Mirror Symmetry, Autoequivalences, and Bridgeland Stability Conditions The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Fan, Yu-Wei. 2019. Mirror Symmetry, Autoequivalences, and Bridgeland Stability Conditions. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:42029475 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA
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Citation Fan, Yu-Wei. 2019. Mirror Symmetry, Autoequivalences, andBridgeland Stability Conditions. Doctoral dissertation, HarvardUniversity, Graduate School of Arts & Sciences.
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3.1 Correspondence among flat surfaces, holomorphic top forms on Calabi–Yaumanifolds, and stability conditions on triangulated categories. . . . . . . . . . 94
vi
List of Figures
1.1 The Atiyah flop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 The local conifold is self-mirror. More precisely for the local conifold, a
resolution and its flop are equivalent, so the upper hemisphere should beidentified with the lower hemisphere. . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 The base and discriminant loci of Lagrangian fibrations in conifold transition. 171.4 Lagrangian S3 seen from the double conic fibration. . . . . . . . . . . . . . . . 201.5 The symplectic reduction of Lagrangian torus fibers before and after the A-flop. 261.6 The discriminant locus before and after the A-flop. . . . . . . . . . . . . . . . 291.7 A-flop shown in the double conic fibration picture. . . . . . . . . . . . . . . . 291.8 Polytopes in the polyhedral decomposition of the affine base of Schoen’s CY. 331.9 Sequence of Lagrangian spheres in Xs=0. . . . . . . . . . . . . . . . . . . . . . 351.10 Special Lagrangian spheres S′k in Xs=1. . . . . . . . . . . . . . . . . . . . . . . . 361.11 Transformation of L0 and L1 by the symplectomorphism ρ. . . . . . . . . . . 371.12 Perturbation of Lc, Li and polygons contributing to mb
1 on CF(L, (Lc, ρ)). . . . 401.13 Two triangles contributing to m1 and m2. . . . . . . . . . . . . . . . . . . . . . 411.14 Triangles contributing to m1 and m2 on CF(Sm[1]⊕Sm+1, Lc) and CF(Lc, Sm[1]⊕
Yu Qiu, Shu-Heng Shao, Arnav Tripathy, Chien-Hsun Wang, Dan Xie, Jingyu Zhao and
Jie Zhou for many helpful and interesting discussions.
I am grateful to Chin-Lung Wang and Hui-Wen Lin for introducing mirror symmetry
to me when I was an undergraduate at National Taiwan University. I also am grateful to
Mboyo Esole, Yuan-Pin Lee and Pei-Yu Tsai for offering very helpful career advice to me.
I thank all the Yau’s seminar participants including Teng Fei, Peter Smillie, Chenglong Yu,
Netanel Rubin-Blaier, Charles Doran, Peng Gao, An Huang, Yoosik Kim, Tsung-Ju Lee,
Hossein Movasati, Sze-Man Ngai, Valentino Tosatti, Yi Xie, Mei-Heng Yueh, Boyu Zhang,
Dingxin Zhang, Xin Zhou, Yang Zhou and Jonathan Zhu for so many interesting talks and
conversations that broaden the scope of my interest.
I want to thank my fellow graduate students in the True Alcove: Chi-Yun Hsu, Koji Shimizu,
Yunqing Tang and Cheng-Chiang Tsai, as well as my friends Hung-Yu Chien, Chieh-
Hsuan Kao and Ya-Ling Kao for their friendship and support.
At the end, I want to express my sincere gratitude to my parents Chin-Hung Fan and
Li-Chun Tseng, and my sister Yu-Chen Fan for their love and support. Last but not least, I
viii
want to express my deepest gratitude to my love Xueyin Shao, who supported me during
my most difficult times. I want to take this chance to say I love you.
ix
Introduction
Calabi–Yau manifolds have very rich geometric structures and have been studied ex-
tensively in mathematics and physics. The present thesis studies three different aspects of
Calabi–Yau manifolds: mirror symmetry, systolic geometry, and dynamical systems.
Mirror symmetry was first introduced by string theorists. It predicts that Calabi–Yau
manifolds come in pairs, in which the complex geometry of one is equivalent to the symplectic
geometry of the other, and vice versa. One mathematical formulation of this prediction was
proposed by Kontsevich [65].
Conjecture 0.0.1 (Homological mirror symmetry conjecture [65]). Let X be a Calabi–Yau
manifold. There exists a Calabi–Yau manifold Y such that there are equivalences between triangulated
categories
DbCoh(X) ∼= DπFuk(Y) and DbCoh(Y) ∼= DπFuk(X).
Here DbCoh(X) and DπFuk(X) denote the derived category of coherent sheaves and the derived
Fukaya category of X, respectively.
Roughly speaking, the objects in the derived category DbCoh(X) are complexes of
holomorphic vector bundles on X, and the objects in the derived Fukaya category DπFuk(X)
are Lagrangian submanifolds in X with certain extra data. Homological mirror symmetry
conjecture has been proved in many cases, see for instance [82, 88, 90].
Mirror symmetry phenomenon naturally leads to the following question.
Question 0.0.2. Given an “object" (e.g. operation, geometric structure, etc.) in complex geometry,
can one construct its counterpart in symplectic geometry, or vice versa?
1
Chapter 1 and Chapter 2 contain results along this line of thought.
Chapter 1: Mirror of Atiyah flop
Flops are fundamental operations in birational geometry. Among them, Atiyah flop
is the simplest and the most well-known one. An Atiyah flop X → X ← X† contracts
a (−1,−1)-rational curve C in a complex threefold X and resolve the resulting conifold
singularity with another (−1,−1)-rational curve C†.
The goal of Chapter 1 is to construct the mirror of Atiyah flop in symplectic geometry
under mirror symmetry.
Theorem 0.0.3 (= Proposition 1.1.1). Given a symplectic sixfold (Y, ω) and a Lagrangian three-
sphere S ⊂ Y, we construct another symplectic sixfold (Y†, ω†) with a corresponding Lagrangian
three-sphere S† ⊂ Y†, together with a symplectomorphism f (Y,S) : (Y, ω) → (Y†, ω†). It has the
property that f (Y†,S†) f (Y,S) = τ−1
S , where τS is the Dehn twist along the Lagrangian sphere S.
The symplectomorphism f (Y,S) in Theorem 0.0.3 is the mirror of Atiyah flop. The
contracted (−1,−1)-rational curve in algebraic geometry corresponds to the Lagrangian
three-sphere in symplectic geometry. Bridgeland [13] shows that threefolds related by a flop
are derived equivalent: Db(X) ∼= Db(X†). The property f (Y†,S†) f (Y,S) = τ−1
S is mirror to
the fact that the composition of two flop functors Db(X)→ Db(X†)→ Db(X) is the inverse
of the Seidel–Thomas spherical twist [89] by OC(−1).
Let DX/X ⊂ Db(X) be the subcategory which consists of objects supported on C.
This subcategory captures the local geometry of the flopping curve. Then Bridgeland’s
equivalence restricts to an equivalence DX/X∼= DX†/X. It is proved by Chan–Pomerleano–
Ueda [21] that DX/X is equivalent to certain derived Fukaya category DbFY. We prove the
following compatibility result between the Atiyah flop and our mirror Atiyah flop.
Theorem 0.0.4 (= Proposition 1.5.4). The symplectomorphism f (Y,S) induces an equivalence
between the derived Fukaya categories DbFY∼= DbFY† . Moreover, the equivalence is the same as the
composition DbFY∼= DX/X
∼= DX†/X∼= DbFY† , where the first and third equivalences are given
by Chan–Pomerleano–Ueda [21], and the second equivalence is Bridgeland’s flopping equivalence.
2
Note that unlike the Atiyah flop which produces different complex manifold, its mirror
f (Y,S) is a symplectomorphism. It is not very surprising since symplectic geometry is much
softer than complex geometry. On the other hand, we can endow more structures so that
the effect of mirror Atiyah flop can be seen. One way to do so is to consider the Bridgeland
stability conditions [14] on DbFY.
Theorem 0.0.5 (= Theorem 1.1.3). Let Y = u1v1 = z + q, u2v2 = z + 1, z 6= 0 be the deformed
conifold and ΩY = dz∧ du1 ∧ du2 be a holomorphic volume form on Y. Then there exists a collection
P of graded special Lagrangian submanifolds which defines a geometric stability condition (Z ,P)
on DbFY. Moreover, the mirror Atiyah flop f (Y,S) defines another geometric stability condition
(Z†,P†) with respect to ( f (Y,S))∗ΩY† . Finally, (Z ,P) and (Z†,P†) are related by a wall-crossing
in the space of Bridgeland stability conditions Stab(DbFY), which matches with Toda’s wall-crossing
in Stab(DX/X) on the mirror [98].
Another way to see the effect of mirror Atiyah flop is by equipping the symplectic sixfold
Y with a Lagrangian fibration. See Theorem 1.1.2 for more details.
Chapter 2: Mirror of Weil–Petersson metric
The moduli space of complex structures Mcpx(Y) on a Calabi–Yau manifold Y has a
canonical Kähler metric, the Weil–Petersson metric. The existence of such a natural metric
often implies strong results that one can not obtain by purely algebraic methods. In the case
of Calabi–Yau threefold, this metric provides a fundamental differential geometric tool, the
special Kähler geometry, to study mirror symmetry.
The goal of Chapter 2 is to construct the mirror object of the Weil–Petersson metric.
Under mirror symmetry,Mcpx(Y) should be identified with the so-called “stringy Kähler
moduli space"MKah(X) of a mirror Calabi–Yau manifold X. When dim(X) ≤ 2,MKah(X)
is well-defined via Bridgeland stability conditions by a work of Bayer–Bridgeland [8].
When dim(X) ≥ 3, it is conjectured by Bridgeland [14, 16] that there is an embedding
MKah(X) → Aut(Db(X))\Stab(Db(X))/C. In other words, the stringy Kähler moduli
space is encoded in the space of Bridgeland stability conditions on the derived category
3
Db(X). Our strategy therefore is to first define the Weil–Petersson geometry on the space of
Bridgeland stability conditions, then restrict to the stringy Kähler moduli space.
Definition 0.0.6 (= Definition 2.3.3). For any Calabi–Yau categoryD, we define the Weil–Petersson
metric on Stab+(D)/C for an appropriate subset Stab+(D) ⊂ Stab(D). The metric descends to
the double quotient space Aut(D)\Stab+(D)/C.
We compute several low-dimensional examples to justify our definition of Weil–Petersson
metric:
Theorem 0.0.7 (= Example 2.3.5 and Theorem 2.1.1). We compute the following examples of
Weil–Petersson metric on the space of Bridgeland stability conditions.
• Let E be an elliptic curve. ThenMKah(E) ∼= Aut(Db(E))\Stab+(Db(E))/C ∼= PSL(2, Z)\H;
our Weil–Petersson metric coincides with the Poincaré metric on H.
• Let A = Eτ × Eτ be the self-product of a generic elliptic curve. Then MKah(A) ∼=
AutCY(Db(A))\Stab+(Db(A))/C ∼= Sp(4, Z)\H2 is the Siegel modular variety; our Weil–
Petersson metric coincides with the Bergman metric on Sp(4, Z)\H2.
The Weil–Petersson metric on Aut(D)\Stab+(D)/C is a degenerate metric in general.
However, the Weil–Petersson metric on the complex moduli Mcpx(Y) is non-degenerate.
Hence we expect the non-degeneracy condition can be used to characterize the stringy
Kähler moduli spaceMKah(X).
Conjecture 0.0.8 (= Conjecture 2.3.6). For dim(X) ≥ 3, there exists an embedding of the stringy
Kähler moduli space
i :MKah(X) → Aut(Db(X))\Stab+(Db(X))/C.
Moreover, the pullback of our Weil–Petersson metric is a non-degenerate Kähler metric onMKah(X).
It should be identified with the Weil–Petersson metric on the complex moduli spaceMcpx(Y) of a
mirror manifold of Y.
4
Chapter 3: Systolic inequality on K3 surfaces
Systolic geometry studies the least length of a non-contractible loop sys(M, g) in a
Riemannian manifold M. Loewner’s torus systolic inequality states that sys(T2, g)2 ≤2√3vol(T2, g) holds for any metric on the two-torus. We propose the following question that
naturally generalizes Loewner’s torus systolic inequality from the perspective of Calabi–Yau
geometry.
Question 0.0.9 (= Question 3.1.2). Let Y be a Calabi–Yau manifold, and let ω be a symplectic
form on Y. Does there exist a constant C > 0 such that
minL:sLag
∣∣∣ ∫L
Ω∣∣∣2 ≤ C ·
∣∣∣ ∫Y
Ω ∧Ω∣∣∣
holds for any holomorphic top form Ω on Y? Here “sLag" denotes the special Lagrangian submanifolds
in Y with respect to ω and Ω.
Motivated by the connection between flat surfaces and stability conditions, as well as the
conjectural description of Bridgeland stability conditions on Fukaya category by Bridgeland
and Joyce [14, 16, 55], we define the categorical analogue of minL:sLag
∣∣∣ ∫L Ω∣∣∣, which can also
be regarded as the categorical analogue of systole (see Table 3.1).
Definition 0.0.10 (= Definition 3.1.3). Let D be a triangulated category, and σ be a Bridgeland
stability condition on D. Its systole is defined to be
sys(σ) := min|Zσ(E)| : E is σ−semistable.
Using the idea in Chapter 2, we also define the categorical analogue of holomorphic
volume∣∣∣ ∫Y Ω ∧Ω
∣∣∣ of a Calabi–Yau manifold Y.
Definition 0.0.11 (= Definition 3.2.7). Let Ei be a basis of the numerical Grothendieck group
N (D) and let σ = (Z ,P) be a Bridgeland stability condition on D. Its volume is defined to be
vol(σ) :=∣∣∣∑
i,jχi,jZ(Ei)Z(Ej)
∣∣∣,where (χi,j) = (χ(Ei, Ej))
−1 is the inverse matrix of the Euler pairings.
5
Then one can ask the following question, which should give the same answer as Question
0.0.9 assuming mirror symmetry.
Question 0.0.12. Let X be a Calabi–Yau manifold and D = DbCoh(X) be its derived category of
coherent sheaves. Does there exist a constant C > 0 such that
sys(σ)2 ≤ C · vol(σ)
holds for any σ ∈ Stab∗(D)? Here Stab∗(D) is a subset of Stab(D) whose double quotient by
Aut(D) and C gives the stringy Kähler moduli spaceMKah(X) of X.
Question 0.0.9 and Question 0.0.12 are mirror to each other in the sense that
• in Question 0.0.9, the symplectic form ω is fixed, and the question is asking for the
supremum of the ratio sys2/vol among all Ω ∈ Mcpx(Y); while
• in Question 0.0.12, the complex structure Ω is fixed, and the question is asking for the
supremum of the ratio sys2/vol among all [σ] ∈ Aut(D)\Stab∗(D)/C ∼= MKah(X).
Note that the ratio sys(σ)2/vol(σ) is invariant under the Aut(D)-action and the free
C-action.
The main result in this chapter is the following theorem.
Theorem 0.0.13 (= Theorem 3.1.5). Let X be a K3 surface of Picard rank one, with Pic(X) = ZH
and H2 = 2n. Then
sys(σ)2 ≤ (n + 1)vol(σ)
holds for any σ ∈ Stab∗(D).
Chapter 4: Entropy of autoequivalences
The topological entropy htop( f ) of an automorphism f is an important dynamical invariant
that measures the complexity of the dynamical system formed by the iterations of f . A
fundamental theorem by Gromov–Yomdin [44, 45, 106] states that if X is a compact Kähler
6
manifold and f : X → X is a holomorphic surjective map, then
htop( f ) = log ρ( f ∗).
Here f ∗ denotes the induced linear map on H∗(X; C), and ρ( f ∗) is its spectral radius.
Let D be a triangulated category and let Φ : D → D be an endofunctor. The notion of
categorical entropy hcat(Φ) has been introduced by Dimitrov–Haiden–Katzarkov–Kontsevich
[29]. It was shown by Kikuta–Takahashi [63] that if D = DbCoh(X) and Φ = Ł f ∗ is induced
by an automorphism f on X, then hcat(Ł f ∗) = htop( f ). They made the following conjecture
which can be viewed as the categorical analogue of Gromov–Yomdin theorem.
Conjecture 0.0.14 (Kikuta–Takahashi [63]). Let X be a smooth projective variety over C and let
Φ : Db(X)→ Db(X) be an autoequivalence. Then
hcat(Φ) = log ρ([Φ]).
Here [Φ] denotes the induced linear map on the numerical Grothendieck group of Db(X).
The main result of this chapter is the discovery of the first counterexamples of Conjecture
0.0.14.
Theorem 0.0.15 (= Theorem 4.1.1 and Proposition 4.1.4). Let X be a Calabi–Yau hypersurface
in CPd+1 and d ≥ 4 be an even integer. Consider the autoequivalence Φ := TOX (−⊗O(−1))
on Db(X), where TOX is the spherical twist [89] with respect to the structure sheaf OX. Then
hcat(Φ) > 0 = log ρ([Φ]).
This gives a counterexample to Conjecture 0.0.14. In fact, hcat(Φ) is the unique positive real number
λ > 0 satisfying ∑k≥1χ(O(k))
ekλ = 1.
One can consider spherical twists as “hidden symmetries" on a Calabi–Yau manifold X:
they are not induced by automorphisms on X, but rather correspond to Dehn twists along
Lagrangian spheres on a mirror Calabi–Yau Y under mirror symmetry [89]. Theorem 0.0.15
shows that because of the existence of hidden symmetries on Calabi–Yau manifolds, the
7
autoequivalences on the derived category Db(X) contain much richer dynamical information
than the automorphisms on X.
On the other hand, it sill is interesting to characterize the autoequivalences that satisfy
Conjecture 0.0.14. Along this direction, we show that the Pd-twists defined by Huybrechts–
Thomas [52] satisfy Conjecture 0.0.14. Note that Pd-twists are mirror to Dehn twists along
Lagrangian complex projective spaces.
Theorem 0.0.16 (= Theorem 4.5.5). Let X be a smooth projective variety of dimension 2d over C,
and let PE be the Pd-twist [52] of a Pd-object E ∈ Db(X). Then
hcat(PE ) = 0 = log ρ([PE ]).
Hence Conjecture 0.0.14 holds for Pd-twists.
8
Chapter 1
Mirror of Atiyah flop1
1.1 Introduction
Flop is a fundamental operation in birational geometry. By the work of Kollár [64], any
birational transformation of compact threefolds with nef canonical classes and Q-factorial
terminal singularities can be decomposed into flops.
Atiyah flop is the most well-known among many different kinds of flops. It contracts a
(−1,−1) curve and resolves the resulting conifold singularity by a small blow-up, producing
a (−1,−1) curve in another direction, see Figure 1.1.
Figure 1.1: The Atiyah flop.
In mirror symmetry, complex and symplectic geometries are dual to each other. Flop is
an important operation in complex geometry. It is natural to ask whether there is a mirror
1Co-authored with Hansol Hong, Siu-Cheong Lau and Shing-Tung Yau. Reference: [35].
9
operation in symplectic geometry. In this chaper we focus on the mirror of Atiyah flop.
SYZ mirror symmetry of a conifold singularity is well-known by the works of [46,
20, 76, 2, 21, 56]. A conifold singularity is given by u1v1 = u2v2 in C4. There are two
different choices of anti-canonical divisors which turn out to be mirror to each other, namely
D1 = u2v2 = 1 and D2 = (u2 − 1)(v2 − 1) = 0. Consider the resolved conifold
OP1(−1)⊕OP1(−1), with the divisor D2 deleted. Its SYZ mirror is given by the deformed
conifold (u1, v1, u2, v2, z) ∈ C4 × C× : u1v1 = z + q, u2v2 = z + 1. Here q is the Kähler
parameter of the resolved conifold, namely q = e−A where A is the symplectic area of the
(−1,−1) curve in the resolved conifold. The deformed conifold contains a Lagrangian
sphere whose image in the z-coordinate projection is the interval [−1,−q] ⊂ C. The
Lagrangian sphere is mirror to the holomorphic sphere in the resolved conifold.
Now take the Atiyah flop. The Kähler moduli of the resolved conifold is the punctured
real line R−0, consisting of two Kähler cones R+ and R− of the resolved conifold and its
flop respectively. A serves as the standard coordinate and flop takes A ∈ R+ to −A ∈ R−.
Thus the Atiyah flop amounts to switching A to −A, or equivalently q to q−1. As a result,
the SYZ mirror changes from u1v1 = z + q, u2v2 = z + 1 to u1v1 = z + q−1, u2v2 = z + 1.
However the above two manifolds are symplectomorphic to each other, and hence they
are just equivalent from the viewpoint of symplectic geometry. Unlike Atiyah flop in
complex geometry, the mirror operation does not produce a new symplectic manifold. It is
not very surprising since symplectic geometry is much softer than complex geometry.
In contrast to complex geometry, the mirror flop is just a symplectomorphism rather
than a new symplectic manifold. First observe that this symplectomorphism is non-trivial
(Section 1.4.1).
Proposition 1.1.1. Given a symplectic threefold (X, ω) and a Lagrangian three-sphere S ⊂ X, we
have another symplectic threefold (X†, ω†) with a corresponding Lagrangian three-sphere S† ⊂
X†, together with a symplectomorphism f (X,S) : (X, ω) → (X†, ω†). It has the property that
f (X†,S†) f (X,S) = τ−1S , where τS is the Dehn twist along the Lagrangian sphere S.
We shall regard X and X† as the same symplectic manifold using the above symplecto-
10
morphism f (X,S).
We need to endow a symplectic threefold with additional geometric structures in order
to make it more rigid, so that the effect of the mirror flop can be seen. In the above local
case, u1v1 = z + q, u2v2 = z + 1 and u1v1 = z + q−1, u2v2 = z + 1 simply have different
complex structures. However in general requiring the existence of a complex structure on a
symplectic manifold would be too restrictive. Friedman [37] and Tian [97] showed that there
are topological obstructions to complex smoothing of conifold points; Smith-Thomas-Yau
[92] found the mirror statement for topological obstructions to Kähler resolution of conifold
points.
In this chapter, we consider two kinds of geometric structures, namely Lagrangian
fibrations, and Bridgeland stability conditions on the derived Fukaya category. First consider
a symplectic threefold X equipped with a Lagrangian fibration π : X → B. Let S ⊂ X be a
Lagrangian sphere. We assume that π around S is given by a local model of Lagrangian
fibration on the deformed conifold, where S is taken as the vanishing sphere under a
conifold degeneration, see Definition 1.4.3. We call such a fibration to be conifold-like
around S. Then we make sense of the mirror flop by doing a local surgery around S and
obtain another Lagrangian fibration π† : X → B. (X and X† have been identified by the
above symplectomorphism ρX,S.)
Theorem 1.1.2. Given a symplectic threefold (X, ω) with a Lagrangian fibration π : X → B which
is conifold-like around a Lagrangian three-sphere S ⊂ X, there exists another Lagrangian fibration
π† : X → B with the following properties.
1. π† is also conifold-like around S.
2. The images of S under π and π† are the same, denoted by S. They are one-dimensional affine
submanifolds in B away from discriminant locus.
3. π† = π outside a tubular neighborhood of S. In particular the affine structures on B induced
from π and π† are identical away from a neighborhood of S.
4. The induced orientations on S from π and π† are opposite to each other.
11
We call the change from π to π† to be the A-flop of a Lagrangian fibration along S.
As a compact example, consider the Schoen’s Calabi–Yau, which admits a conifold-like
Lagrangian fibration around certain Lagrangian spheres by the work of Gross [47] and
Castaño-Bernard and Matessi [76]. Then we can apply the A-flop to obtain other Lagrangian
fibrations.
More generally we can consider the effect of A-flop along S on Lagrangian submanifolds
other than Lagrangian torus fibers. Given a Lagrangian submanifold L ⊂ X which has T2-
symmetry around S (see Definition 1.4.7), we can construct another Lagrangian submanifold
L† (which also has T2-symmetry around S) which we call to be the A-flop of L, with the
property that (L†)† equals to the inverse Dehn twist of L along S.
Then we can take A-flop of special Lagrangian submanifolds with respect to a certain
holomorphic volume form (if it exists). Formally we start with a Bridgeland stability
condition (Z,P) [14] on the derived Fukaya category, where Z is a homomorphism of the K
group to C, and P is a collection of objects in the derived Fukaya category which are said to
be stable. A stability condition (Z,P) is said to be geometric if there exists a holomorphic
volume form Ω such that Z is given by the period∫·Ω and P is a collection of graded
special Lagrangians with respect to Ω. A-flop should be understood as a change of stability
conditions (Z,P) 7→ (Z†,P†).
In this chapter we realize the above for the local deformed conifold in Section 1.6. We
obtain the following theorem in Section 1.6.6.
Theorem 1.1.3. Let X be the deformed conifold u1v1 = z + q, u2v2 = z + 1, z 6= 0 (where
q 6= 1). Equip X with the holomorphic volume form Ω = dz∧ du1 ∧ du2. There exists a collection P
of graded special Lagrangians which defines a geometric stability condition (Z,P) on X. Moreover
the flop (Z†,P†) also defines a geometric stability condition with respect to ( f (X,S))∗ΩX† where
f (X,S) : X → X† = u1v1 = z + 1, u2v2 = z + 1/q : z 6= 0 is the symplectomorphism in
Stability conditions for the derived Fukaya category were constructed for the An case by
Thomas [94], for certain local Calabi–Yau threefolds associated to quadratic differentials by
12
Bridgeland–Smith [17, 91], and for punctured Riemann surfaces with quadratic differentials
by Haiden–Katzarkov–Kontsevich [48]. In this chapter we construct stability conditions
on the derived Fukaya category of the deformed conifold by applying the mirror functor
construction in [24, 23]; in the mirror side we use the results of Nagao–Nakajima [74] about
stability conditions on the noncommutative resolved conifold (see Theorem 1.6.9).
Theorem 1.1.4 (see Theorem 1.6.4). The mirror construction in [23] applied to the deformed
conifold X produces the noncommutative resolved conifold A given by Equation (1.15). In particular,
there is a natural equivalence of triangulated categories
Ψ : DbF → DbnilmodA (1.1)
where F is a subcategory of Fuk(X) generated by Lagrangians spheres, and DbnilmodA is a
subcategory of DbmodA consisting of modules with nilpotent cohomology.
The relation between the mirror construction in [23] and the SYZ construction is sum-
marized in Figure 1.2. The SYZ construction uses Lagrangian torus fibration coming from
degeneration to the large complex structure limit. The noncommutative mirror construction
in [23] uses Lagrangian vanishing spheres coming from degeneration to the conifold point.
LCSL of the deformed conifold
Another LCSL
conifold
B-side moduli of the local conifold
LVL of the resolved conifold
LVL of its op
nc resolution of the conifold
A-side moduli of the local conifold
SYZ
nc mirror construction
SYZ
Figure 1.2: The local conifold is self-mirror. More precisely for the local conifold, a resolution and its flop areequivalent, so the upper hemisphere should be identified with the lower hemisphere.
We shall prove that stable modules in DbnilmodA with respect to a certain stability
condition can be obtained as transformations of special Lagrangians under (1.1); as a result
the corresponding stability condition on DbF is geometric.
13
1.2 Review on flops and Bridgeland stability conditions
In this section, we recall the results by Toda which relate flops with wall-crossings in the
space of Bridgeland stability conditions on certain triangulated categories. For more details
and proofs, see [99].
1.2.1 Bridgeland stability conditions and crepant small resolutions
Let f : Y → Y be a crepant small resolution in dimension three and C the exceptional
locus, which is a tree of rational curves C = C1 ∪ · · · ∪ CN .
Define the triangulated subcategory DY/Y ⊂ Db(Y) to be
DY/Y := E ∈ Db(Y) | Supp(E) ⊂ C. (1.2)
Let pPer(Y/Y) ⊂ Db(Y) (p = 0,−1) be the abelian categories of perverse coherent
sheaves introduced by Bridgeland [13], and
pPer(DY/Y) :=p Per(Y/Y) ∩DY/Y.
Proposition 1.2.1 ([28]). The abelian categories 0Per(DY/Y) and −1Per(DY/Y) are the hearts of
certain bounded t-structures on DY/Y, and are finite-length abelian categories. The simple objects in
0Per(DY/Y) and −1Per(DY/Y) are ωC[1],OC1(−1), . . . ,OCN (−1) and OC,OC1(−1)[1], . . . ,
OCN (−1)[1] respectively.
Theorem 1.2.2 ([13][22]). Let g : Y† → Y be the flop of f , and φ : Y 99K Y† be the canonical
birational map. Then the Fourier-Mukai functor with the kernel OY×YY† ∈ Db(Y × Y†) is an
equivalence
ΦOY×YY†
Y→Y† : Db(Y)∼=−→ Db(Y†).
This equivalence restricts to an equivalenceDY/Y∼=−→ DY†/Y and takes 0Per(Y/Y) to −1Per(Y†/Y).
Such an equivalence is called standard in [99].
Let FM(Y) be the set of pairs (W, Φ), where W → Y is a crepant small resolution, and
Φ : Db(W)→ Db(Y) can be factorized into standard equivalences and the auto-equivalences
14
given by tensoring line bundles. For each (W, Φ) ∈ FM(Y), there is an associated open
subset
U(W, Φ) ⊂ Stabn(Y/Y)
of the space of normalized Bridgeland stability conditions on DY/Y. A Bridgeland stability
condition on DY/Y is called normalized if the central charge Z([Ox]) of the skyscraper sheaf
at each x ∈ C is −1.
Assume in addition that there is a hyperplane section in Y containing the singular point
such that its pullback in Y is a smooth surface, Toda proved the following theorem.
Theorem 1.2.3 ([99]). Let Stabn(Y/Y) be the connected component of Stabn(Y/Y) containing the
standard region U(
Y, Φ = idDb(Y)
). Define the following union of chambers
M :=⋃
(W,Φ)∈FM(Y)
U(W, Φ).
Then M ⊂ Stabn(Y/Y), and any two chambers are either disjoint or equal. Moreover, M =
Stabn(Y/Y).
In other words, we can obtain the whole connected component Stabn(Y/Y) from the
standard region U(Y, id) by sequence of flops and tensoring line bundles.
1.2.2 The conifold
Let Y = Spec C[[x, y, z, w]]/(xy− zw) and f : Y → Y be the blowing up at the ideal
(x, z). As computed in [99],
Stabn(Y/Y)/Aut0(DY/Y)∼= P1 − 3 points .
Let Y† → Y be the blowing up at the other ideal (x, w). Then the three removed points
correspond to the large volume limit points of Y and Y†, and the conifold point.
More precisely, P1− 3 points is obtained by gluing the upper and lower half complex
planes H, H†, and the real line with the origin removed. The hearts of the Bridgeland
stability conditions in H and H† are given by CohY/Y and CohY†/Y respectively. The
15
heart of the Bridgeland stability conditions on the real line is given by the perverse heart
0Per(DY/Y)∼=−1 Per(DY†/Y).
Let C, C† be the exceptional curves of Y → Y, Y† → Y respectively. Then the equivalence
DY/Y −→ DY†/Y satisfies
1. Φ(OC(−1)) = OC†(−1)[1].
2. Φ(OC(−2)[1]) = OC† .
3. For x ∈ C, the cohomology of E := Φ(Ox) ∈ DY†/Y vanish except for H0(E) = OC†
and H−1(E) = OC†(−1).
One can observe the following wall-crossing phenomenon: the skyscraper sheaves
Ox ∈ DY/Y are stable objects with respect to the stability conditions on the upper half plane
H, but are unstable in H†. In fact, its image under Φ is a two term complex E that fits into
the following exact triangle:
OC†(−1)[1]→ E→ OC†[1]→ (1.3)
Note that the usual skyscraper sheaf at a point in C† can be obtained by switching the first
and the third terms in (1.3).
Remark 1.2.4. It is well-known that if C is a (−1,−1)-curve, then the ‘flop-flop’ functor is the same
as the inverse of the spherical twist by OC(−1), i.e. ΦOY×YY†
Y†→YΦ
OY×YY†
Y→Y† = T−1OC(−1). Proposition
1.1.1 is the mirror statement of this fact.
1.3 Review on the SYZ mirror of the conifold
SYZ mirror construction for toric Calabi–Yau manifolds was carried out in [20] using the
wall-crossing techniques of [6]. The reverse direction, namely SYZ construction for blow-up
of V ×C along a hypersurface in a toric variety V was carried out by [2]. In this section we
recall the construction for the conifold Y = (u1, v1, u2, v2) ∈ C4 : u1v1 = u2v2 as a special
16
case in [20, 2]. The statement is that Y− u2v2 = 1 is mirror to Y− (u2 = 1 ∪ v2 = 1).
The study motivates the definition of A-flop for Lagrangian fibrations in the next section.
The resolved conifold Y = OP1(−1)⊕OP1(−1) is obtained from a small blowing-up of
the conifold point (u1, v1, u2, v2) = 0. It is a toric manifold equipped with a toric Kähler form.
We have the T2-action on Y given by (λ1, λ2) · (u1, v1, u2, v2) = (λ1u1, λ−11 v1, λ2u2, λ−1
2 v2),
and we denote the corresponding moment map by (µ1, µ2) : Y → R2. Then from the works
of Ruan [83], Gross [46] and Goldstein [42], there is a Lagrangian fibration
(µ1, µ2, |zw− 1|) : Y → R2 ×R≥0.
It serves as one of the local models of Lagrangian fibrations which were used by Castaño-
Bernard and Matessi [75, 76] to build up global fibrations from a tropical base manifold.
Figure 1.3: The base and discriminant loci of Lagrangian fibrations in conifold transition.
The discriminant locus of this fibration is contained in the hyperplane
(x1, x2, x3) ∈ R2 ×R≥0 : x3 = 1,
see the top left of Figure 1.3. This hyperplane is known as the wall for open Gromov–
Witten invariants of torus fibers as it contains images of holomorphic discs of Maslov index
zero. By studying wall-crossing of holomorphic discs emanated from infinity divisors (of a
compactification of Y), [20] constructed the SYZ mirror of Y− zw = 1.
Theorem 1.3.1 (A special case in [20] and [2]). The SYZ mirror of Y− u2v2 = 1 is
and label them as ωjk. See Figure 1.8. The top triangular face of σj is glued to the bottom
triangular face of σj+1 (j ∈ Z3), and so topologically⋃3
j=1 σj forms a solid torus. Similarly
do the same thing for τk so that⋃3
k=1 τk forms another solid torus. For the nine cubes, glue
the top face of ωjk with the bottom face of ωj+1,k for j ∈ Z3, and glue the right face of
ωjk with the left face of ωj,k+1 for k ∈ Z3. This topologically forms a two-torus times an
interval. Finally glue the front face of ωjk with the j-th square face of τk, and glue the back
face of ωjk with the k-th square face of σj. Here the square faces of σj and τk are ordered
counterclockwisely. This forms S3 as gluing of two solid tori along their boundaries.
The fan structure at every vertex of the polyhedral decomposition is that of P2 ×P1.
Together with the standard affine structure of each polytope, this gives S3 an affine structure
with singularities. The discriminant locus is given by the dotted lines shown in Figure 1.8.
Note that each dotted line in a square face of a prism indeed has multiplicity three. Thus
the discriminant locus is a union of 24 circles counted with multiplicities. Moreover the
dotted lines in cubes form three horizontal and three vertical circles, intersecting with each
32
(0,0,0)
(0,0,1)
(1,0,0)
(1,0,1)
(0,1,0)
(0,1,1)glue
glue
glue
glue
The whole is a three-sphere topologically.
Figure 1.8: Polytopes in the polyhedral decomposition of the affine base of Schoen’s CY.
other at nine points. These are the nine conifold singularities (which are positive nodes).
By gluing local models of Lagrangian fibrations around discriminant locus with the
Lagrangian fibration from the affine structure away from discriminant locus, [76] produced a
symplectic manifold which is homeomorphic to the Schoen’s Calabi–Yau. Moreover by using
the results on symplectic resolution of Smith–Thomas–Yau [92] and complex smoothing of
Friedman [37] and Tian [97], they showed that the existence of certain tropical two-cycles
containing a set of conifold points ensure that the nodes can be simultaneously resolved
(and smoothened in the mirror side). In particular all the nine nodes in this example can be
resolved simultaneously.
In the smoothing the three horizontal and three vertical circles which form part of the
discriminant locus are moved apart so that they no longer intersect with each other. This
gives a symplectic manifold X together with a Lagrangian fibration. The corresponding
affine base coincides with the one in the previous work of Gross [47, Section 4].
The local model for each conifold point in this example is the Lagrangian fibration
(|u1|2 − |v1|2, |u2|2 − |v2|2, |z|) on (u1, v1, u2, v2, z) ∈ C4 × C× : u1v1 = z− a = u2v2 for
a < 0; the local model for its smoothing is the fibration defined by the same expression
on (u1, v1, u2, v2, z) ∈ C4 ×C× : u1v1 = z− a, u2v2 = z− b for a < b < 0. A Lagrangian
fibration corresponding to the simultaneous smoothing can be constructed by gluing these
33
local models. In particular X and the fibration are conifold-like around each of the van-
ishing spheres corresponding to the nine conifold points. Hence we can perform A-flop
around each of these spheres and obtain new Lagrangian fibrations. The operation can be
understood as link surgery in the base S3.
Note that we cannot always keep the circles Ai, Bj in constant levels in the A-flop. For
instance, suppose Ai and Bj are contained in the planes in levels ai, bj respectively with
a1 < a2 < a3 < b1 < b2 < b3. (These planes have normal vectors pointing to the right if
drawn in Figure 1.8.) Now we perform the A-flop along the vanishing sphere between
levels a1 and b1. The resulting fibration is equivalent to the one with these circles in levels
a2 < a3 < b1 < a′1 < b2 < b3 where a′1 is the new level of A1. At this stage all these circles
are still kept in constant levels. Now let’s do the A-flop along the vanishing sphere between
levels a2 and b3. Then the resulting fibration cannot have all these circles in constant levels:
if they were in constant levels, then a2 < b1 < a′1 < b3 < a2, a contradiction!
1.5 Derived Fukaya category of the deformed conifold
In Example 1.4.4, we consider a path of complex structures on the deformed conifold
(with a fixed symplectic form) given by the equations
Xs =
u1v1 = z−
(a + b
2+
b− a2
eπi (1+s))
, u2v2 = z−(
a + b2
+b− a
2eπi s
), z 6= 0
(1.6)
for s ∈ [0, 1]. (Xs=0 and Xs=1 were denoted as X and X† in 1.4.4, respectively.) This
deformation of complex structures parametrized by s is SYZ mirror to the flop operation
on the resolved conifold. In the last section we realized this operation as surgery of a
Lagrangian fibration.
In this section, we study the effect of deformation of complex structures (together with
holomorphic volume forms) on special Lagrangians. This would motivate us to consider
A-flop on stability conditions of the derived Fukaya category.
Recall from Section 1.3 that we have two Lagrangian spheres S0 and S1 in Xs=0. Moreover,
34
there is a sequence of Lagrangian spheres Sn : n ∈ Z in Xs=0 which corresponds to a
collection of non-trivial matching paths in the base of the double conic fibration Xs=0 → C×.
We depict these spheres in the universal cover of C×(3z) as shown in Figure 1.9.
Figure 1.9: Sequence of Lagrangian spheres in Xs=0.
Definition 1.5.1. F is defined to be the full subcategory of Fuk(Xs=0) generated by S0 and S1.
The main purpose of this section is to prove the following.
Theorem 1.5.2. Regular Lagrangian torus fibers of π which have non-empty intersection with S0,
as well as the Lagrangian spheres Si, are contained in F .
The theorem follows from Proposition 1.5.7 and 1.5.9 below.
The torus fibers and spheres Si are special Lagrangians with respect to the holomorphic
volume form
Ω := d log z ∧ du1 ∧ du2
on Xs=0. Here we used d log z instead of d log z to match the ordering of phases on both
sides of the mirror2. In particular, we measure the angle in clockwise direction for phases of
Si. The diagram in the right side of Figure 1.9 compares the phases of Si’s, where S0 has the
2In order to match the phase inequality in the mirror side, we can either impose the mirror functor to becontravariant, or use the complex structure induced by the conjugate volume form d log z ∧ du1 ∧ du2 like here.All Si are still special Lagrangians under this volume form, and we have the phase inequalities θ(Si) > θ(Sj) for0 < i < j or i < j < 1. This matches the ordering of the phases of stable objects in an exact triangle of the mirror
B-side convention. Namely for an exact triangle L1 → L1#L2 → L2[1]→ where Li are special Lagrangians, their
phases should satisfy θ(L1) ≤ θ(L1#L2) ≤ θ(L2).
35
biggest phase in our convention. In Section 1.6 we will see that taking these to be stable
objects defines a Bridgeland stability condition on the derived Fukaya category.
Moreover each Si corresponds to another Lagrangian sphere S†i in F , the flop of Si
constructed in Section 1.4.3. The Lagrangian torus fibers of π† and S†i are special with
respect to the pull-back holomorphic volume form from Xs=1, and they define another
Bridgeland stability condition. In fact, we have S†i = ρ−1(S′−i) where S′n : n ∈ Z is the
set of new special Lagrangian spheres in Xs=1 which map to straight line segments by
z-projection as in Figure 1.10.
Figure 1.10: Special Lagrangian spheres S′k in Xs=1.
For later use we orient these spheres as follows. In conic fiber direction, each Si restricts
to a 2-dimensional torus |u1| = |v1|, |u2| = |v2|. We fix the orientation on the fiber torus
to be dθ1 ∧ dθ2 where θi are the arguments of ui respectively. We orient their matching paths
as in the right side of Figure 1.9.
Set L0 := S0 and L1 := S1. They are distinguished objects in the set Sn of Lagrangian
spheres in the sense that they have minimal/maximal slopes (or phases) as well as they
generate DbF . We will study Lagrangian Floer theory of L0 and L1 intensely in Section
1.6.1.
36
Recall from Section 1.2 that DY/Y is the subcategory of Db(Y) generated by OC(−1)[1]
and OC. [21] proved the following equivalence of subcategories of Fuk(Xs=0) and Db(Y).
Theorem 1.5.3. ([21]) There is an equivalence DbF ' DY/Y, sending
L0 7→ OC(−1)[1] L1 7→ OC. (1.7)
Using the chain model of Abouzaid [1], they explicitly computed the A∞-structure of
the endomorphism algebras of L0 ⊕ L1 to conclude that
End(L0 ⊕ L1) ' End(OC(−1)[1]⊕OC). (1.8)
See [21, Section 5, 7] for more details.
In this chapter, we shall use either the Morse–Bott model in [38] or pearl trajectories
[11, 90] to study Lagrangian torus fibers and the noncommutative mirror functor. They are
conceptually easier to understand.
Figure 1.11: Transformation of L0 and L1 by the symplectomorphism ρ.
The A-flop can be realized by the symplectomorphism ρ from Xs=0 to Xs=1 given in
1.4.4 (see Figure 1.7). Figure 1.11 shows how ρ acts on Li, where the third diagram describes
the moment at which L0 and L1 happen to have the same phases. Observe that Xs=1 (1.6)
is obtained from Xs=0 by swapping two sets of coordinates (u1, v1) and (u2, v2). However,
swapping the coordinates is different from the symplectomorphism that gives A-flop, as its
effect on z-plane shows.
37
As in Figure 1.11, ρ sends L0 and L1 to special Lagrangian spheres in Xs=1 which we
denote by L′0 and L′1 respectively. Let F ′ denote the Fukaya subcategory of Xs=1 consisting
of L′0 and L′1. There is a natural functor ρ∗ : F → F ′ induced by the symplectomorphism ρ.
On the other hand, we can repeat the same argument as in the proof of Theorem 1.5.3 to see
that
DbF ′ ' DY†/Y with L′0 7→ OC†(−1) and L′1 7→ OC†(−2)[1].
Notice that this identification is coherent with the fact that L′0 is somewhat similar to the
orientation reversal of L0, whereas L1ρ7→ L′1 can be understood as a change of winding
number with respect to z = 0.
In fact, we have
End(L′0 ⊕ L′1) ' End(L0 ⊕ L1) (1.9)
as two set of objects are related by a symplectomorphism, and
End (OC†(−1)⊕OC†(−2)[1]) ' End (OC(−1)[1]⊕OC) (1.10)
due to the flop functor (see 1.2.2). It directly implies that the functor ρ∗ induced by the
symplectomorphism is mirror to the flop functor through the identification of A and B side
categories via [21]. Namely,
Proposition 1.5.4. We have a commutative diagram of equivalences:
DbF
ρ∗
' // DY/Y
Φ
DbF ′ ' // DY†/Y
(1.11)
Proof. It obviously commutes on the level of objects by the construction. (1.8), (1.9) and
(1.10) imply that the diagram also commutes on morphism level.
We shall study how the symplectomorphism ρ or its induced functor ρ∗ acts on various
geometric objects in DbF . For that, we should examine what kind of geometric objects are
actually contained in DbF .
38
1.5.1 Geometric objects of DbF
Let us first prove that any torus fiber intersecting L0 and L1 is isomorphic to a mapping
cone Cone(
L0α→ L1
)for some degree-1 element α ∈ HF(L0, L1) in the derived Fukaya
category. In particular this will imply that the category DbF contains those torus fibers as
objects.
One can choose the gradings on Li such that HF∗(L0, L1) = H∗(S1a)[−1]⊕ H∗(S1
b)[−1].
Here, we use the Morse–Bott model, where S1a and S1
b denotes the intersection loci over
z = a and z = b, respectively. Both S1a and S1
b are isomorphic to a circle. Thus degree 1
elements in HF(L0, L1) are given by linear combinations of (Poincarè duals of) fundamental
classes of S1a and S1
b. The cone Cone(L0α→ L1) can be identified with a boundary deformed
object (L0 ⊕ L1, α) (see [38] or [86]).
Let Lc := (for a < c < b) be a Lagrangian torus that intersects L0 at z = c. This condition
determines Lc uniquely, as components of Lc in double conic fiber direction should satisfy
the same equation as those of L0 and L1. We orient Lc as in Figure 1.9 in z-plane components,
and use the standard one (from dθ1dθ2 as for Si) along the conic fiber directions. Lc cleanly
intersects L0 and L1 along 2-dimensional tori which we denote by T0 := Lc ∩ L0 and
T1 := Lc ∩ L1. One can see that CF(Lc, L0) = C∗(T0) and CF(L1, Lc) = C∗(T1) for suitable
choice of a grading on Lc. Similarly, CF(L0, Lc) = C∗(T0)[−1] and CF(Lc, L1) = C∗(T1)[−1].
Let Uρ1,ρ2,ρz be a unitary flat line bundle on Lc whose holonomies along circles in the
double conic fibers are ρ1 and ρ2 and that along the circle in z-plane is ρz.
Proof. One can simply use the Morse–Bott model for each of cohomology groups in (1.12).
Each of this group is simply a singular cohomology of the intersection loci, equipped with
twisted differential. Since the intersection loci are 2-dimensional torus in the double conic
fiber, the twisting is determined by (ρ1, ρ2). Here we only have classical differential, as
there is no holomorphic strip between Li and Tc. One can easily check that the cohomology
39
vanishes if the twisting is nontrivial.
Alternatively, one can perturb Lagrangians to have transversal intersections as in Figure
1.12 to see that the Floer differential has coefficients ρ1 − 1 and ρ2 − 1, which are nonzero
for nontrivial (ρ1, ρ2).
Figure 1.12: Perturbation of Lc, Li and polygons contributing to mb1 on CF(L, (Lc, ρ)).
The lemma implies that (Lc, Uρ1,ρ2,ρz) has no Floer theoretic intersection with L0 or L1
unless ρ1 = ρ2 = 1. From now on, we will only consider flat line bundles of the type U0,0,ρz
on Lagrangian torus fibers, which will be written as Uρz instead of U0,0,ρz for notational
simplicity. Let P0 := PD[T0] ∈ CF∗(Lc, L0) = C∗(T0) and P1 := PD[T1] ∈ CF∗(L1, Lc) =
C∗(T1). We also set αa := PD[S1a ] ∈ CF∗(L0, L1) and αb := PD[S1
b] ∈ CF∗(L0, L1). Notice
that degαa = degαb = 1 whereas degP0 = degP1 = 0.
Lemma 1.5.6. P0 ∈ CF0((Lc, Uρz), (L0⊕L1, α)) and P1 ∈ CF0((L0⊕L1, α), (Lc, Uρz)) are cycles
with respect to m0,α1 and mα,0
1 respectively if and only if α is given as λaαa + λbαb ∈ CF1(L0, L1)
with (λa : λb) = (Tω(∆2)ρz : Tω(∆1))3 where ∆1 and ∆2 are triangles shaded in Figure 1.13.
3It is harmless to put T = e−1 since only finitely many polygons contribute to A∞-structures. Neverthelesswe will keep the notation T to highlight contributions from nontrivial holomorphic polygons.
40
Figure 1.13: Two triangles contributing to m1 and m2.
Proof. We will prove the lemma for P1 only, and the proof for P0 is similar. We pick a point ×
as in Figure 1.13 for representative of Uρz so that when boundary of a holomorphic polygon
passes this point, the corresponding mk-operation is multiplied by ρ±1z depending on the
orientation. (More precisely, the point × represent 2-dimensional subtorus in Lc lying over
this point, which is called a hyper-torus and used to fix the gauge of a flat line bundle in
[25].)
Observe that two holomorphic triangles ∆1 and ∆2 shown in Figure 1.13 contribute to
Moreover, the Weil–Petersson metric on the stringy Kähler moduli space AutCY(DA)\Stab+N (DA)/C×
is identified with the Bergman metric on the Siegel modular variety P(SL(2, Z) × SL(2, Z))o
Z2\(H×H).
This observation is compatible with self-mirror symmetry for the split abelian surfaces.
In fact a lattice polarized version of the global Torelli Theorem asserts that the complex
moduli space of split abelian surfaces are given by the above Siegel modular variety.
Remark 2.4.14. A similar computation can be carried out for M-polarized K3 surfaces for the
lattice M ∼= U⊕2 ⊕ 〈−2〉 or U⊕2. The main difference is that there are spherical objects in the
derived category DX of a K3 surface X and we need to remove the union of certain hyperplanes from
P+. Moreover, the subgroup of Aut0(DX) which preserves the connected component Stab†(DX)
acts freely on Stab†(DX). So one does not need to take the quotient of the group of Calabi–Yau
autoequivalences by Aut0tri(D) as in the abelian surface case (c.f. [15]).
2.4.3 Abelian variety
Let X be an abelian variety of dimension n. Since there is no quantum corrections and
the Chern classes are trivial, the expected central charge at the complexified Kähler moduli
87
ω ∈ H2(X; C) is given by
Zfω(E) = −〈fω, vX(E)〉Muk = −
∫X
e−ωch(E).
The existence of Bridgeland stability condition with this central charge is known for n ≤ 3.
By Proposition 2.4.6, the Weil–Petersson potential is
KWP(τ) = − log(=(ω)n)− log2n
n!.
Fix a polarization H. We think of ω = τH for τ ∈ H as a slice of the stringy Kähler
moduli spaceMKah(X). Then the Weil–Petersson metric on H is essentially the Poincaré
metric. This example is a toy model in the sense that there is no quantum correction.
The above observation is compatible with Wang’s mirror result [102, Remark 1.3], which
says that in the case of infinite distance, the Weil–Petersson metric is asymptotic to a scaling
of the Poincaré metric.
2.4.4 Quintic threefold
Although the existence of a Bridgeland stability condition for a quintic threefold X ⊂ P4
has not yet been proven, we can still compute the Weil–Petersson potential using the central
charge in Equation (2.1) near the large volume limit.
Let τH ∈ H2(X; C) be the complexified Kähler class, where H is the hyperplane class
and τ ∈H. First we observe that
exp∗(τH) = 1 + τH +τ2
2(1 +
15 ∑
d≥1Ndd3qd)H2 +
τ3
6(1 +
15 ∑
d≥1Ndd3qd)H3,
where q = e2π√−1τ and NX
d denotes the genus 0 Gromov–Witten invariant of X of degree d,
and we use the quantum product
H ∗ H = Φ(q)H2 =15(5 + ∑
d≥1NX
d qdd3)H2.
88
Then the central charge computes to be
Z(E) = − 〈exp∗(τH), vX(E)〉Muk
= −∫
Xe−τHvX(E) +
ζ(3)χ(X)
(2π)3 (τ2
10H2ch1(E)− τ3
6ch0(E)) ∑
d≥1NX
d d3qd,
where χ(X) is the topological Euler number of X. Near the large volume limit, the Weil–
Petersson potential is given by
KWP(τ) = − log(
H3(Φ(q)(τ3
6+
ττ2
2)−Φ(q)(
τ3
6+
τ2τ
2))− 2 log
( ζ(3)χ(X)
(2π)3
)∼ − log(
43
H3=(τ)3)− 2 log( ζ(3)χ(X)
(2π)3
)+ O(q).
Therefore the Weil–Petersson metric of a quintic threefold is a quantum deformation of the
Poincaré metric on H as expected. In particular, for sufficiently small q, it is non-degenerate
and the Weil–Petersson distance to the large volume limit is infinite. When there is no
B-field, i.e. τ ∈√−1R, the correction term O(q) is explicitly given by log(Φ(q)).
Remark 2.4.15 ([18]). The stringy Kähler moduli spaceMKah(X) of a quintic Calabi–Yau threefold
X ⊂ P4 is expected to be identified with the suborbifold
[z ∈ C | z5 6= 1/Z5] ⊂ [P1/Z5].
The point z = ∞ is the large volume limit, the point z5 = 1 is the conifold point, and the point z = 0
is the Gepner point. We expect the following properties of the Weil–Petersson metric onMKah(X).
1. The Weil–Petersson distance to the conifold point, which corresponds to a quintic threefold with
a conifold singularity, should be finite. This is based on a result of Wang [102] on the mirror
complex moduli, which asserts that if a Calabi–Yau variety has at worst canonical singularities,
then it has finite Weil–Petersson metric along any smoothing to Calabi–Yau manifolds.
2. The Weil–Petersson metric at the Gepner point should be an orbifold metric. This is because
the auto-equivalence
Φ(−) = TOX ((−)⊗OX(H)
),
where TOX denotes the Seidel–Thomas spherical twist with respect to OX, at the Gepner
89
point satisfies the relation Φ5 = [2]. This descends to Φ5 = id on K(DX). On other hand,
the calculations of Candelas–de la Ossa–Green–Parkes [18] shows that the Weil–Petersson
curvature tends to +∞ as we approach the Gepner point.
It is interesting to investigate the interplay among the geometry of a Calabi–Yau threefold
X, the cubic intersection form on H2(X; Z), and curvature properties of the Weil–Petersson
metric near a large volume limit [101, 103, 105].
On the other hand, probably a more alluring research direction is to examine the Weil–
Petersson metric away from a large volume limit, where central charges are not of the form
(2.1), as the metric is inherently global. For instance, the Weil–Petersson metric around a
Gepner point may be studied via matrix factorization categories via the Orlov equivalence
[78]
DbCoh(X) ∼= HMF(W),
where HMF(W) is the homotopy category of a graded matrix factorization of the defining
equation W of the quintic 3-fold X. Toda studied stability conditions, called the Gepner
type stability conditions, conjecturally corresponding to the Gepner point [100].
90
Chapter 3
Systolic inequality on K3 surfaces1
3.1 Introduction
Let (M, g) be a Riemannian manifold. Its systole sys(M, g) is defined to be the least
length of a non-contractible loop in M. In 1949, Charles Loewner proved that for any metric
g on the two-torus T2,
sys(T2, g)2 ≤ 2√3
vol(T2, g).
Moreover, the equality can be attained by the flat equilateral torus. There are various
generalizations of Loewner’s tours systolic inequality. We refer to [58] for a survey on the
rich subject of systolic geometry.
The first goal of the present chapter is to propose a new generalization of Loewner’s
torus systolic inequality from the perspective of Calabi–Yau geometry. We start with an
observation in the case of two-torus. Suppose that the torus is flat T2 ∼= C/Z + τZ, and is
equipped with the standard complex structure Ω = dz and symplectic structure ω = dx∧ dy.
Then the shortest non-contractible loops must be given by straight lines, which are nothing
but special Lagrangian submanifolds with respect to the complex and symplectic structures.
1Reference: [34].
91
Hence, under these assumptions, one can rewrite Loewner’s torus systolic inequality as:
infL:sLag
∣∣∣ ∫L
dz∣∣∣2 ≤ 1√
3
∣∣∣ ∫T2
dz ∧ dz∣∣∣.
Here “sLag" denotes the compact special Lagrangian submanifolds in T2 with respect to ω
and Ω = dz. The key observation is that the quantities in both sides of the above inequality
can be generalized to any Calabi–Yau manifold.
We propose the following definition of systole of a Calabi–Yau manifold, with respect to its
complex and symplectic structures.
Definition 3.1.1. Let Y be a Calabi–Yau manifold, equipping with a symplectic form ω and a
holomorphic top form Ω. Then its systole is defined to be
sys(Y, ω, Ω) := infL:sLag
∣∣∣ ∫L
Ω∣∣∣.
Here “sLag" denotes the compact special Lagrangian submanifolds in (Y, ω, Ω).
With this definition, we propose the following question that naturally generalizes
Loewner’s torus systolic inequality for Calabi–Yau manifolds.
Question 3.1.2. Let Y be a Calabi–Yau manifold and ω be a symplectic form on Y. Does there exist
a constant C > 0 such that
sys(Y, ω, Ω)2 ≤ C ·∣∣∣ ∫
YΩ ∧Ω
∣∣∣holds for any holomorphic top form Ω on Y?
Note that the ratio |∫
L Ω|2/|∫
Y Ω ∧Ω| has been considered in the context of attractor
mechanism in physics [31, 67, 72], which is of independent interest.
To answer Question 3.1.2, one needs to characterize the middle homology classes of
the Calabi–Yau manifold Y that can support special Lagrangian submanifolds. This is a
difficult problem in general. Nevertheless, if Y has a mirror Calabi–Yau manifold X, then
special Lagrangian submanifolds on Y conjecturally are corresponded to certain Bridgeland
semistable objects in the derived category of coherent sheaves on X [16]. The good news is
that when X is a K3 surface, there is a algebro-geometric result (Proposition 3.4.2) which
92
characterizes the classes that support Bridgeland semistable objects. This motivates us to
turn Question 3.1.2 into its mirror question (Question 3.1.4) and answer it in the case of K3
surfaces (Theorem 3.1.5). The formulation of the mirror question involves the categorical
analogues of systole and volume, which we now describe.
The second goal of the present chapter is to propose the definitions of categorical
systole and categorical volume, in terms of Bridgeland stability conditions on triangulated
categories. For the definition of categorical systole, there are two sources of motivation:
the correspondence between flat surfaces and stability conditions, and the conjectural
description of stability conditions on the Fukaya categories of Calabi–Yau manifolds.
To give a Bridgeland stability condition on a triangulated category D, one needs to
declare a set of objects in D to be semistable, and assign a complex number (central charge Z)
to each object in D. The correspondence between flat surfaces and stability conditions was
studied by Gaiotto–Moore–Neitzke [39], Bridgeland–Smith [17], and Haiden–Katzarkov–
Kontsevich [48]. Roughly speaking, under this correspondence, saddle connections on flat
surfaces are corresponded to semistable objects, and their lengths are corresponded to the
absolute values of central charges.
On the other hand, the systole of a flat surface is defined to be the length of its shortest
saddle connection. Based on the aforementioned correspondence, we propose the following
definition of systole of a Bridgeland stability condition.
Definition 3.1.3 (= Definition 3.2.1). Let D be a triangulated category, and σ be a Bridgeland
stability condition on D. Its systole is defined to be
sys(σ) := min|Zσ(E)| : E is σ−semistable.
This definition is supported by another source of motivation, the conjectural description
of stability conditions on the Fukaya categories of Calabi–Yau manifolds. Let Y be a Calabi–
Yau manifold and ω be a symplectic form on Y. One can associate a triangulated category,
the derived Fukaya category DπFuk(Y, ω), to the pair (Y, ω). Roughly speaking, the objects
in the Fukaya category are Lagrangian submanifolds with some extra data.
93
It is conjectured by Bridgeland [16] and Joyce [55] that a holomorphic top form Ω
on Y should give a Bridgeland stability condition on DπFuk(Y, ω), with central charges
given by period integrals of Ω along Lagrangians, and special Lagrangians are semistable
objects. Assuming this conjecture, Definition 3.1.1 coincides with Definition 3.1.3 for
D = DπFuk(Y, ω) and σ is given by a holomorphic top form Ω.
We summarize the correspondences in the following table.
Surface S Calabi–Yau (Y, ω) Triangulated categoryabelian differentials holomorphic top forms stability conditionssaddle connections special Lagrangians semistable objects
lengths period integrals central chargessys(S) sys(Y, ω, Ω) sys(σ)
|∫
Y Ω ∧Ω| vol(σ)
Table 3.1: Correspondence among flat surfaces, holomorphic top forms on Calabi–Yau manifolds, and stabilityconditions on triangulated categories.
In a previous joint work with Kanazawa and Yau [36], we study the categorical analogue
of the holomorphic volume∣∣∣ ∫Y Ω ∧Ω
∣∣∣ of Calabi–Yau manifolds. It is again motivated from
the conjectural description of stability conditions on their Fukaya categories. We recall the
definition of categorical volume vol(σ) in Section 3.2.
The third goal of the present chapter is to propose the mirror question of Question
3.1.2 under mirror symmetry, in terms of Bridgeland stability conditions. Mirror symmetry
conjecture states that Calabi–Yau manifolds come in pairs, in which the complex geometry of
one is equivalent to the symplectic geometry of the other, and vice versa. One mathematical
formulation of the conjecture, the homological mirror symmetry conjecture, was proposed by
Kontsevich [65] in 1994. It states that if (X, ωX, ΩX) and (Y, ωY, ΩY) is a mirror pair of
Calabi–Yau manifolds, then there are equivalences between triangulated categories
Similarly, we have log sys(σ2) > log sys(σ1)− ε. Hence log sys(σ) is a continuous function
on Stab(D), and so is sys(σ).
Example 3.2.6 (Derived category of A2-quiver). Let Q be the A2-quiver (· → ·) and D =
Db(Rep(Q)) be the bounded derived category of the category of representations of Q. Let E1 and E2
be the simple objects in Rep(Q) with dimension vectors dim(E1) = (1, 0) and dim(E2) = (0, 1).
There is one more indecomposable object E3 that fits into the exact sequence
0→ E2 → E3 → E1 → 0.
Let σ = (Z ,P) be a stability condition on D with z1 := Z(E1) and z2 := Z(E2). Suppose that
P(0, 1] = Rep(Q). Then
• When arg(z1) < arg(z2), the only σ-stable objects are E1 and E2 up to shiftings. Thus
sys(σ) = min|z1|, |z2|.
98
• When arg(z1) > arg(z2), the only σ-stable objects are E1, E2 and E3 up to shiftings. Thus
sys(σ) = min|z1|, |z2|, |z1 + z2|.
Note that in order to compute the categorical systoles of stability conditions in different chambers,
one needs to compute the central charges of different sets of dimensional vectors. However, it is not
the case for the derived categories of elliptic curves and K3 surfaces, see the proof of Theorem 3.3.1
and Proposition 3.4.2.
3.2.2 Categorical volume
Motivated by the discussion in Section 2.3.2, we define the notion of categorical volume of
Bridgeland stability conditions. It is the categorical analogue of the holomorphic volume∣∣∣ ∫Y Ω ∧Ω∣∣∣ of a compact Calabi–Yau manifold Y with holomorphic top form Ω.
Definition 3.2.7. Let Ei be a basis of the numerical Grothendieck groupN (D) and let σ = (Z ,P)
be a Bridgeland stability condition on D. Its volume is defined to be
vol(σ) :=∣∣∣∑
i,jχi,jZ(Ei)Z(Ej)
∣∣∣,where (χi,j) = (χ(Ei, Ej))
−1 is the inverse matrix of the Euler pairings.
One can easily check that the above definition is independent of the choice of the basis
Ei. The following lemma is also straightforward.
Lemma 3.2.8. Let σ be a Bridgeland stability condition on D. Then
1. vol(Φ · σ) = vol(σ) for any autoequivalence Φ ∈ Aut(D).
2. vol(z · σ) = e2x · vol(σ) for any complex number z = x + iy ∈ C.
Remark 3.2.9. Although the categorical volume can be defined for stability conditions on any
triangulated category D, its geometric meaning is not clear unless D comes from compact Calabi–Yau
geometry. Below is an example of Calabi–Yau triangulated category for which the categorical volume
vanishes for some stability conditions.
99
Example 3.2.10 (CY3-category of A2-quiver). Let Q be the A2-quiver (· → ·) and ΓQ be the
Ginzburg Calabi–Yau–3 dg-algebra associated with Q. See [41, 60] for the definition of ΓQ. Let
D(ΓQ) be the derived category of dg-modules over ΓQ and D = Dfd(ΓQ) be the full subcategory of
D(ΓQ) consisting of dg-modules with finite dimensional total cohomology.
By Keller [60, Theorem 6.3], the category D is a Calabi–Yau–3 category, i.e., there is a natural
isomorphism Hom(E, F) ∼= Hom(F, E[3])∗ for any E, F ∈ D. By Smith [91], there is an embedding
of D into the Fukaya category of certain quasi-projective Calabi–Yau threefold.
The numerical Grothendieck group N (D) is generated by two spherical objects S1, S2, each
associated with a vertex in the A2-quiver. The spherical objects satisfy:
• Hom∗D(S1, S1) = Hom∗D(S2, S2) = C⊕C[−3].
• Hom∗D(S1, S2) = C[−1], Hom∗D(S2, S1) = C[−2].
Let σ = (Z ,P) be a stability condition on D with z1 := Z(S1) and z2 := Z(S2). Then its
categorical volume is
vol(σ) = |z1z2 − z2z1| = 2|Im(z1z2)|,
which vanishes if z1z2 ∈ R. This can happen if z1 and z2 are of the same phase, i.e., the stability
condition σ sits on a wall in Stab(D).
3.3 Elliptic curves case
In this section, we give an affirmative answer to Question 3.1.4 in the case of elliptic
curves. Let D = DbCoh(E) be the derived category of an elliptic curve E. In this case,
the ˜GL+(2; R)-action on the space of Bridgeland stability conditions Stab(D) is free and
transitive [14, Theorem 9.1]. Hence,
Stab(D) ∼= ˜GL+(2; R) ∼= C×H,
and the double quotient
Aut(D)\Stab(D)/C ∼= PSL(2, Z)\H
100
is indeed the Kähler moduli space of elliptic curve. Thus we should take Stab∗(D) = Stab(D)
in Question 3.1.4 to be the whole space of stability conditions.
Theorem 3.3.1. Let D = DbCoh(E) be the derived category of an elliptic curve E. Then
sys(σ)2 ≤ 1√3· vol(σ)
holds for any σ ∈ Stab(D).
One can consider this inequality as the mirror of Loewner’s torus systolic inequality in
the introduction.
Proof. Firstly, by Lemma 3.2.3 and Lemma 3.2.8, one notices that the ratio
sys(σ)2
vol(σ)
is invariant under the free C-action on the space of stability conditions. Hence we only need
to compute the ratio on the quotient space Stab(D)/C ∼= H.
The quotient space Stab(D)/C ∼= H can be parametrized by the normalized stability
conditions as follows. Let τ = β + iω ∈ H where β ∈ R and ω > 0. The associated
normalized stability condition στ is given by:
• Central charge: Zτ(F) = −deg(F) + τ · rk(F).
• For 0 < φ ≤ 1, the set of (semi)stable objects Pτ(φ) consists of the slope-(semi)stable
coherent sheaves whose central charge lies in the ray R>0 · eiπφ.
• For other φ ∈ R, define Pτ(φ) by the property Pτ(φ + 1) = Pτ(φ)[1].
Note that there is no wall-crossing phenomenon in the elliptic curve case, i.e., for any
Bridgeland stability condition on DbCoh(E), the set of all Bridgeland (semi)stable objects is
the same as the set of all slope-(semi)stable coherent sheaves up to shiftings. This makes the
computation of categorical systole easier.
To compute the systole of στ, by Lemma 3.2.3 (1), we need to know the central charges
of all the stable objects of στ, which are in this case the slope-stable coherent sheaves. Recall
101
that if F is a slope-stable coherent sheaf on E, then it is either a vector bundle or a torsion
sheaf. The slope-stable vector bundles on an elliptic curve E are well-understood, see for
instance [5, 81].
• Let F be an indecomposable vector bundle of rank r and degree d on an elliptic curve
E. Then F is slope-stable if and only if d and r are relatively prime.
• Fix a point x ∈ E. For every rational number µ = dr , where r > 0 and (d, r) = 1, there
exists a unique slope-stable vector bundle Vµ of rank r and det(Vµ) ∼= OE(dx).
Hence the categorical systole is
sys(στ) = min(d,r)=1
r>0
1, | − d + τr| = min(d,r)∈Z2
(d,r) 6=(0,0)
| − d + τr| = λ1(Lτ),
where λ1(Lτ) denotes the least length of a nonzero element in the lattice Lτ = 〈1, τ〉.
On the other hand, the categorical volume of στ has been computed in Example 2.3.5,
which equals to 2ω. Thus
supτ∈H
sys(στ)2
vol(στ)=
12· sup
τ∈H
λ1(Lλ)2
ω.
Note that ω is the area of the parallelogram spanned by 1 and τ. Hence the quantity
supτ∈H λ1(Lλ)2/ω is the so-called Hermite constant γ2 of lattices in R2. It is classically known
that the Hermite constant γ2 = 2√3
(see for instance [19]). This concludes the proof.
3.4 K3 surfaces case
In this section, we give an affirmative answer to Question 3.1.4 for K3 surfaces of Picard
rank one. This is the main result of the present chapter. We start with recalling some
standard notations.
Let X be a smooth complex projective K3 surface and D = DbCoh(X) be its derived
category. Sending an object E ∈ D to its Mukai vector v(E) = ch(E)√
td(X) identifies the
102
numerical Grothendieck group of D with the lattice
This again contradicts with the assumption that limn→∞1n log Cn = λ since λ2 < λ.
This concludes the proof of the lemma.
4.4 Counterexample of Kikuta-Takahashi
Using Theorem 4.3.1, we can now construct counterexamples of Conjecture 4.1.3.
Proposition 4.4.1. For any even integer d ≥ 4, let X be a Calabi–Yau hypersurface in CPd+1 of
degree (d + 2) and Φ = TO (−⊗O(−1)). Then
h0(Φ) > 0 = log ρ(ΦH∗).
In particular, Conjecture 4.1.3 fails in this case.
120
Proof. By Theorem 4.3.1, we only need to show that the spectral radius ρ(ΦH∗) equals to 1.
Consider another autoequivalence Φ′ := TO (−⊗O(1)) on Db(X). By [7, Proposition
5.8], there is a commutative diagram
Db(X)Φ′ //
Ψ
Db(X)
Ψ
HMFgr(W)τ // HMFgr(W).
Here W is the defining polynomial of X, HMFgr(W) is the associated graded matrix factoriza-
tion category, Ψ is an equivalence introduced by Orlov [78], and τ is the grade shift functor
on HMFgr(W) which satisfies τd+2 = [2]. Hence we have (Φ′)d+2 = [2] and (Φ′)d+2H∗ = idH∗ .
On the other hand, (TO)H∗ is an involution on H∗(X; C) when X is an even dimensional
strict Calabi–Yau manifold ([51, Corollary 8.13]). Thus (TO)H∗ = (TO)−1H∗ . Hence we also
have Φd+2H∗ = idH∗ , which implies that ρ(ΦH∗) = 1.
Remark 4.4.2. The autoequivalence Φ′ that we considered in the proof is the one that corresponds to
the monodromy around the Gepner point (Zd+2-orbifold point) in the Kähler moduli of X.
Remark 4.4.3 (Ouchi). The functor Φ = TO (−⊗O(−1)) does not produce counterexamples
of Conjecture 4.1.3 if X is an odd dimensional Calabi–Yau manifold. When X is of odd dimension,
Lemma 4.3.2 implies
h0(Φ) = limn→∞
1n
log χ(G, Φn(G′)) ≤ log ρ([Φ]).
On the other hand, we have h0(Φ) ≥ log ρ([Φ]) by Kikuta-Shiraishi-Takahashi [62, Theorem 2.13].
4.5 Categorical entropy of P-twists
Despite the fact that we have disproved Conjecture 4.1.3, it still is an interesting problem
to find a characterization of autoequivalences satisfying the conjecture. The conjecture
holds true for autoequivalences on smooth projective curves [61], abelian surfaces [107], and
smooth projective varieties with ample (anti)-canonical bundle [63]. Ouchi [80] also showed
121
that spherical twists satisfy the conjecture.
By slightly modifying Ouchi’s proof, we show that Conjecture 4.1.3 also holds for
P-twists. One can consider P-twists as the categorical analogue of Dehn twists along
Lagrangian complex projective space.
We start with an easy lemma on the complexity function.
Lemma 4.5.1. 1. δt(G, E⊕ F) ≥ δt(G, E).
2. For a distinguished triangle A→ B→ C → A[1], we have
δt(G, B) ≤ δt(G, A) + δt(G, C).
We recall the notion of Pd-objects and Pd-twists introduced by Huybrechts and Thomas
[52]. They are the categorical analogue of the symplectomorphisms associated to a La-
grangian complex projective space [85].
Definition 4.5.2 ([52]). An object E ∈ Db(X) is called a Pd-object if E ⊗ ωX ∼= E and
Hom∗(E , E) is isomorphic as a graded ring to H∗(Pd, C).
For a Pd-object E , a generator h ∈ Hom2(E , E) can be viewed as a morphism h : E [−2]→
E . The image of h under the natural isomorphism Hom2(E , E) ∼= Hom2(E∨, E∨) will be
denoted h∨.
Definition 4.5.3 ([52]). The Pd-twist PE with respect to a Pd-object is an autoequivalence on
Db(X) given by the double cone construction
F 7→ PE (F ) := Cone(
Cone(Hom∗−2(E ,F )⊗ E → Hom∗(E ,F )⊗ E)→ F)
,
where the first right arrow is given by h∨ · id− id · h.
The following fact is crucial for computing the categorical entropy of P-twists in the
next section.
Lemma 4.5.4 ([52]). Let E ∈ Db(X) be a Pd-object. Then
1. PE (E) ∼= E [−2d].
122
2. PE (F ) ∼= F for any F ∈ E⊥ = F : Hom∗(E ,F ) = 0.
Theorem 4.5.5. Let X be a smooth projective variety of dimension 2d over C, and let PE be the
Pd-twist of a Pd-object E ∈ Db(X). Then for t ≤ 0, we have ht(PE ) = −2dt. In particular,
Conjecture 4.1.3 holds for Pd-twists:
h0(PE ) = 0 = log ρ((PE )H∗).
Assume that E⊥ 6= φ, then we have ht(PE ) = 0 for t > 0.
Proof. Fix a split generator G ∈ Db(X), and let
A := Cone(Hom∗−2(E , G)⊗ E → Hom∗(E , G)⊗ E).
By the definition of Pd-twist, we have a distinguished triangle
G → PE (G)→ A[1]→ G[1].
And by Lemma 4.5.4, we have PE (A) = A[−2d].
By applying Pn−1E to the distinguished triangle, we have
δt(G, PnE (G)) ≤ δt(G, Pn−1
E (G)) + δt(G, A[−2d(n− 1) + 1])
= δt(G, Pn−1E (G)) + δt(G, A)e(−2d(n−1)+1)t
≤ · · · (do this inductively)
≤ 1 + δt(G, A)n−1
∑k=0
e(−2dk+1)t
For t ≤ 0, we have δt(G, PnE (G)) ≤ 1 + δt(G, A) · ne(−2d(n−1)+1)t. Hence
ht(PE ) = limn→∞
1n
log δt(G, PnE (G)) ≤ −2dt.
On the other hand, since G⊕ E is also a split generator of D, we can apply Lemma 4.5.1
123
and 4.5.4 to get
ht(PE ) = limn→∞
1n
log δt(G, PnE (G⊕ E))
≥ limn→∞
1n
log δt(G, PnE (E))
= limn→∞
1n
log δt(G, E [−2nd])
= limn→∞
1n
log δt(G, E)e−2ndt
= −2dt.
Hence we have ht(PE ) = −2dt for t ≤ 0.
Note that the induced Fourier–Mukai type action (PE )H∗ is identity on the cohomology
[52]. Thus Conjecture 4.1.3 holds for the Pd-twist PE :
h0(PE ) = 0 = log ρ((PE )H∗).
For t > 0, we have δt(G, PnE (G)) ≤ 1 + δt(G, A)(et + n− 1). Hence ht(PE ) ≤ 0. Assume
that E⊥ 6= φ and take B ∈ E⊥, then we can apply the same trick on the split generator
G⊕ B:
ht(PE ) = limn→∞
1n
log δt(G, PnE (G⊕ B))
≥ limn→∞
1n
log δt(G, PnE (B))
= limn→∞
1n
log δt(G, B)
= 0.
This concludes the proof of the theorem.
Remark 4.5.6. When dim(X) = 2, a P1-object E ∈ Db(X) is also a spherical object (S2 ∼= P1).
For t ≤ 0, we have ht(PE ) = −2t. On the other hand, ht(TE ) = −t for the spherical twist TE [80].
This matches with the fact that T2E∼= PE by Huybrechts and Thomas [52].
124
References
[1] M. Abouzaid. A topological model for the Fukaya categories of plumbings. J.Differential Geom., 87(1):1–80, 2011.
[2] M. Abouzaid, D. Auroux, and L. Katzarkov. Lagrangian fibrations on blowups of toricvarieties and mirror symmetry for hypersurfaces. Publ. Math. Inst. Hautes Études Sci.,123:199–282, 2016.
[3] H. Akrout. Singularités topologiques des systoles généralisées. Topology, 42(2):291–308,2003.
[4] K. Altmann. The versal deformation of an isolated toric Gorenstein singularity. Invent.Math., 128(3):443–479, 1997.
[5] M. F. Atiyah. Vector bundles over an elliptic curve. Proc. London Math. Soc., 3(7):414–452, 1957.
[6] D. Auroux. Mirror symmetry and T-duality in the complement of an anticanonicaldivisor. J. Gökova Geom. Topol. GGT, 1:51–91, 2007.
[7] M. Ballard, D. Favero, and L. Katzarkov. Orlov spectra: bounds and gaps. Invent.Math., 189(2):359–430, 2012.
[8] A. Bayer and T. Bridgeland. Derived automorphism groups of K3 surfaces of Picardrank 1. Duke Math. J., 166(1):75–124, 2017.
[9] A. Bayer and E. Macrì. Projectivity and birational geometry of Bridgeland modulispaces. J. Amer. Math. Soc., 27(3):707–752, 2014.
[10] A. Bayer, E. Macrì, and P. Stellari. The space of stability conditions on abelianthreefolds, and on some Calabi–Yau threefolds. Invent. Math., 206(3):869–933, 2016.
[11] P. Biran and O. Cornea. Quantum Structures for Lagrangian Submanifolds.arXiv:0708.4221, 2007.
[12] A. Bondal and D. Orlov. Reconstruction of a variety from the derived category andgroups of autoequivalences. Compositio Math., 125(3):327–344, 2001.
[13] T. Bridgeland. Flops and derived categories. Invent. Math., 147(3):613–632, 2002.
[14] T. Bridgeland. Stability conditions on triangulated categories. Ann. of Math., 166(2):317–345, 2007.
125
[15] T. Bridgeland. Stability conditions on K3 surfaces. Duke Math. J., 141(2):241–291, 2008.
[16] T. Bridgeland. Spaces of stability conditions. In Algebraic Geometry: Seattle 2005,volume 80 of Proceedings of Symposia in Pure Mathematics, pages 1–21. AmericanMathematical Society, 2009.
[17] T. Bridgeland and I. Smith. Quadratic differentials as stability conditions. Publ. Math.Inst. Hautes Études Sci., 121:155–278, 2015.
[18] P. Candelas, X. C. de la Ossa, P. S. Green, and L. Parkes. A pair of Calabi–Yaumanifolds as an exactly solvable superconformal theory. Nuclear Phys. B, 359(1):21–74,1991.
[19] J. W. S. Cassels. An introduction to the geometry of numbers. Classics in Mathematics.Springer-Verlag Berlin Heidelberg, 1997.
[20] K. Chan, S.-C. Lau, and N. C. Leung. SYZ mirror symmetry for toric Calabi-Yaumanifolds. J. Differential Geom., 90(2):177–250, 2012.
[21] K. Chan, D. Pomerleano, and K. Ueda. Lagrangian torus fibrations and homologicalmirror symmetry for the conifold. Commun. Math. Phys., 341(1):135–178, 2016.
[22] J.-C. Chen. Flops and equivalences of derived categories for threefolds with onlyterminal Gorenstein singularities. J. Differential Geom., 61(2):227–261, 2002.
[23] C.-H. Cho, H. Hong, and S.-C. Lau. Noncommutative homological mirror functor.arXiv:1512.07128, 2015.
[24] C.-H. Cho, H. Hong, and S.-C. Lau. Localized mirror functor for Lagrangian immer-sions, and homological mirror symmetry for P1
a,b,c. J. Differential Geom., 106(1):45–126,2017.
[25] C.-H. Cho, H. Hong, and S.-C. Lau. Localized mirror functor constructed from aLagrangian torus. J. Geom. Phys., 136:284–320, 2019.
[26] A. Caldararu. The Mukai pairing II: the Hochschild–Kostant–Rosenberg isomorphism.Adv. Math., 194:34–66, 2005.
[27] M. Van den Bergh. Non-commutative crepant resolutions. In The legacy of Niels HenrikAbel, pages 749–770. Springer, Berlin, 2004.
[28] M. Van den Bergh. Three-dimensional flops and noncommutative rings. Duke Math. J.,122(3):423–455, 2004.
[29] G. Dimitrov, F. Haiden, L. Katzarkov, and M. Kontsevich. Dynamical systems andcategories. Contemp. Math., 621:133–170, 2014.
[30] I. V. Dolgachev. Mirror symmetry for lattice polarized K3 surfaces. J. Math. Sci.,81(3):2599–2630, 1996.
[31] M. R. Douglas, R. Reinbacher, and S.-T. Yau. Branes, Bundles and Attractors: Bogo-molov and Beyond. arXiv:math/0604597, 2006.
126
[32] Y.-W. Fan. Entropy of an autoequivalence on Calabi–Yau manifolds. Math. Res. Lett.,25(2):509–519, 2018.
[33] Y.-W. Fan. On entropy of P-twists. arXiv:1801.10485, 2018.
[34] Y.-W. Fan. Systoles, Special Lagrangians, and Bridgeland stability conditions.arXiv:1803.09684, 2018.
[35] Y.-W. Fan, H. Hong, S.-C. Lau, and S.-T. Yau. Mirror of Atiyah flop in symplecticgeometry and stability conditions. To appear in Adv. Theor. Math. Phys., 22(5), 2018.
[36] Y.-W. Fan, A. Kanazawa, and S.-T. Yau. Weil–Petersson geometry on the space ofBridgeland stability conditions. arXiv:1708.02161. Accepted by Commun. Anal. Geom..,2017.
[37] R. Friedman. Simultaneous resolution of threefold double points. Math. Ann.,274(4):671–689, 1986.
[38] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono. Lagrangian intersection Floer theory: anomalyand obstruction. Part I and II, volume 46 of AMS/IP Studies in Advanced Mathematics.American Mathematical Society, 2009.
[39] D. Gaiotto, G. W. Moore, and A. Neitzke. Wall-crossing, Hitchin Systems, and theWKB Approximation. Adv. Math., 234:239–403, 2013.
[40] S. Ganatra, T. Perutz, and N. Sheridan. Mirror symmetry: from categories to curvecounts. arXiv:1510.03839, 2015.
[41] V. Ginzburg. Calabi–Yau algebras. arXiv:math/0612139, 2006.
[42] E. Goldstein. Calibrated fibrations on noncompact manifolds via group actions. DukeMath. J., 110(2):309–343, 2001.
[43] V. A. Gritsenko and V. V. Nikulin. Siegel automorphic form corrections of someLorentzian Kac–Moody Lie algebras. Amer. J. Math., 119(1):181–224, 1997.
[44] M. L. Gromov. Entropy, homology and semialgebraic geometry. Astérisque, 145-146(5):225–240, 1987.
[45] M. L. Gromov. On the entropy of holomorphic maps. Enseign. Math., 49(3-4):217–235,2003.
[46] M. Gross. Examples of special Lagrangian fibrations. In Symplectic geometry and mirrorsymmetry (Seoul, 2000), pages 81–109. World Sci. Publ., River Edge, NJ, 2001.
[47] M. Gross. Toric degenerations and Batyrev–Borisov duality. Math. Ann., 333(3):645–688,2005.
[48] F. Haiden, L. Katzarkov, and M. Kontsevich. Flat surfaces and stability structures.Publ. Math. Inst. Hautes Études Sci., 126:247–318, 2017.
127
[49] J. Halverson, H Jockers, J. M. Lapan, and D. R. Morrison. Perturbative Corrections toKahler Moduli Spaces. Commun. Math. Phys., 333:1563–1584, 2015.
[50] S. Hosono, B. H. Lian, K. Oguiso, and S.-T. Yau. c = 2 Rational Toroidal ConformalField Theories via the Gauss Product. Commun. Math. Phys., 241(2-3):245–286, 2003.
[51] D. Huybrechts. Fourier–Mukai transforms in algebraic geometry. Oxford MathematicalMonographs. Oxford University Press, 2006.
[52] D. Huybrechts and R. Thomas. P-objects and autoequivalences of derived categories.Math. Res. Lett., 13(1):87–98, 2006.
[53] H. Iritani. An integral structure in quantum cohomology and mirror symmetry fortoric orbifolds. Adv. Math., 222:1016–1079, 2009.
[54] H. Jockers, V. Kumar, J. M. Lapan, D. R. Morrison, and M. Romo. Two-sphere partitionfunctions and Gromov–Witten Invariants. Commun. Math. Phys., 325(3):1139–1170,2014.
[55] D. Joyce. Conjectures on Bridgeland stability for Fukaya categories of Calabi–Yaumanifolds, special Lagrangians, and Lagrangian mean curvature flow. EMS Surv.Math. Sci., 2(1):1–62, 2015.
[56] A. Kanazawa and S.-C. Lau. Geometric transitions and SYZ mirror symmetry.arXiv:1503.03829, 2015.
[57] A. Kanazawa and S.-C. Lau. Local Calabi–Yau manifolds of affine type A and openYau–Zaslow formula via SYZ mirror symmetry. arXiv:1605.00342, 2016.
[58] M. G. Katz. Systolic geometry and topology, volume 137 of Mathematical Surveys andMonographs. American Mathematical Society, 2007.
[59] L. Katzarkov, M. Kontsevich, and T. Pantev. Hodge theoretic aspects of mirrorsymmetry. Proc. Sympos. Pure Math., 78:87–174, 2008.
[60] B. Keller. Deformed Calabi–Yau completions. J. Reine Angew. Math., 654:125–180, 2011.
[61] K. Kikuta. On entropy for autoequivalences of the derived category of curves. Adv.Math., 308:699–712, 2017.
[62] K. Kikuta, Y. Shiraishi, and A. Takahashi. A note on entropy of autoequivalences:lower bound and the case of orbifold projective lines. Nagoya Math. J., pages 1–18,2018.
[63] K. Kikuta and A. Takahashi. On the categorical entropy and the topological entropy.Int. Math. Res. Not., 23(2):457–469, 2019.
[64] J. Kollár. Flops. Nagoya Math. J., 113:15–36, 1989.
[65] M. Kontsevich. Homological algebra of mirror symmetry. In Proceedings of theInternational Congress of Mathematicians (Zürich, 1994), pages 120–139. Birkhäuser, Basel,1995.
128
[66] M. Kontsevich and Y. Soibelman. Stability structures, motivic Donaldson–Thomasinvariants and cluster transformations. arXiv:0811.2435, 2008.
[67] M. Kontsevich and Y. Soibelman. Wall-crossing structures in Donaldson–Thomasinvariants, integrable systems and mirror symmetry. In Homological mirror symmetryand tropical geometry, volume 15 of Lect. Notes Unione Mat. Ital., pages 197–308. Springer,Cham, 2014.
[68] S.-C. Lau. Open Gromov–Witten invariants and SYZ under local conifold transitions.J. Lond. Math. Soc., 90(2):413–435, 2014.
[69] A. Maciocia and D. Piyaratne. Fourier–Mukai transforms and Bridgeland stabilityconditions on abelian threefolds. Algebr. Geom., 2(3):270–297, 2015.
[70] E. Macrì. Stability conditions on curves. Math. Res. Lett., 14(4):657–672, 2007.
[71] E. Macrì and P. Stellari. Infinitesimal derived Torelli theorem for K3 surfaces. Int.Math. Res. Not., (17):3190–3220, 2009.
[72] G. W. Moore. Arithmetic and Attractors. arXiv:hep-th/9807087, 1998.
[73] S. Mukai. On the moduli space of bundles on K3 surfaces. In Vector bundles on algebraicvarieties (Bombay, 1984), volume 11 of Tata Inst. Fund. Res. Stud. Math., pages 341–413.Tata Inst. Fund. Res., Bombay, 1987.
[74] K. Nagao and H. Nakajima. Counting invariant of perverse coherent sheaves and itswall-crossing. Int. Math. Res. Not., (17):3885–3938, 2011.
[75] R. Casta no Bernard and D. Matessi. Lagrangian 3-torus fibrations. J. Differential Geom.,81(3):483–573, 2009.
[76] R. Casta no Bernard and D. Matessi. Conifold transitions via affine geometry andmirror symmetry. Geom. Topol., 18(3):1769–1863, 2014.
[77] D. Orlov. Equivalences of derived categories and K3 surfaces. J. Math. Sci., 84(5):1361–1381, 1997.
[78] D. Orlov. Derived categories of coherent sheaves and triangulated categories ofsingularities. Progr. Math., 270:503–531, 2009.
[79] D. Orlov. Remarks on generators and dimensions of triangulated categories. Mosc.Math. J., 9(1):153–159, 2009.
[80] G. Ouchi. On entropy of spherical twists. arXiv:1705.01001, 2017.
[81] A. Polishchuk. Abelian varieties, theta functions and the Fourier transform, volume 153 ofCambridge Tracts in Mathematics. Cambridge University Press, 2003.
[82] A. Polishchuk and E. Zaslow. Categorical mirror symmetry: the elliptic curve. Adv.Theor. Math. Phys., 2(2):443–470, 1998.
129
[83] W.-D. Ruan. Lagrangian torus fibrations and mirror symmetry of Calabi–Yau mani-folds. In Symplectic geometry and mirror symmetry (Seoul, 2000), pages 385–427. WorldSci. Publ., River Edge, NJ, 2001.
[84] G. Schumacher. On the geometry of moduli spaces. Manuscripta Math., 50:229–267,1985.
[85] P. Seidel. Graded Lagrangian submanifolds. Bull. Soc. Math. France, 128(1):103–149,2000.
[86] P. Seidel. Fukaya Categories and Picard—Lefschetz Theory, volume 10 of EMS ZürichLectures in Advanced Mathematics. European Mathematical Society, 2008.
[87] P. Seidel. Homological mirror symmetry for the genus two curve. J. Algebraic Geom.,20(4):727–769, 2011.
[88] P. Seidel. Homological mirror symmetry for the quartic surface, volume 236 of Memoirs ofthe American Mathematical Society. American Mathematical Society, 2015.
[89] P. Seidel and R. Thomas. Braid group actions on derived categories of coherentsheaves. Duke Math. J., 108(1):37–108, 2001.
[90] N. Sheridan. Homological mirror symmetry for Calabi–Yau hypersurfaces in projectivespace. Invent. Math., 199(1):1–186, 2015.
[91] I. Smith. Quiver algebras as Fukaya categories. Geom. Topol., 19(5):2557–2617, 2015.
[92] I. Smith, R. Thomas, and S.-T. Yau. Symplectic conifold transitions. J. Differential Geom.,62(2):209–242, 2002.
[93] B. Szendröi. Non-commutative Donaldson–Thomas invariants and the conifold. Geom.Topol., 12(2):1171–1202, 2008.
[94] R. Thomas. Stability conditions and the braid group. Comm. Anal. Geom., 14(1):135–161,2006.
[95] W. Thurston. On the geometry and dynamics of diffeomorphisms of surfaces. Bull.Amer. Math. Soc., 19(2):417–431, 1988.
[96] G. Tian. Smoothness of the universal deformation space of compact Calabi–Yaumanifolds and its Petersson–Weil metric. Adv. Ser. Math. Phys., 1:629–646, 1987.
[97] G. Tian. Smoothing 3-folds with trivial canonical bundle and ordinary double points.In Essays on mirror manifolds, pages 458–479. Int. Press, Hong Kong, 1992.
[98] Y. Toda. Moduli stacks and invariants of semistable objects on K3 surfaces. Adv. Math.,217(6):2736–2781, 2008.
[99] Y. Toda. Stability conditions and crepant small resolutions. Trans. Amer. Math. Soc.,360(11):6149–6178, 2008.
130
[100] Y. Toda. Gepner type stability conditions on graded matrix factorizations. Algebr.Geom., 1(5):613–665, 2014.
[101] T. Trenner and P. M. H. Wilson. Asymptotic curvature of moduli spaces for Calabi–Yauthreefolds. J. Geom. Anal., 21(2):409–428, 2011.
[102] C.-L. Wang. On the incompleteness of the Weil–Petersson metric along degenerationsof Calabi–Yau manifolds. Math. Res. Lett., 4(1):157–171, 1997.
[103] C.-L. Wang. Curvature properties of the Calabi–Yau moduli. Doc. Math., 8:577–590,2003.
[104] P. M. H. Wilson. The Kähler cone on Calabi–Yau threefolds. Invent. Math., 107(3):561–583, 1992.
[105] P. M. H. Wilson. Sectional curvatures of Kähler moduli. Math. Ann., 330(4):631–664,2004.
[106] Y. Yomdin. Volume growth and entropy. Israel J. Math., 57(3):285–300, 1987.
[107] K. Yoshioka. Categorical entropy for Fourier–Mukai transforms on generic abeliansurfaces. arXiv:1701.04009, 2017.