Bayer-Macr\`i decomposition on Bridgeland moduli spaces over surfaces

0.3 MB·English

by  Wanmin Liu

BAYER-MACR`I DECOMPOSITION ON
BRIDGELAND MODULI SPACES OVER SURFACES
WANMIN LIU
6 Abstract. We find a decomposition formula of the local Bayer-Macr`ı
1
mapfortheneflinebundletheoryontheBridgelandmodulispaceover
0
surface. Ifthereisaglobal Bayer-Macr`ı map,suchdecomposition gives
2
a precise correspondence from Bridgeland walls to Mori walls. As an

v application, we compute the nef cone of the Hilbert scheme S[n] of n-
o points overspecial kindsof fibered surface S of Picard rank two.
N

0
3

] Contents
G
1. Introduction 1
A
2. Bridgeland stability conditions 3
.
h 3. Bayer-Macr`ı’s nef line bundle theory 9
t
a 4. Bayer-Macr`ı decomposition 12
m 5. A toy model: fibered surface over P1 with a global section 18
[ Appendix A. Twisted Gieseker stability and the large volume limit 20

2 Appendix B. Bayer-Macr`ı decomposition on K3 surfaces by using
v σˆ 21
ω,β
7
References 22
9
3
6
0
.
1 1. Introduction
0
5 LetS beasmoothprojective surfaceover C. LetM betheGieseker mod-
1 uli space of semistable sheaves with fixed Chern character ch over S. One
:
v point-view of studying the birational geometry of M is to study the classifi-
i cation of line bundles on M, and different cones inside its real N´eron-Severi
X
group N1(M), such as nef cone Nef(M), pseudo-effective cone Eff(M). An-
r
a other point-view is introduced by Bridgeland [Bri07] by the idea of enlarg-
ing the category of coherent sheaves Coh(S) to its bounded derived ver-
sion Db(S), and studying moduli spaces of Bridgeland semistable objects in
Db(S). Let σ be a Bridgeland stability condition and M (ch) be the moduli
σ
space of σ-semistable objects in Db(S) with the same invariant ch. The
collection of all stability conditions forms an interesting parameter space
Stab(S), which is a C-manifold. Bridgeland identified M as M (ch), for σ
σ
Date: December 1, 2016.
2010 Mathematics Subject Classification. 14D20, (Primary); 18E30, 14E30
(Secondary).
Key words and phrases. Bayer-Macr`ı map, Bridgeland stability condition, minimal
model program, moduli space of complexes, wall-crossing.
1
2 W.-M.LIU
in a special chamber inside Stab(S). He envisioned that the wall-chamber
structures of Stab(S) will recover birational models of M.
When S is the projective plane P2 and ch = (1,0, n), the moduli space
−
M is the Hilbert scheme P2[n] of n-points over P2. Arcara, Bertram, Coskun
and Huizenga [ABCH13] found a precise relation between Bridgeland walls
inside Stab(P2) with respect to the (1,0, n) and Mori walls inside the
−
Eff(P2[n]), see Corollary 4.14. Bertram and Coskun generalized the specu-
lation to other rational surfaces [BC13]. Bayer and Macr`ı [BM14a, BM14b]
linked the two point-views by establishing a line bundle theory on Bridge-
land moduli spaces. Let σ = σ be the stability condition constructed by
ω,β
Arcara and Bertram [AB13], which depends on an ample line bundle ω and
another line bundle β over S. Assume that σ is in a chamber C. Bayer and
Macr`ı constructed a map by sending σ to a nef line bundle ℓ on M (ch),
σ σ
which is called the local Bayer-Macr`ı map. The line bundleℓ only depends
σ
on the chamber C. If S is a K3 surface and ch is primitive, they then con-
structed a global Bayer-Macr`ı map (by gluing the local Bayer-Macr`ı maps,
see Definition 3.6),
ℓ : Stab (S) N1(M),
†
→
sending a stability condition σ to a line bundle ℓ on M. The existence of
σ
the global Bayer-Macr`ı map is also known for the projective plane P2 with
primitive Chern character ch by Li and Zhao [LZ16].
Inthispaper,wefindadecompositionofthelinebundleℓ . Thedecom-
σω,β
positionisclassifiedintotwocasesaccordingtothegivenCherncharacterch.
Thecase for objects supportedin dimensiononeis given inLemma4.5. The
case for objects supported in dimension two is given in Lemma 4.8. In this
case, an equivalent decomposition is also obtained by Bolognese, Huizenga,
Lin, Riedl, Schmidt, Woolf and Zhao [BHL+16, Proposition 3.8.]. If there
is a global Bayer-Macr`ı map, we then obtain the precise correspondences
from Bridgeland walls to Mori walls for such two cases, see Theorem 4.10
and Theorem 4.13 respectively. By Mori walls, we mean the walls appear
on the stable base locus decomposition of the pseudo-effective cone Eff(M).
As an application of the main theorem, we compute the nef cone of the
Hilbert scheme S[n] of n-points over special kinds of fibered surface S in
Theorem 5.2. Here S is either the Hirzebruch surface or an elliptic surface
over P1 with a global section of Picard rank two. Theexample suggests that
to obtain extremal nef line bundle, we cannot assume that the ω is parallel
to β.
Some of the techniques discussed in this paper have been partially gener-
alized by Coskun and Huizenga [CH15] to compute the nef cone of certain
Gieseker moduli spaces.
Outline of the paper. Section 2 is a brief review of the notion of Bridge-
land stability condition. Section 3 is abrief review of Bayer and Macr`ı’s line
bundle theory on Bridgeland moduli spaces. Main theorems on the Bayer-
Macr`ı decomposition are given in Section 4. In section 5, we provide an
application of the main theorem. Some backgrounds on the large volume
limit are given as Appendix A. Some parallel computations by using Zˆ
ω,β
(B.1) for K3 surface are given in Appendix B.
BAYER-MACR`I DECOMPOSITION ON BRIDGELAND MODULI SPACES 3
Acknowledgement. This work is part of the author’s PhD thesis at the
Hong Kong University of Science and Technology. The author thanks his
PhD supervisor Wei-Ping Li for encouragement and guidance; thanks Huai-
LiangChang,NaichungConanLeung,ChunyiLi,JasonLo,andZhenboQin
and Xiaolei Zhao for helpful conversations and suggestions; thanks Izzet
Coskun for the invitation to the Graduate Student Workshop on Moduli
Spaces and Bridgeland Stability atUniversityofIllinoisatChicagoonMarch
2013; thanks Changzheng Li for the invitation to Kavli IPMU on October
2013; thankstheorganizersandparticipantsoftheworkshopGeometry from
Stability Conditions atUniversity of Warwick on February2015; andthanks
the anonymous referees for their valuable suggestions and comments. This
work was supported by IBS-R003-D1.
2. Bridgeland stability conditions
Let S be a smooth projective surface over C and Db(S) be the bounded
derived category of coherent sheaves on S. Denote the Grothendieck group
of Db(S) by K(S). A Bridgeland stability condition ([Bri07, Proposition
5.3]) σ = (Z, ) on Db(S) consists of a pair (Z, ), where Z :K(S) C is
A A →
a group homomorphism (called central charge) and Db(S) is the heart
A ⊂
of a bounded t-structure, satisfying the following three properties.
(1) Positivity. For any 0 = E the central charge Z(E) lies in the
6 ∈ A
semi-closed upper half-plane R e(0,1]iπ.
>0 ·
·
Let E 0 . Define the Bridgeland slope (might be + valued) and
∈ A\{ } ∞
the phase of E as
(Z(E)) 1
µ (E) := −ℜ ; φ(E) := arg(Z(E)) (0,1].
σ
(Z(E)) π ∈
ℑ
For nonzero E, F , we have the equivalent relation:
∈ A
µ (F) <( )µ (E) φ(F) < ( )φ(E).
σ σ
≤ ⇐⇒ ≤
For 0 = E , we say E is Bridgeland (semi)stable if for any subobject
6 ∈ A
0 = F ( E (0 = F E) we have µ (F) < ( )µ (E).
σ σ
6 6 ⊆ ≤
(2) Harder-Narasimhan property. Every object E has a Harder-
∈ A
Narasimhan filtration 0 = E ֒ E ֒ ... ֒ E = E such that
0 1 n
→ → →
thequotients E /E areBridgeland-semistable, withµ (E /E ) >
i i 1 σ 1 0
−
µ (E /E ) > > µ (E /E ).
σ 2 1 σ n n 1
··· −
(3) Support property. There is a constant C > 0 such that, for all
Bridgeland-semistableobjectE ,wehave E C Z(E),where
∈ A k k ≤ | |
is a fixed norm on K(X) R.
k·k ⊗
2.1. Bridgeland Stability Conditions on Surfaces. Let S be a smooth
projective surface. Fix ω,β N1(S) := NS(S) with ω ample. Define
R
∈
Z (E) := e (β+√ 1ω).ch(E).
ω,β − −
−
ZS
For E Coh(S), denote its Mumford slope by
∈
µω(E) := ωc·hch01(E(E)) if ch0(E) 6= 0;
( + otherwise.
∞
4 W.-M.LIU
Let Coh(S)bethesubcategoryofcoherentsheaves whoseHN-factors
ω,β
T ⊂
(with respect to Mumford stability) are of Mumford slope strictly greater
than ω.β. Let Coh(S) be the subcategory of coherent sheaves whose
ω,β
F ⊂
HN-factors (with respect to Mumford stability) are of Mumford slope less
than or equal to ω.β. Then ( , ) is a torsion pair of Coh(S) [AB13].
ω,β ω,β
T F
Define the heart as the tilt of such torsion pair:
ω,β
A
:= E Db(S) :H 1(E) ,H0(E) ,Hp(E) = 0 otherwise .
ω,β − ω,β ω,β
A { ∈ ∈ F ∈ T }
Lemma 2.1. [AB13, Corollary 2.1] Fix ω,β NS(S) with ω ample. Then
R
∈
σ := (Z , ) is a Bridgeland stability condition.
ω,β ω,β ω,β
A
2.2. Logarithm Todd class. Let X be a smooth projective variety over
C. Let us introduce a formal variable t, and write
1 1 3
td(X)(t) := 1+( K )t+ K2 ch (X) t2
−2 X 12 2 X − 2
(cid:18) (cid:19)
1 1
+ K . K2 ch (X) t3+ higher order of t4.
−24 X 2 X − 2
(cid:18) (cid:18) (cid:19)(cid:19)
Taking the logarithm with respect to t, and expressing it in the power series
of t, we obtain
1 1
lntd(X)(t) = K t ch (X)t2 +0 t3+ higher order of t4.
X 2
−2 − 12 ·
In particular, the logarithm Todd class of a smooth projective surface S or
a smooth projective threefold X is given respectively by
1 1
lntd(S) := (0, K , ch (S)); or
S 2
−2 −12
1 1
lntd(X) := (0, K , ch (X),0).
X 2
−2 −12
2.3. The Mukai pairing. We refer to [Huy06, Section 5.2] for the details.
Let X still be a smooth projective variety of dimension n over C. Define
the Mukai vector of an object E Db(X) by
∈
v(E) := ch(E).e12lntd(X) ∈ ⊕Hp,p(X)∩H2p(X,Q) =:Ha∗lg(X,Q).
Let A(X) be the Chow ring of X. The Chern character gives a mapping
ch :K(X) A(X) Q. There is a natural involution : A(X) A(X),
∗
→ ⊗ →
v = (v ,...,v ,...,v ) v := (v ,...,( 1)iv ,...,( 1)nv ).
0 i n ∗ 0 i n
7→ − −
We call v the Mukai dual of v. Denote E := R om(E, ). We have
∗ ∨ S
H O
ch(E∨) = (ch(E))∗, v(E∨)= (v(E))∗.e−21KX.
Define the Mukai pairing for two Mukai vectors w and v by
(2.1) w, v X := w∗.v.e−12KX.
h i −
ZX
The Hirzebruch-Riemann-Roch theorem gives
χ(F,E) = ch(F ).ch(E).td(X) = v(F), v(E) .
∨ X
−h i
ZX
BAYER-MACR`I DECOMPOSITION ON BRIDGELAND MODULI SPACES 5
For a smooth projective surface S, the Mukai vector of E Db(S) is
∈
v(E) = (v (E),v (E),v (E))
0 1 2
1 1 1 1
(2.2) = (ch ,ch ch K ,ch ch .K + ch χ( ) K2 ).
0 1− 4 0 S 2− 4 1 S 2 0 OS − 16 S
(cid:18) (cid:19)
By (2.1) the Mukai paring of w =(w ,w ,w ) and v = (v ,v ,v ) is
0 1 2 0 1 2
1 1 1
(2.3) w, v = w .v w (v v .K ) v (w + w .K ) w v K2.
h iS 1 1− 0 2− 2 1 S − 0 2 2 1 S − 8 0 0 S
2.4. Central charge in terms of the Mukai paring.
Lemma 2.2. The central charge Z has the expression:
ω,β
(2.4) Zω,β(E) = h℧Zω,β,v(E)iS, where ℧Zω,β := eβ−34KS+√−1ω+214ch2(S).
Moreover the vector ℧ (or simply ℧ ) is given by
Zω,β Z
3 1 1 3 1 1
℧ = 1,β K , ω2+ (β K )2 χ( ) K2
Zω,β − 4 S −2 2 − 4 S − 2 OS − 8 S
(cid:18) (cid:18) (cid:19)(cid:19)
3
(2.5) + √ 1 0,ω,(β K ).ω .
S
− − 4
(cid:18) (cid:19)
Proof.
Z (E) = e (β+√ 1ω).ch(E)
ω,β − −
−
ZS
= e (β+lntd(S)+√ 1ω). td(S).ch(E). td(S).
− −
−
ZS
p p
Denote ch(F ):= e (β+lntd(S)+√ 1ω). Then
∨ − −
ch(F)∗ = ch(F∨) = e−(β−21KS+√−1ω)+112ch2(S)
1 1 1 1
= 1, (β K +√ 1ω), (β K +√ 1ω)2+ ch (S) .
S S 2
− − 2 − 2 − 2 − 12
(cid:18) (cid:19)
So
1 1 1 1
ch(F) = 1,(β K +√ 1ω), (β K +√ 1ω)2+ ch (S)
S S 2
− 2 − 2 − 2 − 12
(cid:18) (cid:19)
= e(β−21KS+√−1ω)+112ch2(S).
Therefore,
Z (E) = ch(F ). td(S).ch(E). td(S) = v(F),v(E) .
ω,β ∨ S
− h i
ZS
p p
So
℧Zω,β = v(F) = ch(F).e12lntd(S) =eβ−34KS+√−1ω+214ch2(S).
By using the Noether’s formula
1 1
ch (S)= χ( ) K2.
−12 2 OS − 8 S
and direct computation, we get the concrete expression of ℧ . (cid:3)
Z
6 W.-M.LIU
Denote Stab(S) the collection of all Bridgeland stability conditions. It
is a C-manifold of dimension K (S) C, with two group actions: a left
num
⊗
action by Aut(Db(S)) and a right action by G]L+(R) [Bri07, Lemma 8.2].
2
The stability σ is said to be geometric if all skyscraper sheaves , x S,
x
O ∈
areσ-stableofthesamephase. Wecansetthephasetobe1byarightgroup
action. Denote by Stab (S) Stab(S) the connected component containing
†
⊂
geometric stability conditions. The stability σ is said to be numerical if
the central charge Z takes the form Z(E) = π(σ),v(E) for some vector
S
h i
π(σ) K (S) C. As in [Huy14, Remark 4.33], we further assume the
num
∈ ⊗
numericalBridgelandstabilityfactorsthroughK (S) C H (S,Q)
num Q⊗ → a∗lg ⊗
C. Therefore π(σ) H (S,Q) C. For a numerical geometric stability
∈ a∗lg ⊗
condition with skyscraper sheaves of phase 1, the heart must be of the
A
form (see [Bri07, Proposition 10.3] and Huybrechts [Huy14, Theorem
ω,β
A
4.39]). Therefore, Lemma 2.2 gives
(2.6) π(σ )= ℧ H (S,Q) C.
ω,β Zω,β ∈ a∗lg ⊗
2.5. Bertram’s nested wall theorem. We follow notations in [Mac14,
Section 2] (but use H instead of ω therein). Fix an ample divisor H and
another divisor γ H , i.e. H.γ = 0. Denote
⊥
∈
g := H2, d:= γ2.
−
It is known by Hodge index theorem that d 0, and d = 0 if and only if
γ = 0. Let ch = (ch ,ch ,ch ) beof Bogomolo≥v type, i.e. ch2 2ch ch 0.
0 1 2 1− 0 2 ≥
Write it as
ch = (ch ,ch ,ch ) := (x,y H +y γ+δ,z),
0 1 2 1 2
where y , y are real coefficients, and δ H,γ . Write the potential
1 2 ⊥
∈ { }
destabilizing Chern character as
ch = (ch ,ch ,ch ):= (r,c H +c γ +δ ,χ),
′ ′0 ′1 ′2 1 2 ′
where c , c are real coefficients, and δ H,γ . A potential wall is
1 2 ′ ⊥
∈ { }
defined as
W(ch,ch) := σ Stab(S) µ (ch) = µ (ch) .
′ σ σ ′
{ ∈ | }
A potential wall W(ch,ch) is a Bridgeland wall if there is a σ Stab(S)
′
∈
and objects E, F such that ch(E) = ch, ch(F) = ch and µ (E) =
σ ′ σ
∈ A
µ (F). There is a wall-chamber structure on Stab(S) with respect to ch
σ
[Bri07, Bri08, Tod08]. Bridgeland walls are real codimension 1 in Stab(S),
which separate Stab(S) into chambers. Let E be an object that is σ -stable
0
for a stability condition σ in some chamber C. Then E is σ-stable for any
0
σ C. Choose
∈
ω := tH,
(2.7)
β := sH +uγ,
(cid:26)
for some real numbers t, s, u, with t positive. With a sign choice of γ, we
further assume u 0. There is a half 3-space of stability conditions
≥
Ω = Ω := σ t > 0,u 0 Stab (S),
ω,β tH,sH+uγ tH,sH+uγ †
{ | ≥ } ⊂
which should be considered as the u-indexed family of half planes
Π := σ t > 0, u is fixed. .
(H,γ,u) tH,sH+uγ
{ | }
BAYER-MACR`I DECOMPOSITION ON BRIDGELAND MODULI SPACES 7
Definition 2.3. A frame with respect to the triple (H,γ,u), is a choice of
anampledivisor H on S, another divisorγ H , andnon-negative number
⊥
∈
u, such that the stability conditions σ are on the half plane Π with
ω,β (H,γ,u)
(s,t)-coordinates as equation (2.7). We simply call this as fixing a frame
(H,γ,u), and write σ := σ .
s,t tH,sH+uγ
Theorem 2.4 (Bertram’s nested wall theorem in (s,t)-model). [Mac14,
Section2] Fix a frame (H,γ,u). The potential walls W(ch,ch) (for the fixed
′
ch and different potential destabilizing Chern character ch)in the (s,t)-half-
′
plane Π (t > 0) are given by nested semicircles with center (C,0) and
(H,γ,u)
radius R = √D+C2:
(2.8) (s C)2+t2 = D+C2,
−
where C = C(ch,ch) and D = D(ch,ch) are given by
′ ′
xχ rz+ud(xc ry )
2 2
(2.9) C(ch,ch′) := − − ,
g(xc ry )
1 1
−
2zc 2c udy xu2dc +2y udc 2χy +ru2dy
1 2 1 1 2 1 1 1
(2.10) D(ch,ch′) := − − − .
g(xc ry )
1 1
−
If ch = x = 0, we have
0
• 6
2y ud(2y ux)+2z
1 2
(2.11) D = C + −
− x gx
2y y2
(2.12) = 1C +( 1 F),
− x x2 −
where F = F(ch) is independent of ch:
′
d y 2 1
(2.13) F(ch) := u 2 + (y2g y2d 2xz).
g − x x2g 1 − 2 −
(cid:16) (cid:17)
Moreover, if ch is of Bogomolov type, i.e. ch2 2ch ch 0, then
1 − 0 2 ≥
F(ch) 0 for all u.
≥
If ch = 0 and ch .H > 0, i.e. x = 0 and y > 0, then ch = r = 0,
• 0 1 1 ′0 6
and C = z+duy2 is independent of ch. We have
gy1 ′
2c ud(2c ur)+2χ
1 2
(2.14) D = C + −
− r gr
2c c2
(2.15) = 1C +( 1 F ),
′
− r r2 −
where F = F (ch) is independent of ch:
′ ′ ′
d c 2 1
(2.16) F (ch):= u 2 + (c2g c2d 2rχ).
′ ′ g − r r2g 1 − 2 −
(cid:16) (cid:17)
Moreover, if ch is of Bogomolov type, then F(ch) 0 for all u.
′ ′
≥
Proof. We refer to Maciocia’s paper [Mac14, Section 2]. The only unproved
parts are equations (2.14, 2.15). It is an easy exercise to check them. (cid:3)
8 W.-M.LIU
2.6. From (s,t)-model to (s,q)-model. We follow the idea of Li-Zhao
[LZ16], and consider a G]L+(R) action on σ . The potential walls in the
2 ω,β
(s,q)-plane are semi-lines.
1 0
Definition 2.5. Define σ = (Z , ) as the right action of
ω′,β ω′,β A′ω,β s 1
(cid:18)−t t(cid:19)
on σ , i.e. = and
ω,β A′ω,β Aω,β
s 1
(2.17) Z (E) := Z (E) Z (E) + i Z (E).
′ω,β ω,β ω,β ω,β
ℜ − tℑ t ℑ
(cid:16) (cid:17)
Lemma 2.6. Fix a frame (H,γ,u). The above right action does not change
the potential walls W(ch,ch) in the (s,t)-plane Π .
′ (H,γ,u)
Proof. This is a direct computation because the potential wall relation for
Z is equivalent to the potential wall relation for Z by using (2.17):
ω′,β ω,β
Z (ch) Z (ch) Z (ch) Z (ch) = 0
′ ′ ′ ′ ′ ′
ℜ ℑ −ℜ ℑ
Z(ch) Z(ch) Z(ch) Z(ch) = 0.
′ ′
⇔ ℜ ℑ −ℜ ℑ
(cid:3)
Definition 2.7. Fix a frame (H,γ,u). We change the (s,t)-plane Π
(H,γ,u)
to the (s,q)-plane Σ by keeping the same s and defining
(H,γ,u)
s2+t2
(2.18) q := .
2
Denote σ := σ . The central charge (2.17) becomes
s,q t′H,sH+uγ
1
Z (E) = ( ch (E)+ch (E)H2q)+ ch (E)γ2u2+uch (E).γ
s,q 2 0 0 1
− −2
(cid:18) (cid:19)
+i(ch (E).H ch (E)H2s).
1 0
−
Corollary 2.8 (Bertram’snestedwalltheoremin(s,q)-model). Fix a frame
(H,γ,u) and use notations as above. The potential walls W(ch,ch) in the
′
(s,q)-plane Σ are given by semi-lines
(H,γ,u)
1 s2
q =Cs+ D, (q > ).
2 2
If x = 0, then the potential walls are given by semi-lines passing
• 6
through a fixed point (y1, 1 y12 F ) with slope C = C(ch,ch):
x 2 x2 − ′
y 1(cid:16) y2 (cid:17) s2
(2.19) q = C(s 1)+ 1 F , (q > ),
− x 2 x2 − 2
(cid:18) (cid:19)
where F = F(ch) as in equation (2.13) is independent of ch.
′
If x = 0 and y > 0, then r = 0. The potential walls are given by
1
• parallel semi-lines with consta6nt slope C = z+duy2:
gy1
c 1 c2 s2
(2.20) q = C(s 1)+ 1 F , (q > ),
′
− r 2 r2 − 2
(cid:18) (cid:19)
where F = F (ch) as in equation (2.16) is independent of ch.
′ ′ ′
Proof. This is a direct computation by using equations (2.8, 2.18). (cid:3)
BAYER-MACR`I DECOMPOSITION ON BRIDGELAND MODULI SPACES 9
Remark 2.9. In the case of P2, the condition q > s2 is relaxed, and q could
2
be a little negative and the boundary is given by a fractal curve [LZ16].
2.7. Duality induced by derived dual.
Lemma 2.10. [Mar13, Theorem 3.1] The functor Φ() := R om(, )[1]
S
· H · O
induces an isomorphism between the Bridgeland moduli spaces M (ch) and
ω,β
M ( ch ) provided these moduli spaces exist and Z (ch) belongs to the
ω, β ∗ ω,β
− −
open upper half plane.
Proof. This is a variation of Martinez’s duality theorem [Mar13, Theorem
3.1], where the duality functor is taken as R om(,ω )[1]. (cid:3)
S
H ·
Corollary 2.11. Fix the Chern character ch =(ch ,ch ,ch ). Assume that
0 1 2
Z (ch) belongs to the open upper half plane. The wall-chamber structures
ω,β
of σ w.r.t. ch is dual to the wall-chamber structures of Φ(σ ) w.r.t.
ω,β ω,β
Φ(ch) = ch = ( ch ,ch , ch ) in the sense that
∗ 0 1 2
− − −
Φ(σ ) = σ .
ω,β ω, β
−
Applying Φ again, we have Φ Φ(σ ) = σ . Moreover, if we fix a frame
ω,β ω,β
◦
(H,γ,u), then σ Π with coordinates (s,t) is dual to Φ(σ )
ω,β (H,γ,u) ω,β
∈ ∈
Π with coordinates ( s,t).
(H, γ,u)
− If σ C, where C−is a chamber w.r.t. ch in Π , then we have
ω,β (H,γ,u)
• Φ(σ )∈ DC, where DC is the corresponding chamber w.r.t. Φ(ch)
ω,β
∈
in Π .
(H, γ,u)
−
If σ := σω,β W(ch,ch′) in Π(H,γ,u), then Φ(σ) W( ch∗, ch′∗)
• ∈ ∈ − −
in Π , and there are relations:
(H, γ,u)
−
µΦ(σ)( ch∗) = µσ(ch); CΦ(σ)( ch∗, ch′∗) = Cσ(ch,ch′);
− − − − −
DΦ(σ)( ch∗, ch′∗)= Dσ(ch,ch′); RΦ(σ)( ch∗, ch′∗)= Rσ(ch,ch′).
− − − −
Proof. The proof is a direct computation. (cid:3)
Remark 2.12. The assumption that Z (ch) belongs to the open upper
ω,β
half plane means exactly that we exclude the case Z (ch) = 0, which is
ω,β
ℑ
equivalent to one of the following three subcases:
ch = (0,0,n) for some positive integer n; or
•
ch > 0 and Z (ch)= 0; or
0 ω,β
• ℑ
ch < 0 and Z (ch)= 0.
0 ω,β
• ℑ
We call the first subcase as the trivial chamber, the second subcase as the
Uhlenbeck wall and the third subcase as the dual Uhlenbeck wall, see Defi-
nition A.2.
3. Bayer-Macr`ı’s nef line bundle theory
3.1. The local Bayer-Macr`ı map. Let S be a smooth projective surface
over C. Let σ = (Z, ) Stab(S) be a stability condition, and ch =
A ∈
(ch ,ch ,ch ) be a choice of Chern character. Assume that we are given a
0 1 2
flat family [BM14a, Definition 3.1] Db(M S) of σ-semistable objects
E ∈ ×
of class ch parametrized by a proper algebraic space M of finite type over
C. Denote N1(M) = NS(M) as the group of real Cartier divisors modulo
R
numerical equivalence. Write N (M) as the group of real 1-cycles modulo
1
numerical equivalence with respect to the intersection paring with Cartier
10 W.-M.LIU
divisors. TheBayer-Macr`ı’s numerical Cartier divisor class ℓ N1(M) =
σ,
E ∈
Hom(N (M),R) is defined as follows: for any projective integral curve C
1
⊂
M,
Z Φ ( ) Z (p )
C S C S
(3.1) ℓσ, ([C]) = ℓσ, .C := E O = ∗E| × ,
E E ℑ − (cid:0)Z(ch) (cid:1)! ℑ − (cid:0) Z(ch) (cid:1)!
whereΦ : Db(M) Db(S) is the Fourier-Mukai functor with kernel , and
E → E
is the structure sheaf of C.
C
O
Theorem 3.1. [BM14a, Theorem 1.1] The divisor class ℓ is nef on M.
σ,
E
Inaddition, we have ℓ .C = 0 if and only if for two general points c,c C,
σ, ′
E ∈
the corresponding objects c, c′ are S-equivalent.
E E
Here two semistable objects are S-equivalent if their Jordan-H¨older fil-
trations into stable factors of the same phase have identical stable factors.
Definition 3.2. LetC beaBridgeland chamberwith respecttoch. Assume
the existence of the moduli space M (ch) for σ C with a universal family
σ
. Then MC(ch) := Mσ(ch) is constant for σ ∈ C. Theorem 3.1 yields a
E ∈
map,
ℓ : C Nef(MC(ch))
−→
σ ℓ
σ,
7→ E
which is called the local Bayer-Macr`ı map for the chamber C w.r.t. ch.
For any σ Stab (S), after a G]L+(R)-action, we assume that σ = σ ,
∈ † 2 ω,β
i.e. skyscraper sheaves are stable of phase 1. Denote
v := v(ch)= ch e12lntd(S).
·
The local Bayer-Macr`ı map is the composition of the following three maps:
Stab†(S) −→π Ha∗lg(S,Q)⊗C−→I v⊥ −θ−C,→E N1(MC(ch)).
The map π forgets the heart: π(σ ) := ℧ as (2.6).
• ω,β Zω,β
For any ℧ H (S,Q) C,define (℧):= ℧ . Onecan check
• ∈ a∗lg ⊗ I ℑ ℧,v S
−h i
that (℧) v (this also follows from the Lemma 3.4), where the
⊥
I ∈
perpendicular relation is with respect to the Mukai paring:
(3.2) v := w H (S,Q) R w,v = 0 .
⊥ { ∈ a∗lg ⊗ | h iS }
ThethirdmapθC, isthealgebraic Mukai morphism. Moreprecisely,
• E
for a fixed Mukai vector w v⊥, and an integral curve C MC(ch),
∈ ⊂
θC, (w).[C] := w,v(Φ ( C)) S.
E h E O i
Definition 3.3. Define w (ch) := ℧ ,v ℧ . We simply write
σω,β −ℑ h Z iS · Z
it as wω,β or wσ. (cid:16) (cid:17)
Lemma 3.4. Fix the Chern character ch = (ch ,ch ,ch ). The line bundle
0 1 2
class ℓ N1(M (ch)) (if exists) is given by
σω,β ∈ σω,β
(3.3) ℓ =R=+= θ (w ),
σω,β σ, ω,β
E