{S}},\mathcal{P}(1))=0$.

  3. (3)

$\operatorname{Ext}_{Y_{-}}^{\geq
1}(\mathcal{P}(-1),\operatorname{\mathcal{S}})=0$.

  4. (4)

$\operatorname{\mathcal{S}}$ is a partial tilting bundle.

###### Proof.

(1). First note that
$\operatorname{Ext}_{Y_{-}}^{i}(\operatorname{\mathcal{S}},\mathcal{P})\simeq\operatorname{Ext}_{Y_{-}}^{i}(\mathcal{P},\operatorname{\mathcal{S}})$
since $\mathcal{P}^{\vee}\simeq\mathcal{P}(1)$ and
$\operatorname{\mathcal{S}}^{\vee}\simeq\operatorname{\mathcal{S}}(1)$. Let us
consider the exact sequence

$0\to\mathcal{O}_{Y_{-}}(-2)\to\mathcal{P}\to\mathcal{O}_{Y_{-}}(1)\to 0.$

Then, applying the functor
$\operatorname{RHom}_{Y_{-}}(\operatorname{\mathcal{S}},-)\simeq\operatorname{R\Gamma}(Y_{-},\operatorname{\mathcal{S}}(1)\otimes-)$
we have an exact triangle

$\operatorname{R\Gamma}(Y_{-},\operatorname{\mathcal{S}}(-1))\to\operatorname{RHom}_{Y_{-}}(\operatorname{\mathcal{S}},\mathcal{P})\to\operatorname{R\Gamma}(Y_{-},\operatorname{\mathcal{S}}(2)).$

Now the result follows from the lemma above. The proof of (2) is similar. (3)
follows from (2). Indeed, since
$\operatorname{\mathcal{S}}^{\vee}\simeq\operatorname{\mathcal{S}}(1)$ and
$(\mathcal{P}(1))^{\vee}\simeq\mathcal{P}$, we have

$\operatorname{Ext}_{Y_{-}}^{i}(\mathcal{P}(-1),\operatorname{\mathcal{S}})\simeq\operatorname{Ext}_{Y_{-}}^{i}(\mathcal{P},\operatorname{\mathcal{S}}(1))\simeq\operatorname{Ext}^{i}_{Y_{-}}(\operatorname{\mathcal{S}},\mathcal{P}(1)).$

Now let us prove (4). Recall that there is an exact sequence

$0\to\mathcal{P}(1)\to\operatorname{\mathcal{S}}\to
Rq_{*}(p^{*}\operatorname{\mathcal{R}}(H-h))\to 0.$

By (2), we have

$\operatorname{Ext}_{Y_{-}}^{i}(\operatorname{\mathcal{S}},\operatorname{\mathcal{S}})\simeq\operatorname{Ext}_{Y_{-}}^{i}(\operatorname{\mathcal{S}},Rq_{*}(p^{*}\operatorname{\mathcal{R}}(H-h)))\simeq\operatorname{Ext}_{Y}^{i}(q^{*}\operatorname{\mathcal{S}},p^{*}\operatorname{\mathcal{R}}(H-h))$

for $i\geq 1$. Let us consider an exact sequence

$0\to p^{*}\operatorname{\mathcal{R}}\to q^{*}\operatorname{\mathcal{S}}\to
p^{*}\operatorname{\mathcal{R}}(H-h)\to 0.$

Then we have

$\operatorname{Ext}_{Y}^{i}(p^{*}\operatorname{\mathcal{R}}(H-h),p^{*}\operatorname{\mathcal{R}}(H-h))\simeq\operatorname{Ext}_{Y_{+}}^{i}(\operatorname{\mathcal{R}},\operatorname{\mathcal{R}})=0$

for $i\geq 1$ and

$\operatorname{Ext}_{Y}^{i}(p^{*}\operatorname{\mathcal{R}},p^{*}\operatorname{\mathcal{R}}(H-h))\simeq\operatorname{Ext}_{Y}^{i}(p^{*}\operatorname{\mathcal{R}},p^{*}\operatorname{\mathcal{R}}(2H+E))\simeq\operatorname{Ext}_{Y_{+}}^{i}(\operatorname{\mathcal{R}},\operatorname{\mathcal{R}}(2))=0$

for $i\geq 1$. Thus we have
$\operatorname{Ext}_{Y}^{i}(q^{*}\operatorname{\mathcal{S}},p^{*}\operatorname{\mathcal{R}}(H-h))=0$
for $i\geq 1$. ∎

Next we show the following:

######  Lemma 3.16.

We have

  1. (1)

$\operatorname{Ext}_{Y_{-}}^{\geq
1}(\mathcal{P}(1),\operatorname{\mathcal{S}})=0$ and

  2. (2)

$\operatorname{Ext}_{Y_{-}}^{\geq
1}(\operatorname{\mathcal{S}},\mathcal{P}(-1))=0$.

###### Proof.

(2) follows from (1). Let us prove (1). Recall that
$Rq_{*}(p^{*}\operatorname{\mathcal{R}})\simeq\mathcal{P}(1)$. Therefore by
the Grothendieck duality we have

$\operatorname{Ext}_{Y_{-}}^{i}(\mathcal{P}(1),\operatorname{\mathcal{S}})\simeq\operatorname{Ext}_{Y}^{i}(p^{*}\operatorname{\mathcal{R}},q^{*}\operatorname{\mathcal{S}}(E)).$

Let us consider the exact sequence

$0\to p^{*}\operatorname{\mathcal{R}}(E)\to
q^{*}\operatorname{\mathcal{S}}(E)\to
p^{*}\operatorname{\mathcal{R}}(2H+2E)\to 0.$

First we have

$\operatorname{Ext}_{Y}^{i}(p^{*}\operatorname{\mathcal{R}},p^{*}\operatorname{\mathcal{R}}(E))\simeq\operatorname{Ext}_{Y_{+}}^{i}(\operatorname{\mathcal{R}},\operatorname{\mathcal{R}})=0$

for $i\geq 1$. Hence it is enough to show the vanishing of

$\operatorname{Ext}_{Y}^{i}(p^{*}\operatorname{\mathcal{R}},p^{*}\operatorname{\mathcal{R}}(2H+2E))\simeq\operatorname{Ext}_{Y_{+}}^{i}(\operatorname{\mathcal{R}},\operatorname{\mathcal{R}}(2)\otimes
Rp_{*}\mathcal{O}_{Y}(2E)).$

Let us consider a spectral sequence

$E_{2}^{k,l}=\operatorname{Ext}_{Y_{+}}^{k}(\operatorname{\mathcal{R}},\operatorname{\mathcal{R}}(2)\otimes
R^{l}p_{*}\mathcal{O}_{Y}(2E))\Rightarrow\operatorname{Ext}_{Y_{+}}^{k+l}(\operatorname{\mathcal{R}},\operatorname{\mathcal{R}}(2)\otimes
Rp_{*}\mathcal{O}_{Y}(2E)).$

Note that

$E_{2}^{k,0}=\operatorname{Ext}_{Y_{+}}^{k}(\operatorname{\mathcal{R}},\operatorname{\mathcal{R}}(2))=0$

for $k\geq 1$. Thus we have

$\displaystyle\operatorname{Ext}_{Y_{+}}^{i}(\operatorname{\mathcal{R}},\operatorname{\mathcal{R}}(2)\otimes
Rp_{*}\mathcal{O}_{Y}(2E))$
$\displaystyle\simeq\operatorname{Ext}_{Y_{+}}^{i-1}(\operatorname{\mathcal{R}},\operatorname{\mathcal{R}}(2)\otimes
R^{1}p_{*}\mathcal{O}_{Y}(2E))$
$\displaystyle\simeq\operatorname{Ext}_{Y_{+}}^{i-1}(\operatorname{\mathcal{R}},\operatorname{\mathcal{R}}(2)\otimes\mathcal{O}_{\operatorname{\mathbf{G}}_{0}}(-3))$
$\displaystyle\simeq\operatorname{Ext}_{\operatorname{\mathbf{G}}}^{i-1}(R,R(-1))$

for $i\geq 1$. This is zero. ∎

Combining all $\operatorname{Ext}$-vanishings in the present subsection, we
obtain the following consequence.

######  Theorem 3.17.

The following vector bundles on $Y_{-}$ are tilting bundles.

  1. (1)

$T_{-}^{\spadesuit}:=\mathcal{O}_{Y_{-}}(-1)\oplus\mathcal{O}_{Y_{-}}\oplus\mathcal{O}_{Y_{-}}(1)\oplus\mathcal{P}\oplus\mathcal{P}(1)\oplus\operatorname{\mathcal{S}}(1)$

  2. (2)

$T_{-}^{\clubsuit}:=\mathcal{O}_{Y_{-}}(-1)\oplus\mathcal{O}_{Y_{-}}\oplus\mathcal{O}_{Y_{-}}(1)\oplus\mathcal{P}\oplus\mathcal{P}(1)\oplus\operatorname{\mathcal{S}}$

  3. (3)

$T_{-}^{\heartsuit}:=\mathcal{O}_{Y_{-}}(-1)\oplus\mathcal{O}_{Y_{-}}\oplus\mathcal{O}_{Y_{-}}(1)\oplus\mathcal{P}(1)\oplus\mathcal{P}(2)\oplus\operatorname{\mathcal{S}}(1)$

  4. (4)

$T_{-}^{\diamondsuit}:=\mathcal{O}_{Y_{-}}(-1)\oplus\mathcal{O}_{Y_{-}}\oplus\mathcal{O}_{Y_{-}}(1)\oplus\mathcal{P}(-1)\oplus\mathcal{P}\oplus\operatorname{\mathcal{S}}$

We note that these bundles are generators of
$\mathrm{D}(\operatorname{Qcoh}(Y_{-}))$ because they split-generate another
generators

$\displaystyle\mathcal{O}_{Y_{-}}(-2)\oplus\mathcal{O}_{Y_{-}}(-1)\oplus\mathcal{O}_{Y_{-}}\oplus\operatorname{\mathcal{S}}(1)\oplus\mathcal{O}_{Y_{-}}(1)\oplus\mathcal{O}_{Y_{-}}(2),$
$\displaystyle\mathcal{O}_{Y_{-}}(-2)\oplus\mathcal{O}_{Y_{-}}(-1)\oplus\operatorname{\mathcal{S}}\oplus~{}\mathcal{O}_{Y_{-}}\oplus\mathcal{O}_{Y_{-}}(1)\oplus\mathcal{O}_{Y_{-}}(2),$
$\displaystyle\mathcal{O}_{Y_{-}}(-1)\oplus\mathcal{O}_{Y_{-}}\oplus\operatorname{\mathcal{S}}(1)\oplus\mathcal{O}_{Y_{-}}(1)\oplus\mathcal{O}_{Y_{-}}(2)\oplus\mathcal{O}_{Y_{-}}(3),~{}\text{~{}and}$
$\displaystyle\mathcal{O}_{Y_{-}}(-3)\oplus\mathcal{O}_{Y_{-}}(-2)\oplus\mathcal{O}_{Y_{-}}(-1)\oplus\operatorname{\mathcal{S}}\oplus~{}\mathcal{O}_{Y_{-}}\oplus\mathcal{O}_{Y_{-}}(1)$

respectively, that are obtained from tilting bundles on
$\operatorname{\mathbf{Q}}$. We also note that the pair $T_{-}^{\spadesuit}$
and $T_{-}^{\clubsuit}$ are dual to each other, and the pair
$T_{-}^{\heartsuit}$ and $T_{-}^{\diamondsuit}$ are dual to each other.

###  3.3. Derived equivalences

According to Lemma 2.3, in order to show the derived equivalence between
$Y_{+}$ and $Y_{-}$, it is enough to show that there are tilting bundles
$T_{+}$ and $T_{-}$ on $Y_{+}$ and $Y_{-}$ respectively, such that they give
the same vector bundle on the common open subset $U$ of $Y_{+}$ and $Y_{-}$,
which is isomorphic to the smooth locus of $X$. Using tilting bundles that we
constructed in this article, we can give four derived equivalences for the
Abuaf-Ueda flop.

######  Lemma 3.18.

On the common open subset $U$, we have the following.

  1. (1)

$\mathcal{O}_{Y_{+}}(a)|_{U}\simeq\mathcal{O}_{Y_{-}}(-a)|_{U}$ for all
$a\in\operatorname{\mathbb{Z}}$.

  2. (2)

$\mathcal{R}|_{U}\simeq\mathcal{P}(1)|_{U}$.

  3. (3)

$\Sigma(1)|_{U}\simeq\operatorname{\mathcal{S}}|_{U}$.

###### Proof.

(1) follows from the fact that $\mathcal{O}_{Y}(E)\simeq\mathcal{O}_{Y}(-H-h)$
since $\mathcal{O}_{Y}(E)|_{U}\simeq\mathcal{O}_{U}$. (2) follows from the
isomorphism $\mathcal{P}(1)\simeq Rq_{*}(p^{*}\mathcal{R})$.

Let us proof (3). To see this, we show that
$Rp_{*}(q^{*}\operatorname{\mathcal{S}})\simeq\Sigma(1)$. By Lemma 3.11, there
is an exact sequence

$0\to p^{*}\mathcal{R}\to q^{*}\operatorname{\mathcal{S}}\to
p^{*}\mathcal{R}(2H+E)\to 0$

on $Y$. Since we have $Rp_{*}\mathcal{O}_{Y}(E)\simeq\mathcal{O}_{Y_{+}}$, by
projection formula, we have an exact sequence

$0\to\mathcal{R}\to
Rp_{*}(q^{*}\operatorname{\mathcal{S}})\to\mathcal{R}(2)\to 0$

on $Y_{+}$. Note that this short exact sequence is not split. Thus the
uniqueness of such a non-trivial sequence shows that the desired isomorphism
$Rp_{*}(q^{*}\operatorname{\mathcal{S}})\simeq\Sigma(1)$. ∎

######  Corollary 3.19.

The pair of bundles $T_{+}^{\ast}$ and $T_{-}^{\ast}$ give the same bundle on
the common open subset $U$ for any
$\ast\in\\{\spadesuit,\clubsuit,\heartsuit,\diamondsuit\\}$.

As a consequence, we have the following theorem.

######  Theorem 3.20.

Let $\ast\in\\{\spadesuit,\clubsuit,\heartsuit,\diamondsuit\\}$ and put

$\Lambda^{\ast}:=\operatorname{End}_{Y_{+}}(T_{+}^{\ast})\simeq\operatorname{End}_{Y_{-}}(T_{-}^{\ast}).$

Then we have derived equivalences

$\displaystyle\Phi^{\ast}:=\operatorname{RHom}_{Y_{+}}(T^{\ast}_{+},-)\otimes^{\mathbb{L}}_{\Lambda^{\ast}}T_{-}^{\ast}:\operatorname{D^{b}}(Y_{+})\xrightarrow{\sim}\operatorname{D^{b}}(Y_{-})$
$\displaystyle\Psi^{\ast}:=\operatorname{RHom}_{Y_{-}}(T^{\ast}_{-},-)\otimes^{\mathbb{L}}_{\Lambda^{\ast}}T_{+}^{\ast}:\operatorname{D^{b}}(Y_{-})\xrightarrow{\sim}\operatorname{D^{b}}(Y_{+})$

that are quasi-inverse to each other.

######  Remark 3.21.

Composing $\Phi^{\ast}$ and $\Psi^{\star}$ for two different
$\ast,\star\in\\{\spadesuit,\clubsuit,\heartsuit,\diamondsuit\\}$, we get some
non-trivial autoequivalences on $\operatorname{D^{b}}(Y_{+})$ (resp.
$\operatorname{D^{b}}(Y_{-})$) that fix line bundles
$\mathcal{O}_{Y_{+}}(-1)$, $\mathcal{O}_{Y_{+}}$ and $\mathcal{O}_{Y_{+}}(1)$
(resp. $\mathcal{O}_{Y_{-}}(-1)$, $\mathcal{O}_{Y_{-}}$ and
$\mathcal{O}_{Y_{-}}(1)$). It would be an interesting problem to find a
(sufficiently large) subgroup of
$\operatorname{Auteq}(\operatorname{D^{b}}(Y_{+}))$
($\simeq\operatorname{Auteq}(\operatorname{D^{b}}(Y_{-}))$) that contains our
autoequivalences.

##  4\. Moduli problem

In this section we study the Abuaf-Ueda from the point of view of non-
commutative crepant resolutions and moduli.

###  4.1. Non-commutative crepant resolution and moduli

######  Definition 4.1.

Let $R$ be a normal Gorenstein domain and $M$ a reflexive $R$-module. Then we
say that $M$ gives a non-commutative crepant resolution (= NCCR) of $R$ if the
endomorphism ring $\operatorname{End}_{R}(M)$ of $M$ is maximal Cohen-Macaulay
as an $R$-module and $\operatorname{End}_{R}(M)$ has finite global dimension.
When $M$ gives an NCCR of $R$ then the endomorphism ring
$\operatorname{End}_{R}(M)$ is called an NCCR of $R$.

In many cases, an NCCR is constructed from a tilting bundle on a (commutative)
crepant resolution using the following lemma.

######  Lemma 4.2.

Let $X=\operatorname{Spec}R$ be a normal Gorenstein affine variety that admits
a (commutative) crepant resolution $\phi:Y\to X$. Then for a tilting bundle
$T$ on $Y$, the double-dual $(\phi_{*}T)^{\vee\vee}$ of the module $\phi_{*}T$
gives an NCCR
$\operatorname{End}_{Y}(T)\simeq\operatorname{End}_{R}(\phi_{*}T)$ of $R$. If
one of the following two conditions are satisfied, then
$(\phi_{*}T)^{\vee\vee}$ is isomorphic to $\phi_{*}T$, i.e. we do not have to
take the double-dual.

  1. (a)

The tilting bundle $T$ contains $\mathcal{O}_{Y}$ as a direct summand.

  2. (b)

The resolution $\phi$ is small, i.e. the exceptional locus of $\phi$ does not
contain a divisor.

When we find an NCCR $\Lambda=\operatorname{End}_{R}(M)$ of an algebra $R$, we
can consider the moduli spaces of modules over $\Lambda$.

In the following we recall the result of Karmazyn [Kar17]. Let $Y\to
X=\operatorname{Spec}R$ be a projective morphism and $T$ a tilting bundle on
$Y$. Assume that $T$ has a decomposition $T=\bigoplus_{i=0}^{n}E_{i}$ such
that (i) $E_{i}$ is indecomposable for any $i$, (ii) $E_{i}\neq E_{j}$ for
$i\neq j$, and (iii) $E_{0}=\mathcal{O}_{Y}$. Then we can regard the
endomorphism ring $\Lambda:=\operatorname{End}_{Y}(T)$ as a path algebra of a
quiver with relations such that the summand $E_{i}$ corresponds to a vertex
$i$.

Now we define a dimension vector $d_{T}=(d_{T}(i))_{i=0}^{n}$ by

$d_{T}(i):=\operatorname{rank}E_{i}.$

Note that, since we assumed that $E_{0}=\mathcal{O}_{Y}$, we have
$d_{T}(0)=1$. We also define a stability condition $\theta_{T}$ associated to
the tilting bundle $T$ by

$\theta_{T}(i):=\begin{cases}-\Sigma_{i\neq
0}\operatorname{rank}E_{i}&\text{if $i=0$}\\\ 1&\text{otherwise.}\end{cases}$

Then we can consider King’s moduli space
$\mathcal{M}_{\Lambda,d_{T},\theta_{T}}^{\mathrm{ss}}$ of $\theta$-semistable
(right) $\operatorname{End}_{Y}(T)$-modules with dimension vector $d_{T}$. It
is easy to see that there is no strictly $\theta_{T}$-semistable object with
dimension vector $d_{T}$ (see [Kar17]), and thus the moduli space
$\mathcal{M}_{\Lambda,d_{T},\theta_{T}}^{\mathrm{ss}}$ is isomorphic to a
moduli space $\mathcal{M}_{\Lambda,d_{T},\theta_{T}}^{\mathrm{s}}$ of
$\theta_{T}$-stable objects.

In this setting, Karmazyn [Kar17] proved the following.

######  Theorem 4.3.

Assume that a morphism $\mathcal{O}_{Y}\to\mathcal{O}_{y}$ is surjective for
all closed points $y\in Y$ in the abelian category $\mathcal{A}$ that
corresponds to $\operatorname{mod}(\Lambda)$ under the derived equivalence

$\operatorname{RHom}_{Y}(T,-):\operatorname{D^{b}}(Y)\to\operatorname{D^{b}}(\Lambda).$

Then there is a monomorphism
$f:Y\to\mathcal{M}_{\Lambda,d_{T},\theta_{T}}^{\mathrm{s}}$.

The condition in the theorem above can be interpreted as the following
geometric condition for the bundle.

######  Lemma 4.4.

The assumption in Theorem 4.3 is satisfied if the dual $T^{\vee}$ of the
tilting bundle $T$ is globally generated.

###### Proof.

We show the surjectivity of the morphism
$\operatorname{Hom}_{Y}(T,\mathcal{O}_{Y})\to\operatorname{Hom}_{Y}(T,\mathcal{O}_{y})$.
This morphism coincides with $H^{0}(Y,T^{\vee})\to T^{\vee}\otimes k(y)$.

By assumption, there is a surjective morphism $\mathcal{O}_{Y}^{\oplus r}\to
T^{\vee}$ for some $r$. Now we have the following commutative diagram

${H^{0}(Y,\mathcal{O}_{Y})^{\oplus r}}$${k(y)^{\oplus
r}}$${H^{0}(Y,T^{\vee})}$${T^{\vee}\otimes k(y).}$

Morphisms $H^{0}(Y,\mathcal{O}_{Y})^{\oplus r}\to k(y)^{\oplus r}$ and
$k(y)^{\oplus r}\to T^{\vee}\otimes k(y)$ are surjective. Thus we have the
desired surjectivity. ∎

Let us discuss the moduli when $Y\to\operatorname{Spec}R$ is a crepant
resolution. Then there is a unique irreducible component $M$ of
$\mathcal{M}_{\Lambda,d_{T},\theta_{T}}^{\mathrm{s}}$ that dominates
$\operatorname{Spec}R$ [VdB04]. We call this component (with reduced scheme
structure) the main component. As a corollary of results above, we have the
following.

######  Corollary 4.5.

Let us assume that a crepant resolution $Y$ of $\operatorname{Spec}R$ admits a
tilting bundle $T$ such that

  1. (a)

$T$ is a direct sum of non-isomorphic indecomposable bundles
$T=\bigoplus_{i=0}^{n}E_{i}$.

  2. (b)

$E_{0}=\mathcal{O}_{Y}$.

  3. (c)

The dual $T^{\vee}$ is globally generated.

Then the main component $M$ of
$\mathcal{M}_{\Lambda,d_{T},\theta_{T}}^{\mathrm{s}}$ is isomorphic to $Y$.

###### Proof.

Since $Y$ and $\mathcal{M}_{\Lambda,d_{T},\theta_{T}}^{\mathrm{s}}$ are
projective over $\operatorname{Spec}R$ (see [VdB04]), the monomorphism
$Y\to\mathcal{M}_{\Lambda,d_{T},\theta_{T}}^{\mathrm{s}}$ is proper. A proper
monomorphism is a closed immersion. Since $Y$ dominants
$\operatorname{Spec}R$, the image of this monomorphism is contained in the
main component $M$. Since $Y$ and $M$ are birational to
$\operatorname{Spec}R$, they coincide with each other. ∎

###  4.2. Application to our situation

First, from the existence of tilting bundles, we have the following.

######  Theorem 4.6.

The affine variety $X=\operatorname{Spec}C_{0}$ that appears in the Abuaf-Ueda
flop admits NCCRs.

Let us consider bundles

$\displaystyle T_{+}$
$\displaystyle:=T_{+}^{\heartsuit}\otimes\mathcal{O}_{Y_{+}}(-1)=\mathcal{O}_{Y_{+}}\oplus\mathcal{O}_{Y_{+}}(-1)\oplus\mathcal{O}_{Y_{+}}(-2)\oplus\mathcal{R}(-1)\oplus\mathcal{R}(-2)\oplus\Sigma(-1)$
$\displaystyle T_{-}$
$\displaystyle:=T_{-}^{\diamondsuit}\otimes\mathcal{O}_{Y_{-}}(-1)=\mathcal{O}_{Y_{-}}\oplus\mathcal{O}_{Y_{-}}(-1)\oplus\mathcal{O}_{Y_{-}}(-2)\oplus\mathcal{P}(-1)\oplus\mathcal{P}(-2)\oplus\operatorname{\mathcal{S}}(-1)$

These bundles satisfy the assumptions in Corollary 4.5. Indeed the globally-
generatedness of dual bundles follows from the following lemma.

######  Lemma 4.7.

  1. (1)

A bundle $\Sigma(a)$ is globally generated if and only if $a\geq 2$.

  2. (2)

A bundle $\mathcal{P}(a)$ is globally generated if and only if $a\geq 2$.

###### Proof.

First we note that $\mathcal{R}(a)$ is globally generated if and only if
$a\geq 1$. Recall that $\Sigma(a)$ is defined by an exact sequence

$0\to\mathcal{R}(a-1)\to\Sigma(a)\to\mathcal{R}(a+1)\to 0.$

If $a\geq 2$ then $\mathcal{R}(a-1)$ and $\mathcal{R}(a+1)$ are globally
generated and $H^{1}(Y_{+},\mathcal{R}(a-1))=0$. Hence we have the following
commutative diagram

${0}$${H^{0}(Y^{+},\mathcal{R}(a-1))\otimes_{\operatorname{\mathbb{C}}}\mathcal{O}_{Y^{+}}}$${H^{0}(Y^{+},\Sigma(a))\otimes_{\operatorname{\mathbb{C}}}\mathcal{O}_{Y^{+}}}$${H^{0}(Y^{+},\mathcal{R}(a+1))\otimes_{\operatorname{\mathbb{C}}}\mathcal{O}_{Y^{+}}}$${0}$${0}$${\mathcal{R}(a-1)}$${\Sigma(a)}$${\mathcal{R}(a+1)}$${0}$

Thus the five-lemma implies that the bundle $\Sigma(a)$ is also globally
generated.

Next let us assume that $\Sigma(a)$ is globally generated for some $a$. Then
the restriction $\Sigma(a)|_{\operatorname{\mathbf{G}}_{0}}$ of $\Sigma(a)$ to
the zero-section $\operatorname{\mathbf{G}}_{0}$ is also globally generated.
Since there is a splitting
$\Sigma(a)|_{\operatorname{\mathbf{G}}_{0}}=R(a-1)\oplus R(a+1)$ on
$\operatorname{\mathbf{G}}_{0}$, we have that $R(a-1)$ is also globally
generated. Thus we have $a\geq 2$.

The proof for $\mathcal{P}(a)$ is similar. ∎

######  Corollary 4.8.

The bundles $T_{+}^{\vee}$ and $T_{-}^{\vee}$ are globally generated.

Let us regard the endomorphism ring
$\Lambda_{+}:=\operatorname{End}_{Y_{+}}(T_{+})$ as a path algebra of a quiver
with relations $(Q_{+},I_{+})$. For $0\leq i\leq 5$, let $E_{+,i}$ be the
$(i+1)$-th indecomposable summand of $T_{+}$ with respect to the order

$T_{+}=\mathcal{O}_{Y_{+}}\oplus\mathcal{O}_{Y_{+}}(-1)\oplus\mathcal{O}_{Y_{+}}(-2)\oplus\mathcal{R}(-1)\oplus\mathcal{R}(-2)\oplus\Sigma(-1).$

The vertex of the quiver $(Q_{+},I_{+})$ corresponding to the summand
$E_{+,i}$ is denoted by $i\in(Q_{+})_{0}$. We define a dimension vector
$d_{+}\in\operatorname{\mathbb{Z}}^{6}$ by

$d_{+}=(d_{0},d_{1},d_{2},d_{3},d_{4},d_{5}):=(1,1,1,2,2,4).$

We also define a stability condition
$\theta_{+}\in\operatorname{\mathbb{R}}^{6}$ by

$\theta_{+}=(\theta_{0},\theta_{1},\theta_{2},\theta_{3},\theta_{4},\theta_{5}):=(-10,1,1,1,1,1).$

Then we have the following as a corollary.

######  Corollary 4.9.

The crepant resolution $Y_{+}$ of $X=\operatorname{Spec}C_{0}$ gives the main
component of the moduli space
$\mathcal{M}_{\Lambda_{+},d_{+},\theta_{+}}^{\mathrm{s}}$ of representations
of an NCCR $\Lambda_{+}$ of $X$ of dimension vector $d_{+}$ with respect to
the stability condition $\theta_{+}$.

Similarly we define a quiver with relations $(Q_{-},I_{-})$ with
$(Q_{-})_{0}=\\{0,1,2,3,4,5\\}$ whose path algebra is
$\Lambda_{-}=\operatorname{End}_{Y_{-}}(T_{-})$, using the order

$T_{-}=\mathcal{O}_{Y_{-}}\oplus\mathcal{O}_{Y_{-}}(-1)\oplus\mathcal{O}_{Y_{-}}(-2)\oplus\mathcal{P}(-1)\oplus\mathcal{P}(-2)\oplus\operatorname{\mathcal{S}}(-1)$

and put

$\displaystyle d_{-}$ $\displaystyle:=(1,1,1,2,2,4)\in\mathbb{Z}^{6}$
$\displaystyle\theta_{-}$ $\displaystyle:=(-10,1,1,1,1,1)\in\mathbb{R}^{6}.$

Then we have

######  Corollary 4.10.

The crepant resolution $Y_{-}$ of $X=\operatorname{Spec}C_{0}$ gives the main
component of the moduli space
$\mathcal{M}_{\Lambda_{-},d_{-},\theta_{-}}^{\mathrm{s}}$ of representations
of an NCCR $\Lambda_{-}$ of $X$ of dimension vector $d_{-}$ with respect to
the stability condition $\theta_{-}$.

Finally we remark that there is an isomorphism of algebras

$\displaystyle\Lambda_{+}=\operatorname{End}_{Y_{+}}(T_{+})\simeq\operatorname{End}_{Y_{+}}(T_{+}\otimes\mathcal{O}_{Y_{+}}(2))\simeq\operatorname{End}_{Y_{-}}(T_{-}^{\vee})\simeq\operatorname{End}_{Y_{-}}(T_{-})^{\mathrm{op}}=\Lambda_{-}^{\mathrm{op}}.$

Note that this isomorphism does not preserve the order of vertices of the
quivers that we used above.

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