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BRIDGELAND STABILITY OF MINIMAL INSTANTON BUNDLES ON
FANO THREEFOLDS
XUQIANG QIN
Abstract. We prove that minimal instanton bundles on a Fano threefold Xof Picard
rank one and index two are semistable objects in the Kuznetsov component Ku(X),
with respect to the stability conditions constructed by Bayer, Lahoz, Macr`ı and Stellari.
When the degree of Xis at least 3, we show torsion free generalizations of minimal
instantons are also semistable objects. As a result, we describe the moduli space of
semistable objects with same numerical classes as minimal instantons in Ku(X). We also
investigate the stability of acyclic extensions of non-minimal instantons.
Contents
1. Introduction 1
2. Review on Fano threefolds and minimal instanton bundles 3
3. Review on stability conditions 5
4. Bridgeland stability of minimal instanton bundles 9
5. Description of the moduli space Mσ(Ku(X),2[Il]) 11
6. Non-minimal instanton bundles 16
References 20
1. Introduction
Instanton bundles first appeared on the 4-sphere S4as a way to describe Yang-Mills
instantons. They serve as bridges between algebraic geometry and mathematical physics.
The notion of mathematical instanton bundles was first introduced on P3, then generalized
to Fano threefolds by Faenzi[Fa14] and Kuznetsov[Ku12].
Definition 1.1. [Ku12] Let Xbe a Fano threefold of Picard rank 1 and index 2. An
instanton bundle of charge non Xis a stable vector bundle Eof rank 2 with c1(E) =
0, c2(E) = n, enjoying the instantonic vanishing condition:
H1(X, E(−1)) = 0.
Any instanton bundle Ewill have charge c2(E)>2. The instanton bundles of charge
2 are called the minimal instantons. By definition, the moduli space of minimal instanton
bundles on Xis an open subscheme of the moduli space Mof Gieseker-semistable rank 2
sheaves with c1= 0, c2= 2 and c3= 0. When Xis a cubic threefold, [Dr00] classified
sheaves in Mand described Mas a blow-up of the intermediate Jacobian of X. His work
2020 Mathematics Subject Classification. Primary 14F08, 14J30, 14J45; Secondary 14D21.
Key words and phrases. Instanton bundles, Bridgeland stability conditions, moduli spaces, Fano three-
folds, semiorthogonal decompositions.
1
2 XUQIANG QIN
was generalized to Xof degree 5 and 4 by the author[Qin1][Qin2].
On the other hand, Bridgeland[Br07] introduced the notion of stability conditions on a
triangulated category. He showed that the set parametrizing stability conditions on a trian-
gulated category has a natural structure of a complex manifold. Since then stabilities on the
bounded derived category Db(X) of a smooth projective variety Xhave been intensely stud-
ied. [Br07][Ma07][Ok06] gave complete descriptions of the stability manifold Stab(X) when
Xis a smooth projective curve. When Xis a smooth projective surface, stability conditions
and related moduli spaces were studied in [Br08][ABCH][AB13][LZ18][Nu16], among many
other papers. When the dimension of Xis at least three, the construction of Bridgeland
stability conditions becomes challenging. We refer the readers to [BMT][BMSZ][Li19] for
information on stabilities on Fano threefolds.
Bayer, Lahoz, Macr`ı and Stellari[BLMS] provided a criterion to define stability condition
on the right orthogonal complements of an exceptional collection in a triangulated category.
They applied the criterion to Fano threefolds of Picard rank one and cubic fourfolds and
induced stability conditions on them. See [LLMS][LPZ1][LPZ2] for some applications on
cubic fourfolds. For our purpose, let Xbe a Fano threefold of Picard rank one and index
two. The bounded derived category Db(X) has the following semiorthogonal decomposition:
Db(X) = hKu(X),OX,OX(H)i
where His the ample generator of the Picard group. The triangulated subcategory Ku(X) is
called the Kuznetsov component. Explicit computations in [PY20] shows that the criterion
of [BLMS] induces a family σ(α, β) of stability conditions on Ku(X) where (α, β ) lies in a
triangular region in the half plane R>0×R. Moreover, [PY20] showed that the family lies
in a single orbit Kwith respect to the standard action of ˜
GL+
2(R) on the stability manifold
Stab(Ku(X)) of Ku(X).
Pertusi and Yang[PY20] showed that ideal sheaves of lines on X(which are easily checked
to belong to Ku(X)) are stable objects with respect to any σ∈ K. Using this, they were
able to identify the Fano surface of lines on Xwith (if the degree of Xis 1, an irreducible
component of) the moduli space of σ(α, β)-stable objects with the numerical class [Il] in
Ku(X).
Minimal instantons are known to be objects in Ku(X) and they have twice the numerical
class of [Il]. It is natural to ask if one can generalize the results of Pertusi and Yang[PY20]
to minimal instantons. Our first result establishes their (semi)stability.
Theorem 1.2. Let Xbe a Fano threefold of Picard rank one, index two and degree d. Let
Ebe a minimal instanton bundle on X. Then Eis σ-semistable for any σ∈ K. If d>2,
then Eis σ-stable.
When d>3, the classifications in [Dr00][Qin1][Qin2] showed that torsion free (but not
locally free) generalizations of minimal instanton bundles are also objects in Ku(X). Our
second result compares the moduli space of Gieseker-semistable rank 2 sheaves with c1=
0, c2= 2, c3= 0 with the moduli space of σ-semistable objects of numerical class 2[Il] in
Ku(X).
Theorem 1.3. Let Xbe a Fano threefold of Picard rank one, index two and degree d. If
d>3, then for any σ∈ K, the moduli space Mdof Gieseker-semistable sheaves on Xwith
Chern character (2,0,−2,0) and satisfying H1(E(−1)) = 0 is isomorphic to a moduli space
Mσ(Ku(X),2[Il]) of σ-semistable objects in Ku(X)with numerical class twice of that of an
ideal sheaf of a line in X.
BRIDGELAND STABILITY OF MINIMAL INSTANTON BUNDLES ON FANO THREEFOLDS 3
As a result, Mσ(Ku(X),2[Il]) is a smooth projective variety of dimension 5 by the descrip-
tion of Mdin [Dr00][Qin1][Qin2] for d>3 (see Section 2). We mention that by [Dr00][Qin2],
the vanishing H1(E(−1)) = 0 is satisfied by any Gieseker-semistable sheaves Ewith Chern
character (2,0,−2,0) for d= 3 and 4.
We note that most of our arguments for Theorem 1.3 work for d= 1,2. However there is
no complete description of M2or M1to the author’s knowledge.
In the last section, we consider non-minimal instantons. Kuznetsov[Ku12] showed that
one can associate to such an instanton a unique vector bundle in Ku(X), called its acyclic
extension (for details see Section 6). We establish the (semi)stability of these acyclic exten-
sions of instantons with charge 3.
Theorem 1.4. Assume d6= 1. Let ˜
Ebe the acyclic extension of an instanton bundle Eof
charge 3. Then ˜
Eis σ-semistable for any stability condition σ∈ K. If d>3,˜
Eis σ-stable.
Other related work. Lahoz, Macr`ı and Stellari[LMS] constructed the first examples of
Bridgeland stability conditions on the Kuznetsov component of a cubic threefold. They
proved that the moduli space of semistable objects with numerical class 2[Il] is isomorphic to
M3for properly chosen stabilities. We note it is not known whether the stability constructed
there is in K.
In a recent work [BBF+], the authors studied the moduli space of semistable objects with
numerical class [Il] + [S(Il)] with respect to some σ∈ K on a cubic threefold, where Sis
the Serre functor of Ku(X). They showed the moduli space is isomorphic to an blow-up of
the theta divisor of the intermediate Jacobian.
Petkovic and Rota[PR20] classified the stable objects in the moduli space containing the
Fano surface of lines when d= 1.
During the completion of this paper, the author was made aware by Zhiyu Liu and Shizhuo
Zhang of their independent preprint[LZ21] in which they claim similar results to Theorem
1.2 and 1.3.
Acknowledgement. I am very grateful to Laura Pertusi for answering my questions on
Bridgeland stabilities, for many interesting discussions, as well as her comments on an early
draft of this paper. I would like to thank Justin Sawon for interesting discussions and his
support. I thank Zhiyu Liu and Shizhuo Zhang for informing me of their result and sending
their draft. I am very grateful to an anonymous referee for suggesting Lemma 6.4 and its
proof. Finally, I thank the referees for careful reading of the paper and useful suggestions.
2. Review on Fano threefolds and minimal instanton bundles
2.1. Fano threefolds of Picard rank 1and index 2.A Fano variety Xis a smooth
projective variety whose anticanonical divisor −KXis ample. Its index i(X) is defined to
be the largest integer so that −KX=i(X)Hfor some ample divisor H. It is well-known
that i(X)6dim(X) + 1 (see [IP99]), with equality holds only if Xis Pdim(X). For a Fano
threefold X, this means i(X) is either 1,2,3 or 4. If i(X) = 3, then Xis a smooth quadric
in P4. If i(X) = 1 or 2, the most interesting cases are those with Pic(X)∼
=Zand they were
classified by Iskovskih[IP99].
In this paper, we will only be interested in Fano threefolds Xwith Picard rank 1 and
index 2. Let Hbe the ample generator of Pic(X). Let d:= H3denote the degree of X.
Then Iskovskih’s classification[IP99] asserts 1 6d65, and:
•if d= 5, X5→P6is a codimension 3 linear section of Gr(2,5) in its Pl¨ucker
embedding;
•if d= 4, X4→P5is a complete intersection of two smooth quadrics;
4 XUQIANG QIN
•if d= 3, X3→P4is a cubic threefold;
•if d= 2, X2→P3is a double cover of P3ramified in a quartic surface;
•if d= 1, X1is a hypersurface of degree 6 in a weighted projective space P(1,1,1,2,3).
Let Xbe a Fano threefold of Picard rank 1 and index 2. Then
H2(X, Z) = H4(X, Z) = H6(X, Z) = Z
and they are generated by the class of a hyperplane, a line and a point respectively. As a
result, we will refer to the Chern classes of coherent sheaves on Xas integers. The ample
generator of the Picard group will be denoted by OX(1), thus ωX∼
=OX(−2). It is an
elementary computation to check that
td(TX) = (1,1,1 + d
3,1).
2.2. Derived categories of of Fano threefolds of index 2.Let Xbe a Fano threefold of
Picard rank 1 and index 2. In this section, we review some facts about the bounded derived
category of coherent sheaves on X. For basic notions on derived category and semiorthogonal
decomposition, we refer the readers to [Huy].
Definition 2.1. [Ku09] The collection of line bundles {OX,OX(1)}is exceptional. The
Kuznetsov component Ku(X) is defined by the semiorthogonal decomposition
Db(X) = hKu(X),OX,OX(1)i.
We describe Ku(X) for 2 6d65:
•if d= 5, Ku(X5)∼
=Db(Q3) where Q3is the Kronecker quiver with three arrows;
•if d= 4, Ku(X4)∼
=Db(C) where Cis a smooth projective curve of genus 2;
•if d= 3, Ku(X3) has a Serre functor SKu(X3)satisfying S3
Ku(X3)= [5];
•if d= 2, Ku(X2) has a Serre functor SKu(X2)satisfying SKu(X3)=ι[2] where ιis the
involution induced by the double cover.
2.3. Instanton bundles. Let Xbe a Fano threefold of Picard rank 1, index 2 and degree
d.
Definition 2.2. [Ku12] An instanton bundle of charge non Xis a stable vector bundle E
of rank 2 with c1(E)=0, c2(E) = n, enjoying the instantonic vanishing condition:
H1(X, E(−1)) = 0.
We mention that the charge c2(E)>2 [Ku12, Corollary 3.2]. Instanton bundles of charge
2 are called the minimal instantons. We also mention here that if Eis a minimal instanton,
then E∈Ku(X) by [Ku12, Lemma 3.1].
By definition, the moduli space of minimal instanton bundles is an open subscheme of the
moduli space of Gieseker-semistable sheaves with Chern character (2,0,−2,0). The classi-
fication of Gieseker-semistable sheaves with Chern character (2,0,−2,0) was first achieved
by Druel[Dr00] for d= 3, and later generalized to d= 5,4 by the author[Qin1][Qin2].
Proposition 2.3. Assume 36d65. Let Ebe a Gieseker-semistable sheaf on Xwith
Chern character (2,0,−2,0). If Eis stable, then either Eis locally free or Eis associated
to a smooth conic Y⊂Xso that we have an exact sequence:
0→E→H0(θ(1)) ⊗ OX→θ(1) →0
where θis the theta-characteristic of Y.
If Eis strictly Gieseker-semistable, then Eis the extension of two ideal sheaves of lines.
BRIDGELAND STABILITY OF MINIMAL INSTANTON BUNDLES ON FANO THREEFOLDS 5
Remark 2.4.Along with the above classification, it was proved in [Dr00][Qin2] that for
d= 3 or 4, any Gieseker-semistable sheaf Ewith Chern character (2,0,−2,0) satisfies the
vanishing condition:
H1(E(−1)) = 0.(2.5)
In [Qin1], the author conjectured that the same thing holds for d= 5. We also mention
that for a Gieseker-semistable sheaf Ewith Chern character (2,0,−2,0), satisfying (2.5) is
equivalent to E∈Ku(X).
Using this description, the moduli space of Gieseker-semistable sheaves with Chern char-
acter (2,0,−2,0) were studied in [Dr00][Qin1][Qin2].
Theorem 2.6. Assume 36d65. The moduli space Mdof Gieseker-semistable sheaves
with Chern character (2,0,−2,0) and satisfying the vanishing condition (2.5) is a smooth
projective variety of dimension 5. More specifically:
•if d= 5,M5is isomorphic to P5;
•if d= 4,M4is a P3-bundle over the Jacobian of the genus 2curve Cmentioned in
Section 2.2;
•if d= 3,M3is isomorphic to the blow-up of the intermediate Jacobian in (minus)
the Fano surface of lines.
3. Review on stability conditions
3.1. Slope-stability and Gieseker-stability. Let Xbe a smooth projective threefold.
Fix an ample line bundle Hon X. When Xis a Fano threefold of Picard rank one, we will
always take the ample generator of the Picard group as H.
Definition 3.1. For any coherent sheaf Fon X, the (Mumford-Takemoto)-slope is defined
as
µH(F) := (H2ch1(F)
H3ch0(F)if ch0(F)6= 0;
+∞if ch0(F) = 0.
A coherent sheaf Fis slope-(semi)stable if for any non-trivial proper subsheaf F0⊂F, we
have µH(F0)<(6)µH(F/F 0).
Next we recall the notion of Gieseker-stability. We will also use the refined notion of
2-Gieseker-stability introduced in [BBF+, Section 4] and [JM, Section 2.5]. We follow the
exposition of [BBF+].
Definition 3.2. We define a pre-order on the polynomial ring R[m] as follows:
(1) For all non-zero f∈R[m], we have f≺0.
(2) If deg(f)>deg(g) for non-zero f, g ∈R[m], then f≺g.
(3) If deg(f) = deg(g) for non-zero f, g ∈R[m] and let lfand lgbe the leading coefficient
of fand g, then fgif and only if f(m)/lf6g(m)/lgfor m0.
For any F∈Coh(X), we denote its Hilbert polynomial by P(F) := χ(F(mH)) =
P3
i=0 aimi. Moreover, let P2(F) := P3
i=1 aimi.
Definition 3.3. (1) We say Fis Gieseker-(semi)stable if for all nontrivial proper sub-
sheaf F0⊂F, the inequality P(F0)≺()P(F) holds.
(2) We say Fis 2-Gieseker-(semi)stable if for all nontrivial proper subsheaf F0⊂F,
the inequality P2(F0)≺()P2(F/F 0) holds.
6 XUQIANG QIN
The three notions imply each other in the following way:
slope-stable ⇒2-Gieseker-stable ⇒Gieseker-stable
⇓
slope-semistable ⇐2-Gieseker-semistable ⇐Gieseker-semistable
3.2. (Weak) stability conditions on triangulated categories. In this section we review
elements in the theory of (weak) stability conditions on threefolds which are essential for
our discussion. We refer the readers to [Br07] for basic notions of t-structure and slicing.
Let Dbe a triangulated category. We first recall the notion of a heart.
Definition 3.4. Aheart of a bounded t-structure on Dis a full additive subcategory Asuch
that:
(1) if i>j are integers, then HomD(A[i], B[j]) = 0 for all A, B ∈ A.
(2) for any nonzero object F∈ D, there exists a sequence of morphisms
0 = F0
φ1
−→ F1
φ2
−→ · · · φm
−−→ Fm=F
so that Cone(φi) is of the form Ai[ki] for Ai∈ A and integers k1> k2>· · · km.
Note a heart Aof a t-struecture is an abelian category.
Definition 3.5. Let Abe an abelian category. A group homomorphism Z:K(A)→Cis
called a weak stability function if for any nonzero object F∈ A,=Z(F)>0 and =Z(F)=0
only if <Z(F)60.
We call Zastability function if in addition, =Z(F) = 0 implies <Z(F)<0 for F6= 0.
Fix a finite rank lattice Λ and surjective group homomorphism v:K(A)→Λ.
Definition 3.6. Aweak stability condition on Dwith respect to Λ is a pair σ= (A, Z ) where
Ais the heart of a bounded t-structure on Dand Z: Λ →Cis a group homomorphism,
such that the following conditions hold:
(1) The composition Av
−→ ΛZ
−→ Cis a weak stability function. For F∈ A, we write
Z(F) := Z(v(F)) for simplicity. We define
µσ(F) = (−<Z(F)
=Z(F)if =Z(F)>0;
+∞otherwise
We call F σ-(semi)stable if for all nonzero subobject F0⊂F, we have
µσ(F0)<(6)µσ(F /F 0).
(2) Any object of Ahas a Harder-Narasimhan(HN) filtration in σ-semistable objects
(called HN factors).
(3) There exists a quadratic form Qon Λ ⊗Rsuch that Q(F)>0 for any σ-semistable
object F∈ A and Qis negative definite when restricted to the kernel of Z.
If in addition Z◦v:K(A)→Cis a stability function, we call σaBridgeland stability
condition.
We use Stab(D) to denote the set of Bridgeland stability conditions on D. Bridgeland[Br07]
showed that Stab(D) has the structure of a complex manifold. Moreover, if we use ˜
GL+
2(R)
to denote the universal cover of GL+
2(R), then there is a right group action of ˜
GL+
2(R) on
Stab(D). We refer the readers to [Br07] for details of this action.
BRIDGELAND STABILITY OF MINIMAL INSTANTON BUNDLES ON FANO THREEFOLDS 7
Given a weak stability condition σ= (A, Z) on D, one can construct a new heart of a
bounded t-structure via the method of tilting : let µ∈R, define
Tµ
σ={E∈ A : all HN factors Fof Ehave slope µσ(F)> µ};
Fµ
σ={E∈ A : all HN factors Fof Ehave slope µσ(F)6µ}.
Proposition 3.7. [HRS] The category
Aµ
σ=<Tµ
σ,Fµ
σ[1] >
is the heart of a bounded t-structure on D.
Next we review the notion of tilt-stability. We will follow the exposition of [BLMS, Section
2]. For j= 0,...,3, define Λj
H∼
=Zj+1 as the lattice generated by vectors of the form
(H3ch0(F), H2ch1(F), . . . , H 3−jchj(F)) ∈Qj+1
together with the natural map vj
H:K(X)→Λj
H.
Then the pair (Coh(X), ZH) with
ZH(F) = −H2ch1(F) + iH3ch0(F)
defines a weak stability condition with respect to Λ1
H. In this case we can take the quadratic
form Q= 0 since Zis injective. This notion of stability coincide with (Mumford-Takemoto)-
slope-stability and will be referred to as such. We will use µHto denote the slope to H.
Any slope-semistable sheaf Fsatisfies the Bogomolov-Gieseker inequality:
∆H(F) := (H2ch1(F))2−2H3ch0(F)·Hch2(F)>0.(3.8)
Choose a parameter β∈R. By Proposition 3.7,
Definition 3.9. We denote by Cohβ
H(X)⊂Db(X) the heart of a bounded t-structure
obtained by tilting the slope stability at µH=β.
Remark 3.10.When Xis a Fano threefold of Picard rank 1, we will always take the ample
generator of the Picard group as Hand drop the subscript Hfrom related notations.
For a coherent sheaf F, we consider the twisted Chern character chβ(F) = e−βH ch(F).
More explicitly:
chβ
0= ch0
chβ
1= ch1−βHch0
chβ
2= ch2−βHch1+β2
2H2ch0
chβ
3= ch3−βHch2+β2
2H2ch1−β3
6H3ch0.
Proposition 3.11. [BLMS, Proposition 2.12] Given α > 0,β∈R, the pair σα,β =
(Cohβ(X), Zα,β )with
Zα,β =1
2α2H3chβ
0(F)−Hchβ
2(F) + iH2chβ
1(F)
defines a weak stability condition on Db(X)with respect to Λ2. The quadratic form Qcan
be given by the discriminant ∆Hdefined in (3.8).
These weak stability conditions vary continuously as (α, β)∈R>0×Rvaries.
When the choices of (α, β) are clear, σα,β -(semi)stability is usually referred to as tilt-
(semi)stability. The notion of 2-Gieseker-stability occurs as limit of tilt stability:
8 XUQIANG QIN
Proposition 3.12. [BBF+, Proposition 4.8][Br08, Proposition 14.2][JM, Theorem 5.2] Let
F∈Db(X)and β < µ(F). Then F∈Cohβ(X)and Fis σα,β -(semi)stable for α0if
and only if F∈Coh(X)and Fis 2-Gieseker-(semi)stable.
We will also need to following result of [Li19, Proposition 3.2]. The original proposition
in [Li19] is more general, our version follows from it by explicit computation.
Proposition 3.13. [Li19, Proposition 3.2] Let Xbe a Fano threefold of Picard rank 1and
index 2and Fbe σα,β-stable with ch0(F)6= 0 for some α > 0, β ∈R.
(1) If d= 5 and −q3
20 6µH(F)6q3
20 , then Hch2(F)
H3ch0(F)60.
(2) If d= 4 and √3
46|µH(F)|61−√3
4, then Hch2(F)
H3ch0(F)61
2(µH(F))2−3
32 .
(3) If d= 3 and −1
26µH(F)61
2, then Hch2(F)
H3ch0(F)60.
(4) If d= 3 and 1
2<|µH(F)|61, then Hch2(F)
H3ch0(F)6|µH(F)| − 1
2.
(5) If d= 2 and −1
26µH(F)61
2, then Hch2(F)
H3ch0(F)60.
Moreover, if the equality holds in any of the cases above, then ch0(F)is 1or 2.
Finally we recall a variant of the tilt stability conditions, which is needed in the next
section. Fix µ∈R, apply Proposition 3.7 to σα,β , we obtain a heart, which we denote by
Cohµ
α,β (X) := Aµ
σα,β .
Let u∈Cbe the unit vector in the upper half plane with µ=−<u
=u.
Proposition 3.14. [BLMS, Proposition 2.15] The pair σµ
α,β := (Cohµ
α,β (X), Zµ
α,β ), where
Zµ
α,β := 1
uZα,β
is a weak stability condition on Db(X)with respect to Λ2.
3.3. Stability conditions on the Kuznetsov component. Let Xbe a Fano threefold
of Picard rank 1 and index 2. We will induce stability conditions on Ku(X) from the
weak stability conditions σ0
α,β of Proposition 3.14. Set A(α, β ) := Coh0
α,β (X)∩Ku(X) and
Z(α, β) = Z0
α,β |Ku(X).
Theorem 3.15. [BLMS, Theorem 6.8][PY20, Theorem 3.3] Suppose −1
26β < 0,0< α <
−β, or −1< β < −1
2,0< α 61 + β. Then the pair
σ(α, β) := (A(α, β ), Z(α, β))
is a Bridgeland stability condition on Ku(X)with respect to Λ2
Ku(X), where
Λ2
Ku(X):= Im(K(Ku(X)) →K(X)→Λ2)∼
=Z2.
We mention the existence of such a Bridgeland stability condition on Ku(X) was first
proved in [BLMS, Theorem 6.8] with β=−1
2. The specific triangular region was computed
in [PY20, Theorem 3.3].
In fact, the stability conditions constructed above are the same up to the action of
˜
GL+
2(R).
Proposition 3.16. [PY20, Proposition 3.6] Fix 0< α0<1
2. For any (α, β)with −1
26
β < 0,0< α < −β, or −1< β < −1
2,0< α 61 + β, there is ˜g∈˜
GL+
2(R)such that
σ(α, β) = σ(α0,−1
2)·˜g.
BRIDGELAND STABILITY OF MINIMAL INSTANTON BUNDLES ON FANO THREEFOLDS 9
In light of this result, we set
K:= σ(α0,−1
2)·˜
GL+
2(R)⊂Stab(Ku(X))
Then all stability conditions constructed in Theorem 3.15 are in K.
4. Bridgeland stability of minimal instanton bundles
For the rest of this paper, let Xbe a Fano threefold of Picard rank one, index two and
degree d. Let Ebe a minimal instanton bundle on X. The Chern character of Eis
ch(E) = (2,0,−2,0),
and the twisted Chern character with respect to −1
2till degree 2 is
ch−1
2
62(E) = (2, H, d−8
4).
Proposition 4.1. Let Ebe a minimal instanton bundle on X. Then Eis σ(α0,−1
2)-
semistable for some 0< α0<1
2. As a result, Eis semistable for any stability condition in
K.
Proof. Eis stable by definition. By [Sa14, Lemma 1.23], Eis µ-stable. As µH(E)=0>−1
2,
we have E∈Coh−1
2(X). Since H2ch−1
2
1>0, Eis σα,−1
2-stable for α0 by [BMS, Lemma
2.7(c)]
Next we show that Eis σα,−1
2-semistable for some α < 1
2. A wall would be given by a
short exact sequence in the heart Coh−1
2(X) of the form
0→E0→E→E00 →0,
such that the following conditions hold:
(1) µα,−1
2(E0) = µα,−1
2(E) = µα,−1
2(E00);
(2) ∆H(E0)>0, ∆H(E00)>0;
(3) ∆H(E0)6∆H(E), ∆H(E00)6∆H(E);
(4) ch−1
2
62(E0)6= (1,1
2H, d−8
8),ch−1
2
62(E0)6= (2, H, d−8
4).
Note we require the last condition because ch(E) is not primitive and we are considering
walls that might break the semistability of E.
The truncated twisted characters of E0and E00 have to satisfy
(2, H, d−8
4)=(a, b
2H, c
8) + (2 −a, 2−b
2H, 2d−16 −c
8)
for some a, b, c ∈Z. Since E0and E00 are in Coh−1
2(X), we have b>0 and 2 −b>0. Thus
b= 0,1 or 2.
As µα,−1
2(E) = d−8−4dα2
4dand ∆H(E)=8d, the first three conditions become
(1) 1
b(c
4d−α2a) = d−8−4dα2
4d=1
2−b(2d−16−c
4d−α2(2 −a));
(2) ( b
2)2−ac
4d>0, (2−b
2)2−(2−a)(2d−16−c)
4d>0
(3) ( b
2)2−ac
4d68
d, (2−b
2)2−(2−a)(2d−16−c)
4d68
d.
Suppose b= 0. If a6= 0, then
4dα2=c
a>0 and ac 60
which is impossible. If a= 0, then c= 0 by the first equation. This means either E0or
E00 has twisted Chern character (2, H, d−8
4). The case that E0has twisted Chern character
10 XUQIANG QIN
(2, H, d−8
4) is excluded by condition (4). If E00 has twisted Chern character (2, H, d−8
4),
then Ewould have a subobject with infinite slope, again contradicting the stability of Efor
α0. The case b= 2 can be excluded by symmetry and the above argument.
Suppose b= 1, then 2 −b= 1. By condition (4), we can assume a6= 1, then either aor
2−awill be greater than or equal to 2. We assume a>2 without loss of generality. We
have then either ch(E0) or ch(E00) being equal to
(a, 1
2(1 −a)H, .....)
which implies amust be odd and a>3. Using (1) and the assumption b= 1, we obtain
c=d−8+4dα2(a−1).
Combine this with the second equation in (2), we obtain
(a−2)(4dα2(a−1) −d+ 8)
4d61
4
which simplifies to
4dα2(a−1) 6d
a−2+d−862d−8.
This leads to contradictions when 1 6d64, since the left hand side is positive. In these
cases there are no walls for the tilt-semistability of E.
If d= 5, the above equation becomes
α261
10(a−1) 61/20.
In this case there are no walls for α > 1
√20 .
As a result, for all 1 6d65, we can choose 1
√20 < α0<1
2so that Eis σα0,−1
2-semistable.
Since Eis torsion free, this implies Eis σ0
α0,−1
2
-semistable and thus σ(α0,−1
2)-semistable
for the same 0 < α0<1
2. By [PY20, Section 3.3], Eis semistable for all stability conditions
in K ⊂ Stab(Ku(X)).
Corollary 4.2. Let Ebe a minimal instanton bundle on X. If the degree of Xsatisfy d>2,
then Eis stable for any stability condition in K.
Proof. It suffices to show the stability of Efor σ(α0,−1
2). Suppose Eis strictly σ(α0,−1
2)-
semistable. By [PY20, Lemma 3.9], it has two Jordan-H¨older factors, each having numerical
class as that of the ideal sheaf of a line. By [PY20, Proposition 4.6], the Jordan-H¨older
factors are ideal sheaves of lines. This contradicts the µ-stability of E. Thus Eis σ(α0,−1
2)-
stable.
For the rest of this section, assume d>3. We look at non-locally free generalizations of
minimal instanton bundles. Let Cbe a smooth conic on Xwith d>3. Note C'P1and
we denote by θits theta characteristic (dual of the ample generator in Pic(C)). Let Ebe
the torsion free sheaf defined by the short exact sequence
0→E→H0(θ(1)) ⊗ OX→θ(1) →0.(4.3)
Then by [Dr00][Qin1]][Qin2], Eis Gieseker-stable and E∈Ku(X).
Proposition 4.4. Let Ebe a sheaf on Xas defined in (4.3). Then Eis stable for any
stability condition in K.
BRIDGELAND STABILITY OF MINIMAL INSTANTON BUNDLES ON FANO THREEFOLDS 11
Proof. The proof is essentially the same as above. Although Eis not slope-stable, we note
since Eis Gieseker-stable, it is 2-Gieseker-semistable. By Proposition 3.12, Eis σα,−1
2-
semistable for α0. We can then apply the arguments for walls in Proposition 4.1 to
show Eis semistable for stability conditions in K. The fact that Eis stable follows from
the arguments of Corollary 4.2.
Remark 4.5.By [PY20, Proposition 4.4], extensions of ideal sheaves of lines on Xare strictly
σ-semistable for every σ∈ K. In fact, the proof of Corollary 4.2 shows that they are the
only strictly σ-semistable objects with numerical class 2[Il].
5. Description of the moduli space Mσ(Ku(X),2[Il])
In this section we will show that for σ∈ K,σ-semistable objects in Ku(X) with numerical
class 2[Il]∈ N (Ku(X)) are Gieseker-semistable sheaves with Chern character (2,0,−2,0)
up to shifts when d>3. We will then use this result to describe Mσ(Ku(X),2[Il]). We need
the following two lemmas.
Lemma 5.1. Let Ebe a 2-Gieseker-semistable sheaf with rk(E)=2, c1(E)=0, c2(E)=2
and c3(E)=0on X. Then Eis Gieseker-semistable.
Proof. Suppose Eis Gieseker-unstable. Then the Harder-Narasimhan filtration of Ein
Coh(X) is of the form
0→E0→E→E00 →0
with E0, E00 Gieseker-semistable of rank 1 and p(E0)p(E). Note ch(E) = (2,0,−2,0).
Since Eis 2-Gieseker-semistable, it is easy to see that we must have ch1(E0) = ch1(E00)=0
and ch2(E0) = ch2(E00) = −1. Then E0and E00 are ideal sheaves of closed subschemes of
dimension 1, which we denote by D0and D00 respectively. Since E0is Gieseker-destabilizing,
ch(E0) = (1,0,−1, m) with m>1. The Hilbert polynomial of D0is thus PD0(t) = t+ 1 −m,
this contradicts [Sa14, Corollary 1.38(1)].
Lemma 5.2. Assume d>3. Let E∈ O⊥
Xbe an object with Chern character (2,0,−2,0).
Then the vertical line β=−1does not intersect any actual wall for the σ0
α,β -stability of E.
Proof. The proof is similar to that of Proposition 4.1. We provide it for the convenience of
the readers. The twisted Chern character of Eis
ch−1
62(E) = (2,2H, d −2)
A wall will provide a equation of the twisted Chern characters:
(2,2H, d −2) = (a, bH, c
2) + (2 −a, (2 −b)H, d −2−c
2).(5.3)
where a, b, c ∈Zsatisfying
(1) 1
b(c
2−1
2dα2a) = d−2−dα2
2=1
2−b(d−2−c
2−1
2dα2(2 −a));
(2) b2−ac
d>0 (2 −b)2−(2−a)(2d−4−c)
d>0
(3) b2−ac
d68
d, (2 −b)2−(2−a)(2d−4−c)
d68
d.
By the definition of a stability function on an abelian category, band 2 −bcannot have
opposite signs, so b(2 −b)>0 and thus 0 6b62.
Suppose b= 0. If a6= 0, then
1
2dα2=c
2a>0 and ac
d60
12 XUQIANG QIN
which is impossible. If a= 0, then c= 0. If the twisted Chern character (a, bH, c
2)
corresponds to a subobject, then d−2−dα2
0= 0. So α2
0= 1 −2
d. In this case we have a
short exact sequence in Coh0
α0,−1
0→ OZ→E[s]→B→0
where s∈Zis a potential shift and Zis 0-dimensional subscheme on X. One can easily
check OX∈Coh0
(α0,−1). By the assumption that E∈ O⊥
X, we have Hom(OX,OZ) =
Hom(OX, B[−1]) = 0, this is absurd if Zis non-empty. On the other hand, having a
quotient object with twisted Chern character (0,0,0) does not affect semistability. The case
b= 2 follows by symmetry.
Suppose b= 1, we have c=dα2(a−1) + d−2. Without loss of generality, we assume
a>1. Note if a= 1, the two terms on the right hand side of equation (5.3) are equal, which
means such a wall will not affect semistability. For a>2, the first equation of (2) provides
a(a−1)α261−a(1 −2
d).
For d>4, we get
a(a−1)α261−2(1 −1
2) = 0.
which is absurd since the left hand side is positive.
For d= 3, we obtain similar contradiction as above if a>3. If a= 2, α261
6. In this
case ccannot be an integer.
Proposition 5.4. Assume d>3. If F∈Ku(X)is σ-stable for some σ∈ K with [F] =
2[Il]∈ N (Ku(X)), then F∼
=E[2k]for some Gieseker-stable sheaf Eof rank 2, c1(E) =
0, c2(E)=2, c3(E) = 0 and k∈Z.
Proof. We follow the idea of the proof of [PY20, Proposition 4.6]. Set ch(F) := (a0, a1, a2, a3).
As [F] = 2[Il]∈ N (Ku(X)) and χ(Il,Il) = −1, the following conditions hold:
χ(OX, F )=0, χ(OX(1), F )=0, χ(Il, F ) = −2, χ(F, Il) = −2
By Hirzebruch-Riemann-Roch Theorem and a similar computation as [PY20], we obtain
ch(F) = 2ch(Il) = (2,0,−2,0).
By [PY20, Proposition 3.6] and the assumption, Fis σ(α, β )-stable for every (α, β)∈V.
In particular, for all (α, β)∈Vsatisfying β2−α262/d, we have F[2k+ 1] ∈ A(α, β) for
some integer k. Up to shifting, we assume G:= F[1] ∈ A(α, β ) is σ(α, β)-stable for (α, β)
satisfying β2−α262/d. Then Ghas slope
σ0
α,β (G) = −β
1
d+1
2α2−1
2β2.
In particular, σ0
α,β (G) = +∞if
β2−α2= 2/d.(5.5)
Since d>3, then there exists pairs (¯α, ¯
β)∈Vso that (5.5) holds. Gis σ0
¯α, ¯
βsemistable
as it has the largest slope (+∞) in heart. By Lemma 5.2, β=−1 is not on a wall for the
σ0
α,β -stability of G. Moreover, the semicircle Cwith center (0,−d+2
2d) and radius d−2
2dgives
a numerical wall for G, potentially realized by OX(−1)[2] ∈Coh0
α,β (X).
Assume that Cis not an actual wall for G. All other walls would be nested semicircles in
C. Thus we may choose (¯α, ¯
β) = (d−2
2d,−d+2
2d)∈ C, so that Gis σ0
¯α, ¯
β-semistable and remains
so for ¯
βapproaching −1
2.
BRIDGELAND STABILITY OF MINIMAL INSTANTON BUNDLES ON FANO THREEFOLDS 13
By definition of Coh0
α,−1
2(X), we have a triangle
A[1] →G→B
where A(resp. B) is in Coh−1
2(X) with σα,−1
2-semistable factors having slope µα,−1
2
6
0(resp. >0). Since Gis σ0
α,−1
2
-semistable, Bis either 0 or supported on points. In any
case, A[1] is also σ0
α,−1
2
-semistable, which implies Ais σα,−1
2-semistable. Note ch(A) =
(2,0,−2, l), where l>0 is the length of the support of B. The wall computation in Propo-
sition 4.1 shows that Ais σα,−1
2-semistable for every α > 0 if d= 3,4 and α > 1
√20 if d= 5.
By Proposition 3.12, Ais a 2-Gieseker-semistable sheaf. When d= 5, applying [BLMS,
Conjecture 4.1],[Li19] for α=1
√20 β=−1
2, we obtain
l64
3α2+1
3+8
3d
64
3·1
20 +1
3+8
15 =14
15
Thus l= 0 if d= 5. When d6= 5, applying [BLMS, Conjecture 4.1],[Li19] for α= 0 β=−1
2,
we obtain
l61
3+8
3d
for d>3. Then lis 0 or 1 for 3 6d64. We claim l= 0. Suppose otherwise, since
G∈Ku(X), we have Hi(A) = 0 for all iexcept when i= 2 and H2(A) = C. Let qbe a
generic point on Xand consider the elementary modification A0of Adefined by the short
exact sequence
0→A0→A→ Oq→0.(5.6)
Then A0is a 2-Gieseker-semistable sheaf with ch(A0) = (2,0,−2,0). By Lemma 5.1, A0is
Gieseker-semistable. By [Dr00],[Qin2], A0∈Ku(X). Along with (5.6), we see H2(A) = 0,
which contradicts our computation above. Hence l= 0 for all d= 3,4,5. Consequently,
G=A[1] and F=A∈Ku(X) is a 2-Gieseker-semistable sheaf. By Lemma 5.1, Fis a
Gieseker-semistable sheaf with rank 2, c1(E)=0, c2(E) = 2, c3(E) = 0. Fis Gieseker-stable
since otherwise Fwould be an extension of two ideal sheaves of lines, making it strictly
σ-semistable.
Assume Cdefines an actual wall for Gand Gbecomes unstable for β→ −1
2. Set ( ¯α, ¯
β) =
(d−2
2d,−d+2
2d). Gis then strictly σ0
¯α, ¯
β-semistable and we have a short exact sequence
0→P→G→Q→0
in Coh0
¯α, ¯
β(X), where P, Q are σ0
¯α, ¯
β-semistable with infinite slope. It is a tedious exercise to
find the potential cases using
• =(Z0
¯α, ¯
β(P)) = =(Z0
¯α, ¯
β(Q)) = 0;
• <(Z0
¯α, ¯
β(P)) 60,<(Z0
¯α, ¯
β(Q)) 60;
•∆(P)>0, ∆(Q)>0
We list them here:
(1) d= 5,ch62(P) = (1,−1,5
2),ch62(Q)=(−3,1,−1
2);
(2) d= 4,ch62(P) = (0,−1,3),ch62(Q)=(−2,1,−1));
(3) d= 4,ch62(P) = (1,−1,2),ch62(Q)=(−3,1,0);
(4) d= 4,ch62(P) = (2,−2,4),ch62(Q)=(−4,2,−2);
(5) d= 3,ch62(P) = (0,−1,5
2),ch62(Q)=(−2,1,−1
2);
14 XUQIANG QIN
(6) d= 3,ch62(P) = (1,−2,4),ch62(Q)=(−3,2,−2);
(7) d= 3,ch62(P) = (2,−3,11
2),ch62(Q)=(−4,3,−7
2);
(8) d= 3,ch62(P) = (1,−1,3
2),ch62(Q)=(−3,1,1
2);
(9) d= 3,ch62(P) = (2,−2,3),ch62(Q)=(−4,2,−1);
(10) d= 3,ch62(P) = (3,−3,9
2),ch62(Q)=(−5,3,−5
2);
(11) d= 3,ch62(P) = (4,−4,6),ch62(Q)=(−6,4,−4).
All the cases are ruled out by Lemmas 5.7 to 5.10. This completes the proof.
Lemma 5.7. Cases (2)(6) can be ruled out.
Proof. Note in both cases ∆(Q) = 0 and gcd(ch0(Q),ch1(Q)) = 0. Let Hi(Q) denote the
i-th cohomology of Qin Coh¯
β(X). Since Qis σ0
¯α, ¯
β-semistable, we see H0(Q) is either 0
or supported on points. Moreover, H−1(Q) is σ¯α, ¯
β-semistable with µ¯α, ¯
β(H−1(Q)) = 0. By
[BMS, Corollary 3.10], H−1(Q) is in fact σ¯α,¯
β-stable. Then cases (2)(6) are ruled out by
Proposition 3.13(2) and (4) respectively.
Lemma 5.8. Cases (5)(7) can be ruled out.
Proof. Arguing as in the previous lemma, we get H−1(Q) is σ¯α, ¯
β-semistable with µ¯α, ¯
β(H−1(Q)) =
0. If H−1(Q) is in fact σ¯α, ¯
β-stable, then cases (5)(7) can be ruled out by Proposition 3.13(3)
and (4) respectively.
If H−1(Q) is strictly σ¯α, ¯
β-semistable. Note for any F∈Coh¯
β(X), ∆(F)/d is an integer.
In both cases (5)(7), we have ∆(H−1(Q))/3 = 1. By [BMS, Lemma 3.9] [BMS, Corollary
3.10], there is a short exact sequence
0→Q0→ H−1(Q)→Q00 →0
in Coh¯
βwhere Q0and Q00 are σ¯α, ¯
β-semistable, with µ¯α, ¯
β(Q0) = µ¯α, ¯
β(Q00) = 0 and ∆(Q0) =
∆(Q00) = 0. It is a straightforward computation to check that we will have either Q0or
Q00 with ch62= (3,−2,2). We can then draw a contradiction using the arguments in the
previous lemma for case (6).
Lemma 5.9. Case (1) can be ruled out.
Proof. As before, H−1(Q) is σ¯α,¯
β-semistable with µ¯α, ¯
β(H−1(Q)) = 0. If H−1(Q) is in fact
σ¯α, ¯
β-stable, then cases (1) can be ruled out by Proposition 3.13(1).
If H−1(Q) is strictly σ¯α, ¯
β-semistable. We have ∆(H−1(Q))/5 = 2. By [BMS, Lemma
3.9] [BMS, Corollary 3.10], there is a short exact sequence
0→Q0→ H−1(Q)→Q00 →0
in Coh¯
βwhere Q0and Q00 are σ¯α, ¯
β-semistable, with µ¯α,¯
β(Q0) = µ¯α, ¯
β(Q00) = 0 and ∆(Q0),∆(Q00)
either 0 or 5. We denote ch62(Q0)=(a, bH, c
2), where a, b, c ∈Z. Then ch62(Q00) =
(3 −a, (−1−b)H, 1−c
2)
µ¯α, ¯
β(Q0)=0⇐⇒ 2a+ 7b+c= 0,
∆(Q0)/5 = 0 or 1 ⇐⇒ 5b2−ac = 0 or 1,
∆(Q00)/5 = 0 or 1 ⇐⇒ 5(1 + b)2−(3 −a)(1 −c) = 0 or 1.
It is then straightforward but tedious to check the above equations have no integer solutions
in (a, b, c).
Lemma 5.10. Cases (3)(4)(8)(9)(10)(11) can be ruled out.
BRIDGELAND STABILITY OF MINIMAL INSTANTON BUNDLES ON FANO THREEFOLDS 15
Proof. In these cases ch62(P) = n·(1,−H, H2
2). Let Hi(P) denote the i-th cohomology
of Pin Coh¯
β(X). Since Pis σ0
¯α, ¯
β-semistable, we see H0(P) is either 0 or supported on
points. Moreover, H−1(P) is σ¯α, ¯
β-semistable. Denote ch(H−1(P)) = (−n, nH, −nH2
2, m).
By [BMS, Conjecture 4.1][Li19], m6nH3
6. Note ∆(H−1(P)) = 0. By [BMS, Corollary
3.10], either H−1(P) is σ¯α,¯
β-stable or it is strictly σ¯α, ¯
β-semistable and its Jordan-H¨older
factors have ch62proportional to ch62(H−1(P)). Let Ri(1 6i6N) be the factors in a
Jordan-H¨older filtration of H−1(P), then ch(Ri) = (−ki, kiH, −kiH2
2, ri) for positive integers
kisuch that k1+· · · +kN=n. By [BMS, Corollary 3.11(c)] and using Hj(Ri) to denote the
j-th cohomology of Riin Coh(X), H0(Ri) has zero dimensional support (say of length li) and
H−1(Ri) is a slope-semistable sheaf with Chern character (ki,−kiH, kiH2
2,−ri+li). Then
H−1(Ri)⊗ OX(1) is a slope-semistable sheaf with Chern character (ki,0,0, kiH3
6−ri+li).
By [BBF+, Proposition 4.18(i)],
ki
H3
6−ri+li60(5.11)
for each 1 6i6N. Note
N
X
i=1
ri=m6nH3
6
Combined with (5.11), we have PN
i=0 li= 0. Since all liare nonnegative integers, we have
li= 0 and ri=kiH3
6for all i. By [BBF+, Proposition 4.18(i)], we see Ri∼
=OX(−1)⊕ki[1]
for all 1 6i6N. Hence H−1(P) = OX(−1)⊕n[1]. Next we claim H0(P) = 0. Suppose
otherwise, then we have a sequence
0→ OX(−1)⊕n[2] →G→Q0→0
in Coh0
¯α, ¯
β(X), where Q0is defined by 0 → H0(P)→Q0→Q→0. Since G∈Ku(X) and
Hi(OX(−1)) = 0 for all i, we have Hom(OX, Q0) = 0. We have the long exact sequence
→Hom(OX, Q[−1]) →Hom(OX,H0(P)) →Hom(OX, Q0)→
Note the first term is 0 since OXand Qare in the same heart, thus Hom(OX,H0(P)) = 0
and in turn H0(P) = 0, i.e. P=OX(−1)⊕n[2].
We can assume that nis maximal, that is, OX(−1)[2] is not a subob ject of Qin
Coh0
¯α, ¯
β(X). It is easy to compute that χ(OX(−1)[2], Q) = −2d−nand χ(P, Q) = −2nd −
n2<0. Note Hom(P, Q[i]) is 0 if i < 0 or i > 3. We have
Hom(P, Q[3]) = Hom(Q, OX(−3)⊕n[2]) = 0.
Then Hom(P, Q[1]) >0. Moreover, since Hom(OX(−1)[2], Q[1]) >2d+n>n, we can define
G0as a extension
0→Q→G0→P→0
in Coh0
¯α, ¯
β(X) which is determined by nindependent vectors in Hom(OX(−1)[2], Q[1]).
We claim G0is σ0
α,β -semistable above the wall Cin a neighbourhood of (¯α, ¯
β). By our
previous three lemmas and the first paragraph of this proof, it suffices to show OX(−1)[2]
is not a subobject of G0in Coh0
¯α, ¯
β(X). Suppose otherwise. Since nis maximal, the induced
map from the subobject OX(−1)[2] to P=OX(−1)⊕n[2] is nontrivial. However, the com-
position of this induced map OX(−1)[2] →Pwith P→Q[1](defined by G0) is trivial, which
contradicts our construction of G0.
16 XUQIANG QIN
Since Hom(OX, G0[i]) = 0 for all i, by Lemma 5.2, G0is σ0
α,β -semistable for β→ −1
2.
We can argue as in the previous case and conclude that G0=F0[1] where F0is a Gieseker-
semistable sheaf with Chern character (2,0,−2,0). Then F0∈Ku(X), so Hom(G0, P )=0
gives a contradiction.
Remark 5.12.We note the same arguments work for case (1) until the last line when we
conclude F0∈Ku(X). See Remark 2.4.
Theorem 5.13. Let Xbe a Fano threefold of Picard rank one, index two and degree d.
If d>3, then for any σ∈ K, the moduli space of Gieseker-semistable sheaves on Xwith
Chern character (2,0,−2,0) and satisfying H1(E(−1)) = 0 is isomorphic to a moduli space
Mσ(Ku(X),2[Il]) of σ-semistable objects in Ku(X)with numerical class twice of that of an
ideal sheaf of a line in X.
Proof. By Propositions 4.1, 4.4, 5.4, Corollary 4.2 and Remark 4.5, we see the notion of
Gieseker-semistable sheaves on Xwith Chern character (2,0,−2,0) and satisfying H1(E(−1)) =
0 is the same as σ-semistable objects in Ku(X) with numerical class 2[Il] up to shifts. More-
over, the S-equivalences are compatible. As a result, having a family of Gieseker-semistable
sheaves on Xwith Chern character (2,0,−2,0) and satisfying H1(E(−1)) = 0 is equiva-
lent to having a family of σ-semistable objects in Ku(X) with numerical class 2[Il] up to
shifts. We can then identify their respective moduli functors. Since the first functor is
co-represented by the moduli spaces in Theorem 2.6, so is the second.
Remark 5.14.For d= 4, there is another way to understand the previous theorem. Recall
there is an equivalence Ku(X)∼
=Db(C) for a smooth projective curve Cof genus 2 (see
[Ku12, Section 5] for the precise definition of the equivalence). By [Ma07, Theorem 2.7],
we have K∼
=Stab(Ku(X)) ∼
=Stab(C) = σ0·˜
GL+
2(R), where σ0is the slope stability on
the curve C. Then for any σ∈ K,σ-semistable objects in Ku(X) with numerical class 2[Il]
corresponds to semistable vector bundles of rank 2 and degree 0 on C. The previous theorem
for d= 4 will then follow from [Qin2, Theorem 1.5].
6. Non-minimal instanton bundles
In this section, we explore some applications of our methods from the previous sections
to non-minimal instanton bundles. Let Ebe an instanton bundle and let nbe its charge.
Then Eis an object in Ku(X) if and only if n= 2. On the other hand, one can associate to
Ea unique vector bundle of rank nwhich is an object in Ku(X):
Lemma 6.1. [Ku12, Lemma 3.5 and 3.6] For each instanton bundle Ethere exists a unique
short exact sequence
0→ E → ˜
E → On−2
X→0,(6.2)
where ˜
E ∈ Ku(X)is a simple slope-semistable vector bundle with Chern character ch( ˜
E) =
(n, 0,−n, 0).˜
Eis called the acyclic extension of E.
Remark 6.3.As remarked in [Ku12], ˜
Eis nothing but the universal extension of H1(E)⊗OX
by E. It can be also viewed as the left mutation LOXEof Ethrough OX. It is clear that if
n= 2, E=˜
E.
The next lemma and its proof are suggested by an anonymous referee.
BRIDGELAND STABILITY OF MINIMAL INSTANTON BUNDLES ON FANO THREEFOLDS 17
Lemma 6.4. Let EOXbe a slope-stable vector bundle with µ(E) = 0 on X. Let ˜
Ebe
the vector bundle defined by the universal extension
0→E→˜
E→H1(E)⊗ OX→0.
Then ˜
Eis Gieseker-stable.
Proof. Since Eis slope-stable, ch2(E)≤0 by Bogomolov inequality. If ch2(E) = 0, by
[BBF+, Proposition 4.18(i)] and our assumption that EOX, ch3(E)<0. In particular
P(E)≺P(OX) and h0(E) = 0. We note that ˜
Eis a slope-semistable vector bundle with
µ(˜
E) = 0. Moreover, h0(˜
E) = h1(˜
E) = 0.
Suppose ˜
Eis not Gieseker-stable. Then H1(E)6= 0 and P(E)≺P(˜
E). Let 0 6=G →˜
E
be a Gieseker-stable subsheaf so that P(G)P(˜
E). Then µ(G) = 0. We can further assume
that ˜
E/G is torsion free. Since ˜
Eis locally free, Gis reflexive. Let Kand Ibe the kernel
and image of the composite morphism:
f:G →˜
E→H1(E)⊗ OX
respectively. If I= 0, then Gis a nontrivial proper subsheaf of Ewith µ(G) = 0, this
contradicts the assumption that Eis slope-stable, thus I6= 0. As Iis a subsheaf of H1(E)⊗
OXas well as a quotient sheaf of G,µ(I) = 0, thus either µ(K) = 0 or K= 0. Now K
is a subsheaf of E, if K=E, then one easily checks that P(G)≺P(˜
E), contradicting our
choice of G. So we must have K= 0, and fis injective.
If G →H1(E)⊗ OXis saturated, then [HuyL, Corollary 1.6.11] implies that G∼
=OX,
which contradicts the fact that Gis a subsheaf of ˜
Eand h0(˜
E) = 0.
If G →H1(E)⊗ OXis not saturated, let G0be its saturation. We have the following
commutative diagram
0
T
0G H1(E)⊗ OXQ0
0G0H1(E)⊗ OXQ00
0
where Tis the torsion part of Q, and the short exact sequence
0→G→G0→T→0.
Since H1(E)⊗ OXis locally free and Q0is torsion free, G0is reflexive. If dim(T) = 2,
then c1(T)>0. Since Q0is a quotient of H1(E)⊗ OX,c1(Q0)≥0. Now 0 = c1(Q) =
c1(Q0)+c1(T)>0 leads to a contradiction. It follows that dim(T)61 and Ext1(T , OX) = 0.
Dualizing the short exact sequence above we obtain:
0→ Ext1(G0,OX)→ Ext1(G, OX)→ Ext2(T , OX)→0
18 XUQIANG QIN
and Ext3(T, OX) = 0, i.e Tmust have pure dimension 1. Since Gand G0are reflexive,
Ext1(G0,OX) and Ext1(G, OX) are both 0-dimensional ([HuyL, Proposition 1.1.10]). But
Ext2(T, OX) is 1-dimensional, which leads to a contradiction.
Corollary 6.5. Let Ebe an instanton bundle on X. Then its acyclic extension ˜
Eis Gieseker-
stable.
Proof. By [Sa14, Lemma 1.23], Eis slope-stable. Since ˜
Eis precisely the universal extension
of H1(E)⊗ OXby E, we conclude by Lemma 6.4.
One can now ask for n>3, whether ˜
E ∈ Ku(X) is σ-stable with respect to σ∈ K. We
show that the answer to this question is positive for n= 3.
Proposition 6.6. Assume d6= 1. Let ˜
Ebe the acyclic extension of an instanton bundle E
of charge 3. Then ˜
Eis σ-semistable for any stability condition σ∈ K.
Proof. By Corollary 6.5 and [BBF+, Proposition 4.8], ˜
Eis σα,−1
2semistable for α0. We
proceed as the proof of Proposition 4.1 to show there is no wall that will make ˜
Etilt-unstable
along β=−1
2. A wall would be given by a sequence in Coh−1
2(X):
0→˜
E0→˜
E → ˜
E00 →0
in which the truncated twisted Chern characters satisfy
(3,3
2H, 3
8(d−8)) = (a, b
2H, c
8) + (3 −a, 3−b
2H, 3d−24 −c
8)
for some a, b, c ∈Z. As in Proposition 4.1, the wall condition and Bogomolov inequality
imply
(1) 1
b(c
4d−α2a) = d−8−4dα2
4d=1
3−b(3d−24−c
4d−α2(3 −a));
(2) ( b
2)2−ac
4d>0, (3−b
2)2−(3−a)(3d−24−c)
4d>0
(3) ( b
2)2−ac
4d618
d, (3−b
2)2−(3−a)(3d−24−c)
4d618
d.
Since ˜
E0,˜
E00 are in Coh−1
2(X), we have b= 0,1,2 or 3. We can easily eliminate the cases of
b= 0 and b= 3 as in the proof of Proposition 4.1. Note b−a
2His the first Chern character
of either ˜
E0or ˜
E00. Hence a, b have the same parity. One of aand 3 −awill be at least two.
Without loss of generality, we assume a>2.
Suppose b= 2. Condition (1) implies c= 2d−16 + 4dα2(a−2). We observe that if a= 2,
then (a, b
2H, c
8) = (2, H, 2d−16
8) is proportional to the truncated twisted character of ˜
E. This
case will not affect tilt-semistability and can be ignored. If a>4, the second inequality of
(2) simplifies to
4dα2(a−2) 6d
a−3+d−862d−8.
This immediately leads to a contradiction for d64 since the left hand side is positive while
the right hand side is non-positive. For d= 5, the above equation becomes
α261
10(a−2) 61
20.
In this case there are no walls for α > 1
√20 .
Suppose b= 1. Condition (1) implies c=d−8+4dα2(a−1). If a>5, the second
inequality of (2) simplifies to
4dα2(a−1) 64d
a−3+ 2d−16 64d−16.
BRIDGELAND STABILITY OF MINIMAL INSTANTON BUNDLES ON FANO THREEFOLDS 19
Arguing as above we see there are no walls when d64 and no wall with α > 1
√20 when
d= 5. If a= 3, the formula for calong with the first inequality of (2) implies d−8< c 6d
3.
Moreover, using the formula for twisted Chern character, c= 3d−8c2where c2∈Zis the
second Chern class of either E0or E00. Hence c≡3d(mod 8). As a result, we have:
•for d= 5, c=−1 and α=1
√20 ;
•for d= 4, no solution;
•for d= 3, c= 1 and α=1
2;
•for d= 2, c=−2 and α=1
2.
Next we eliminate the cases when d= 3 and 2. For d= 3, either the destabilizing subobject
or quotient, which we denote by G, has truncated twisted character (3,1
2H, 1
8). Then ∆(G) =
0. By [BMS, Corollary 3.10], Gis σ1
2,−1
2-stable. Now we can exclude this case by Proposition
3.13(3). For d= 2, either the destabilizing subobject or quotient, which we denote by G0,
has truncated twisted character (3,1
2H, −1
4). Then it is easy to see G0is σ1
2,−1
2-stable by
checking the twisted first Chern class of its subobjects and quotients. Now we can exclude
this case by Proposition 3.13(5) and the fact that the rank of G0is 3.
To summarize, for 2 6d64, there are no walls on β=−1
2; for d= 5, there are no walls
with α > 1
√20 on β=−1
2. We conclude using the argument at the end of the proof for
Proposition 4.1.
Corollary 6.7. Assume d>3. The acyclic extension ˜
Eof an instanton bundle Eof charge
3is σ-stable for any stability condition σ∈ K.
Proof. It remains to show ˜
Eis not strictly σ-semistable. Suppose otherwise, by [PY20,
Theorem 1.1] and Theorem 5.13, the Jordan-H¨older factors (with respect to σ) of ˜
Eare
either ideal sheaves of lines on Xor rank 2 semistable sheaves with Chern class c1= 0,
c2= 2 and c3= 0. It suffices to show none of these sheaves can have nonzero morphisms to
˜
E. Let l⊂Xbe a line. We have long exact sequence
→Hom(OX,˜
E)→Hom(Il,˜
E)→Ext1(Ol,˜
E)→
It is straightforward to check the left and right term both vanish, thus Hom(Il,˜
E) = 0.
Let Ebe a minimal instanton bundle. We have long exact sequence
→Hom(E, E)→Hom(E, ˜
E)→Hom(E, OX)→
The left term is 0 by Gieseker-stability, while the right term is H0(X, E) = 0 since Eis
self-dual. Thus Hom(E, ˜
E) = 0.
Let E0be a stable but non-locally free sheaf with Chern character (2,0,−2,0). By
Proposition 2.3, we have long exact sequence
→Hom(O⊕2
X,˜
E)→Hom(E0,˜
E)→Ext1(θ(1),˜
E)→
where θis the theta character of a smooth conic in X. It is straightforward to check the left
and right term both vanish, thus Hom(E0,˜
E) = 0.
Remark 6.8.Unfortunately the author was unable to extend Proposition 6.6 and Corollary
6.7 to the cases when either d= 1 or charge n>4. In each of these cases, there will be
potential walls which we fail to eliminate.
20 XUQIANG QIN
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Department of Mathematics, The University of North Carolina at Chapel Hill, 205 S Columbia
St, Chapel Hill, NC 27514, USA
Email address:qinx@unc.edu
