Topological Methods in Symplectic Geometry
A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY
Chung-I Ho
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy
Tian-Jun Li
July, 2011 c Chung-I Ho 2011 ALL RIGHTS RESERVED Acknowledgements
For all of the people that make this PhD thesis possible, I would like to first thank my advisor Tian-Jun Li from University of Minnesota for his guidance and tremendous support of my work. Dr. Li is an excellent scholar and always very passionate about research. We sometimes spent hours every day for discussion. If not for his enthusiasts, patience, and dedication of his precious time, this work can never be done. Therefore I would like to deliver my greatest thankfulness to Dr. Li, for his help and all those insightful discussions. Also, the faculty members from University of Minnesota have been given me a lot of help. First of all, I would like to thank my committee members: Dr. Alexander Voronov, Dr. Scot Adams and Dr. Anar Akhmedov. They have been giving me a lot of useful comments for my thesis since two years ago during my preliminary oral defense. Because of their expertise in the related fields, I am able to view things in different perspectives hence make this work more complete and solid. Also, I would like to mention Mr. John Guest and Dr. Naichung Conan Leung for not only provide me very useful research discussions, but also kind enough to serve us my recommendation references for my academic job search. This journey cannot be possibly as smooth without their generous help. The support of my friends from the math department is also a key element in this long process. My group members, Weiyi Zhang, Weiwei Wu, Haojie Chen, Xiaolian Nien and Josef G. Dorfmeister are all brilliant mathematicians and I am very lucky to be in the same group with them to exchange research ideas from time to time. Also, my other mathematician friends, Nai-Chia Chen, Tsai-Yin Lin, Hsin-Yuan Huang, Fanbin Bu, Xiaoqin He, Campbell Patrick, Joel Gomez, Joseph Dickinson, Xiaoqin Guo, Jia Huang and Chin-Ann Yang also give me a lot of help not only in mathematics, but also
i their warm friendship to help me get through the winters in Minnesota. I wish all of those now and future mathematicians good luck in their career and personal life. Finally, I would especially want to thank the whole mathematic department in Uni- versity of Minnesota, for giving me financial support and providing a friendly environ- ment for their students. The staff members Carla Claussen, Bonny Fleming Stephanie Lawson, Diane Trager, and Jan Minette are always so helpful. Being an international student, I receive a lot of their help to ease my tension through a lot of process and also clear a lot of confusions due to their kind help.
ii Abstract
In the first part, we study existence of the Lefschetz decomposition for de Rham co- homology, which is characterized by the strong Lefschetz property. A new spectral sequence for symplectic manifolds is also defined. In the second part, we show that the Lagrangian Luttinger surgery preserves the Kodaira dimension. Some constraints on Lagrangian tori in symplectic four manifolds with non-positive Kodaira dimension are also derived.
iii Contents
Acknowledgements i
Abstract iii
1 Introduction 1
2 Primitive cohomology 8 2.1 Linear operators ...... 8 2.2 Lefschetz decomposition and primitive forms ...... 16 Λ 2.3 ∂+, ∂− and d operator ...... 18 2.4 Primitive cohomologies ...... 22 2.5 Example: Kodaira-Thurston manifold ...... 23
3 Strong Lefschetz property 27 3.1 Symplectic and primitive harmonic forms ...... 28 Λ 3.2 dd -lemma and ∂+∂−-lemma ...... 29
3.3 Strong Lefschetz property, primitive harmonic representatives and ∂+∂−- lemma ...... 30 3.4 Lefschetz decomposition for de-Rham cohomology ...... 33 3.5 Symplectic cohomologies ...... 34
4 Symplectic-de Rham spectral sequence 39 4.1 Spectral sequences and degeneration ...... 40 4.2 d0d00-lemma ...... 46 4.3 Symplectic-de Rham spectral sequence ...... 47
iv 4.4 Examples ...... 48 4.4.1 Four dimensional nilpotent Lie algebra ...... 49 4.4.2 Six-dimensional nilmanifolds ...... 49
5 Luttinger surgery and Kodaira dimension 55 5.1 Symplectic Kodaira dimension ...... 55 5.1.1 κ = −∞ ...... 56 5.1.2 κ =0 ...... 57 5.1.3 κ =1 ...... 58 5.1.4 κ =2 ...... 60 5.2 Luttinger surgery ...... 61 5.2.1 Construction ...... 61 5.2.2 Lagrangian fibrations ...... 64 5.3 Preservation of Kodaira dimension ...... 66 5.3.1 Minimality ...... 66 5.3.2 Kodaira dimension ...... 67 5.4 Manifolds with non-positive κ ...... 70 5.4.1 Torus surgery and homology ...... 70 5.4.2 κ = −∞ ...... 73 5.4.3 Luttinger surgery as a symplectic CY surgery ...... 74 5.4.4 Luttinger surgery in high dimension ...... 75
6 Torus surgery and framings 77 6.1 Topological preferred framing and Lagrangian framing ...... 77
6.1.1 Comparing ker i1 and ker i2 ...... 78
6.1.2 Preferred framings via ker i1 ...... 82 6.1.3 (1, k)-surgeries and topological preferred framings ...... 84 6.1.4 Constraints for Lagrangian framings ...... 86
References 89
v Chapter 1
Introduction
The object of this thesis is the topological properties of symplectic manifolds, especially in dimension four. In the first part, we will analyze the relation of primitive cohomologies and the strong Lefschetz property for symplectic manifolds. Let (M, ω) be a symplectic manifold of dimension 2n. The symplectic structure ω induces the Lefschetz operator
L :Ωk(M) → Ωk+2(M), α 7→ ω ∧ α
It is well known that L together with its dual Λ and a projection H give a sl(2) repre- sentation on Ω∗(M):
[Λ,L] = H, [H, Λ] = 2Λ, [H,L] = −2L
If Ik is the direct sum of irreducible representation of dimension k + 1, Ω∗(M) can be decomposed to the vector subspace
Lp,q = Ω2p+q(M) ∩ In−q
For each α ∈ Ωk(M), there is a unique decomposition
X 1 α = LrBα ,Bα ∈ L0,k−2r. r! k−2r k−2r r≥max{k−n,0}
This decomposition is called the Lefschetz decomposition of differential forms. The el- ements of highest weight for the sl(2) representation are called primitive forms. The
1 2 Lefschetz decomposition shows that each differential form can be generated by primi- tive forms and the operator L. An interesting fact about the decomposition is that it decomposes the exterior derivative to
d : Lp,q → Lp,q+1 + Lp+1,q−1 and there are differential operators
p,q p,q+1 p,q p,q−1 ∂+ : L → L , ∂− : L → L which commute with L and satisfy d = ∂+ + L∂−. If we restrict our attention to the k 0,k ∗ primitive forms P = L , the complexes (P , ∂±) define two cohomologies
k k k ker ∂+ ∩ P k ker ∂− ∩ P PH+(M) = k , PH−(M) = k Im∂+ ∩ P Im∂− ∩ P Question 1.0.1. Can each de Rham cohomology class be decomposed as direct sum of primitive cohomology classes and their wedge product with ω?
Unfortunately, this is not the case for any symplectic manifold. We show that these question is characterized by the strong Lefschetz property, which states that the natural homomorphism Lk : Hn−k(M) → Hn+k(M), L([α]) = [α ∧ ωk] is an isomorphism for 0 ≤ k ≤ n.
Theorem 1.0.2. There is a canonical isomorphism
X r k−2r k L PH+ (M) → H (M) r≥max{k−n,0} for any k if and only if M satisfies the strong Lefschetz property.
The actions of the differential d and the Koszul-Brylinski operator dΛ = [d, Λ] give other descriptions for the strong Lefschetz property. A smooth form α is called symplec- tic harmonic if dα = dΛα = 0. It is proved by Cavalcanti [13], Mathieu [35], Merkulov [39] and Yan [54] that the following conditions are equivalent:
1. M satisfies the strong Lefschetz property. 3 2. Imd ∩ ker dΛ =ImdΛ ∩ ker d = ImddΛ.
3. Each cohomology class of H∗(M) has a symplectic harmonic representative.
The second condition is called the ddΛ-lemma. There are similar concepts for ddΛ- lemma and symplectic harmonic forms in primitive forms, which we call ∂+∂−-lemma and primitive harmonic forms respectively. We show that
Λ Proposition 1.0.3. 1. The dd -lemma holds for M if and only if M has the ∂+∂−- lemma.
2. α is symplectic harmonic if and only if each primitive term of its Lefschetz de- composition is primitive harmonic.
Hence, the strong Lefschetz property has equivalent conditions in primitive forms.
Theorem 1.0.4. The following properties are equivalent for a symplectic manifold M:
1. M satisfies the strong Lefschetz property.
2. The ∂+∂−-lemma holds for M.
3. Each cohomology class of H∗(M) has a primitive harmonic representative.
We use more elementary techniques to prove Theorem 1.0.4 directly.
Question 1.0.5. How far is M from being a strong Lefschetz manifold if it is not?
When the strong Lefschetz property fails, we can estimate the discrepancy between the dimension of Hn−k(M) and the rank of Lk. It turns out that rank(Lk) is bounded by the dimension of symplectic cohomologies defined by Tseng and Yau in [47]. They consider the operators d and dΛ = [d, Λ] together to define cohomologies
ker(d + dΛ) ∩ Ωk(M) Hk (M) = d+dΛ ImddΛ ∩ Ωk(M)
ker ddΛ ∩ Ωk(M) Hk (M) = ddΛ (Imd + ImdΛ) ∩ Ωk(M) Λ k k ker(d + d ) ∩ Ω (M) H Λ (M) = d∩d dΩ˜ k−1(M) + dΛΩ˜ k+1(M) 4 where Ω˜ ∗(M) is the space of ddΛ-closed forms. Using the corresponding operators for primitive forms, there is a cohomology for primitive forms analogous to each of the above cohomologies. It turns out that these cohomologies have the Lefschetz decomposition.
Theorem 1.0.6. [47] The cohomology Hd+dΛ has the Lefschetz decomposition:
k r k−2r Hd+dΛ (M) = ⊕rL PHd+dΛ (M)
Similarly for HddΛ and Hd∩dΛ .
k k Tseng and Yau also show that Hd+dΛ (M) is canonically isomorphic to H (M) for any k if and only if M satisfies the strong Lefschetz property. We prove that
n−k k n−k Theorem 1.0.7. dimHd∩dΛ (M) ≤rankL ≤dimH (M). In the low dimensional case, the bound is sharp:
Proposition 1.0.8. Assume M is a symplectic 4-manifold. Then
n−k k dimHd∩dΛ (M) = rankL .
We notice that the ddΛ-lemma strongly relates to the degeneration of spectral se- quences. In [12], Brylinski used the canonical complex to construct a spectral sequence for manifolds with Poisson structure and showed that it degenerates at E1 for symplectic manifolds. Motivated by the Fr¨olicher spectral sequence for complex manifolds, we con- p,q ∗ struct a spectral sequence Er from the double complex (Ω (M), ∂+, L∂−). Moreover, we give a complete description of the degeneration for symplectic 4-manifolds. We use the famous Kodaira-Thurston manifold to show that this spectral sequence is not necessarily degenerating at E1. In the computation, we also raise the issue in the spirit of the Nomizu theorem. It says that the computation of cohomology for nilmani- folds can be reduced to the Chevalley-Eilenberg cohomology for the corresponding nilpo- tent Lie algebra. The coincidence of the new cohomologies for the Kodaira-Thurston manifold and the nilpotent Lie algebra ch3 suggests the validity of the Nomizu type theorem. Furthermore, it is known that the current of Lagrangian submanifolds is primitive. In symplectic 4-manifold, we can use the Luttinger surgery along a Lagrangian torus to construct new symplectic manifolds. Hence we can compare the primitive cohomologies 5 of two symplectic 4-manifolds connected by the Luttinger surgery. In the second part of this thesis, we focus on the discussion of the Luttinger surgery and its applications. Let (X, ω) be a symplectic 4-manifold with a Lagrangian torus L. It was discovered by Luttinger in [34] that there is a family of surgeries along L that produce symplectic 4-manifolds. This family is countable and indexed by the pairs ([γ], k), where [γ] is an 4 isotopy class of simple closed curves on L and k is an integer. When X = R and ω is the standard symplectic form ω0 = dx1 ∧ dy1 + dx2 ∧ dy2, he also applied Gromov’s celebrated work in [23] to show that, for any Lagrangian torus L, all the resulting 4 symplectic manifolds are symplectomorphic to (R , ω0). This does not occur in general; a Luttinger surgery often fails even to preserve homology. As a matter of fact, many new exotic small manifolds are constructed via this surgery. We observe that the Luttinger surgery preserves one basic invariant:
Theorem 1.0.9. The Luttinger surgery preserves the symplectic Kodaira dimension.
The symplectic Kodaira dimension of a symplectic 4-manifold (X, ω) is defined by 2 the products Kω and Kω · [ω], where Kω is the symplectic canonical class; if (X, ω) is minimal, then 2 −∞ K < 0 or Kω · [ω] < 0 ω 2 0 K = 0 and Kω · [ω] = 0 κ(X, ω) = ω 2 1 K = 0 and Kω · [ω] > 0 ω 2 2 Kω > 0 and Kω · [ω] > 0 For a general symplectic 4-manifold, the Kodaira dimension is defined as the Kodaira dimension of any of its minimal models. According to [31], κ(X, ω) is independent of the choice of symplectic form ω and hence is denoted by κ(X). Theorem 1.0.9 is related to a question of Auroux in [6] (see Remark 5.3.4). Further- more, together with the elementary analysis of the homology change, the invariance of κ implies that
Theorem 1.0.10. Let (X, ω) be a symplectic 4-manifold with κ(X) = −∞ and (X,˜ ω˜) be constructed from (X, ω) via a Luttinger surgery. Then (X,˜ ω˜) is symplectomorphic to (X, ω). 6 For minimal symplectic manifolds of Kodaira dimension zero, i.e., symplectic Calabi- Yau surfaces, we conclude that the Luttinger surgery is a symplectic CY surgery. More- over, together with the homology classification of such manifolds in [32], we have
Theorem 1.0.11. Suppose (X, ω) is a symplectic 4-manifold with κ(X) = 0 and χ(X) > 0. If (X,˜ ω˜) is constructed from (X, ω) under a Luttinger surgery, then X and X˜ have the same integral homology type.
In fact, we conjecture that X˜ and X in Theorem 1.0.11 are diffeomorphic to each other. For symplectic CY surfaces with χ = 0, the only known examples are torus bundles over torus. We conjecture that they all can be obtained from T 4 via Lutttinger surgeries (Conjecture 5.4.9). Theorems 1.0.10 and 1.0.11 provide topological constraints, phrased in terms of topo- logical preferred framings (see Definition 6.1.1), on the existence of exotic Lagrangian tori in such manifolds.
Theorem 1.0.12. Let L be a Lagrangian torus in (X, ω). If κ(X) = −∞, or L is null- homologous, κ(X) = 0 and χ(X) > 0, then the Lagrangian framing of L is topological preferred. In particular, the invariant λ(L) in [17] vanishes whenever it is defined.
The organization of this thesis is as follows. In Chapter 2, we first introduce some basic concepts regarding linear operators. It includes the Lefschetz operator and its dual together with the symplectic star operator. Then we introduce the operator ∂+, ∂− and primitive cohomology. Chapter 3 focus on the strong Lefschetz property. We first define the concepts of
∂+∂−-lemma and primitive harmonic forms. Then we use more elementary technique to prove the equivalent relations of the strong Lefschetz property. We also show that the strong Lefschetz property can be characterized by primitive forms and primitive co- homology can realize normal cohomology theories when the strong Lefschetz property holds. In the last section, we introduce symplectic cohomologies and discuss their com- patible properties with the Lefschetz operator. Eventually, we show that for symplectic manifolds without strong Lefschetz property, the symplectic cohomologies can be used to detect the rank of the Lefschetz operator. In Chapter 4, we focus on the concept of spectral sequences. After reviewing the definition briefly, we discuss the relation between dδ-lemma and the degeneration of 7 spectral sequences. We also find elements in the double complex which can measure the degeneration of the spectral sequence. Then we construct the symplectic-de Rham spectral sequence and show that it has different properties from Brylinski’s canonical spectral sequence. The last section includes the computation of spectral sequences for some symplectic nilmanifolds. We move on to the Luttinger surgery and its implications in Chapters 5 and 6. In Chapter 5, we first introduce the concept of the symplectic Kodaira dimension. Then we review the construction of the Luttinger surgery and show that the Luttinger surgery does not change the Kodaira dimension for symplectic 4-manifold. The effect of the Luttinger surgery for symplectic Calabi-Yau 4-manifolds is discussed in detail. In Chapter 6, we generalize the Luttinger surgery and introduce the torus surgery construction. We focus on the choice of framings and give the constraints on related homology classes . Chapter 2
Primitive cohomology
In this chapter, we define the primitive cohomology induced by the Lefschetz decompo- sition. We first recall several linear operators for smooth differential forms in Section 2.1. They include the Lefschetz operator and the symplectic star operator given by the symplectic structure. Section 2.2 devotes to the sl(2) representation structure for the Lefschetz operator and the definition of primitive forms. The Lefschetz decomposition of differential forms is also mentioned. Section 2.3 introduces a decomposition of the differential d. In Section 2.4, we define the primitive cohomologies. The explicit com- putation of primitive cohomology for the Kodaira-Thurston manifold is given in Section 2.5.
2.1 Linear operators
Let V be a vector space. A symplectic structure on V is given by a nondegenerate 2-form ω ∈ V2 V ∗. ω defines a mapω ¯ : V → V ∗ as
ω¯(X) = ω(X, ·)
Since ω is nondegenerate,ω ¯ is an isomorphism of vector spaces. The inverseω ¯−1 : V ∗ → V can be extended to a multilinear map in the exterior algebra
k k −1 ^ ∗ ^ −1 −1 −1 ω¯ : V → V, ω¯ (ϕ1 ∧ · · · ∧ ϕk) =ω ¯ (ϕ1) ∧ · · · ∧ ω¯ (ϕk)
8 9 The symplectic structure naturally induces a linear operator on the exterior algebra of V :
Definition 2.1.1. Let (V, ω) be a symplectic vector space. The Lefschetz operator L : Vk V ∗ → Vk+2 V ∗ is defined as L(ϕ) = ω ∧ ϕ.
The Lefschetz operator L has a dual operator defined by the interior product. Let ϕ ∈ Vk V ∗ and X ∈ Vl V with 2n ≥ k ≥ l ≥ 1. The interior product
l k k−l ^ ^ ^ ι : V × V ∗ → V ∗ is given by k−l ^ ιX ϕ(Z) = ϕ(X ∧ Z), ∀Z ∈ V ∼ V0 We also define ιcϕ := cϕ for X = c ∈ R = V , . It is easy to show that
|X||Y | ιX ιY = ιY ∧X = (−1) ιX∧Y
Definition 2.1.2. Let (V, ω) be a symplectic vector space. The dual Lefschetz operator Λ: Vk V ∗ → Vk+2 V ∗ is defined as
Λϕ = ιω¯−1(ω)(ϕ).
∗ Remark 2.1.3. Assume V is endowed with a metric ( , ) and e1, e2, ··· , e2n is an ∗ V∗ ∗ Vk ∗ orthonormal basis of V . It induces a metric on V := ⊕k∈Z V with orthonormal basis
{ei1 ∧ ei2 ∧ · · · ∧ eik |1 ≤ i1 < i2 < ··· < ik ≤ 2n} The dual Lefschetz operator Λ also can be defined as the adjoint operator satisfying
(Lα, β) = (α, Λβ) for any α, β ∈ V∗ V ∗.
p 2n−p The other important linear operator is the symplectic star operator ∗s :Ω → Ω . It is defined as ωn ∗ ϕ = ι −1 . s ω¯ (ϕ) n! 10 It is easy to show that
n ω pq ∗ (ϕ ∧ %) = ι −1 = (−1) ι −1 (∗ %) = ι −1 (∗ ϕ) s ω¯ (ϕ∧%) n! ω¯ (ϕ) s ω¯ (%) s for any ϕ ∈ Vp V ∗ and % ∈ Vq V ∗.
Canonical basis
A canonical basis of a symplectic vector space (V, ω) is a basis X1,Y1, ··· ,Xn,Yn of V ∗ with dual basis x1, y1, ··· , xn, yn ∈ V satisfying
xi(Yj) = 0, xi(Xj) = yi(Yj) = δij
P such that ω = xi ∧ yi. It is an easy exercise that ωn Y = x ∧ y ∧ x ∧ y ∧ · · · ∧ x ∧ y =: x ∧ y n! 1 1 2 2 n n and −1 −1 −1 X ω¯ (yi) = Xi, ω¯ (xi) = −Yi, ω¯ (ω) = Xi ∧ Yi.
Since any symplectic vector space possesses a canonical basis, it is useful to give an explicit description of the linear operators in canonical basis. Let I := {1, 2, ··· , n}. For any I ⊂ I, we use Ic := I − I to denote the complement.
For I = {i1, i2, ··· , ik}, we define
(x ∧ y)I := xi1 ∧ yi1 ∧ · · · ∧ xik ∧ yik
xI := xi1 ∧ · · · ∧ xik
yI := yi1 ∧ · · · ∧ yik
Given mutually disjoint subset I,I0,I00 ⊂ I, we define J = I ∪ I0 ∪ I00 and use the following notation to denote the monomial
cI,I0,I00 := (x ∧ y)I ∧ xI0 ∧ yI00
V∗ ∗ It is clear that {cI,I0,I00 } generates the vector space V . Hence we can use it to show that 11 Lemma 2.1.4. For any α ∈ Vq V ∗,
(ΛL − LΛ)α = (n − q)α. (2.1)
Proof. It is enough to show the identity for monomials when ω is canonical. Let q = 0 00 |cI,I0,I00 | := deg(cI,I0,I00 ) = 2|I| + |I | + |I |. From the following relations X LcI,I0,I00 = ω ∧ cI,I0,I00 = (x ∧ y)I+{j} ∧ xI0 ∧ yI00 j∈Jc X 0 00 P 0 00 0 00 ΛcI,I ,I = ι Xi∧Yi cI,I ,I = (x ∧ y)I−{k} ∧ xI ∧ yI k∈I we have
X X ΛLcI,I0,I00 = (x ∧ y)I+{j}−{k} ∧ xI0 ∧ yI00 j∈Jc k∈(I+{j})c X c = (x ∧ y)I−{k}+{j} ∧ xI0 ∧ yI00 + |J |cI,I0,I00 k∈I,j∈Jc X X LΛcI,I0,I00 = (x ∧ y)I−{k}+{j} ∧ xI0 ∧ yI00 k∈I j∈(J−{k})c X = (x ∧ y)I−{k}+{j} ∧ xI0 ∧ yI00 + |I|cI,I0,I00 k∈I,j∈Jc
So
c 0 00 (ΛL − LΛ)cI,I0,I00 = (|J | − |I|)cI,I0,I00 = (n − |I| − |I | − |I | − |I|)cI,I0,I00 = (n − q)cI,I0,I00
V∗ ∗ We introduce another basic elements of V now. Assume I = (i1, i2, ··· , i2k), ij ∈ (I0 ∪ I00)c is an ordered set with even elements. We define
0 00 ap,(i1,i2),··· ,(i2k−1,i2k),I ,I X k 0 00 := (x ∧ y)M ∧ (∧j=1(xi2j−1 ∧ yi2j−1 − xi2j ∧ yi2j )) ∧ xI ∧ yI M∩J=∅,|M|=p and
0 00 0 00 B := {ap,(i1,i2),··· ,(i2k−1,i2k),I ,I |I,I ,I ⊂ I mutually disjoint, |I| : even} 12 For instance,
a0,(1,3),(2,5),{4},{6} = (x1 ∧ y1 − x3 ∧ y3) ∧ (x2 ∧ y2 − x5 ∧ y5) ∧ x4 ∧ y6
0 00 0 00 If ϕ = a0,(i1,i2),··· ,(i2k−1,i2k),I ,I , we also use ap,ϕ to denote ap,(i1,i2),··· ,(i2k−1,i2k),I ,I 0 00 0 00 and sometimes I,I and I are denoted as Iϕ,Iϕ,Iϕ to emphasize their relations. Note that a0,ϕ is defined for 0 ≤ p ≤ n − |ϕ|.
Lemma 2.1.5. Let (V, ω) be a symplectic vector space of dimension 2n with a canonical basis X1,Y1, ··· ,Xn,Yn and a dual basis x1, y1, ··· , xn, yn. Let ϕ = a0,ϕ ∈ B and 0 00 q = |ϕ| = |Iϕ| + |Iϕ| + |Iϕ| = |J|. We also define a−1,ϕ = an−q+1,ϕ = 0.
(1)
Lap,ϕ = (p + 1)ap+1,ϕ
Λap,ϕ = (n − p − q + 1)ap−1,ϕ
1 p In particular, ap,ϕ = p! L ϕ.
(2) (|I0|+|I00|)(|I0|+|I00|−1) ∗scI,I0,I00 = (−1) 2 cJc,I0,I00
q(q−1) ∗sap,ϕ = (−1) 2 an−p−q,ϕ
Proof. (1)
X Lap,ϕ = ω ∧ (x ∧ y)M ∧ ϕ M∩J=∅,|M|=p X X = (x ∧ y)k ∧ (x ∧ y)M ∧ ϕ M∩J=∅,|M|=p k∈(M∪J)c X = (p + 1)(x ∧ y)M1 ∧ ϕ
M1∩J=∅,|M1|=p+1
= (p + 1)ap,ϕ 13 By (2.1),
ΛLpϕ = LΛLp−1ϕ + (n − 2p + 2 − q)Lp−1ϕ = L(LΛLp−2ϕ + (n − 2p + 4 − q)Lp−2ϕ) + (n − 2p + 2 − q)Lp−1ϕ . . = LpΛϕ + (n − q)Lp−1ϕ + (n − q − 2)Lp−1ϕ + ··· + (n − q − 2p + 2)Lp−1ϕ = (pn − pq − p(p − 1))Lp−1ϕ
So 1 n − q − p + 1 Λa = ΛLpϕ = Lp−1ϕ = (n − p − q + 1)a p,ϕ p! (p − 1)! p−1,ϕ
−1 |I0| (2) It is easy to show that ω (cI,I0,I00 ) = (−1) (X ∧ Y )I ∧ YI0 ∧ XI00 .
∗scI,I0,I00 ((X ∧ Y )Jc ∧ XI0 ∧ YI00 ) Y = ι −1 x ∧ y((X ∧ Y ) c ∧ X 0 ∧ Y 00 ) ω (cI,I0,I00 ) J I I |I0| Y = (−1) x ∧ y((X ∧ Y )I ∧ YI0 ∧ XI00 ∧ (X ∧ Y )Jc ∧ XI0 ∧ YI00 ) |I0| Y = (−1) x ∧ y((X ∧ Y )Jc+I ∧ YI0 ∧ XI00 ∧ XI0 ∧ YI00 ) |I0|+|I0|(|I0|+|I00|) Y = (−1) x ∧ y((X ∧ Y )Jc+I ∧ XI0 ∧ YI0 ∧ XI00 ∧ YI00 ) 0 0 00 00 |I0||I00|+ |I |(|I |−1) + |I |(|I |−1) Y = (−1) 2 2 x ∧ y((X ∧ Y )Jc+I ∧ (X ∧ Y )I0 ∧ (X ∧ Y )I00 )
(|I0|+|I00|)(|I0|+|I00|−1) = (−1) 2
Since ωn is a monomial, we can conclude that
(|I0|+|I00|)(|I0|+|I00|−1) (|I0|+|I00|)(|I0|+|I00|−1) ∗s(cI,I0,I00 ) = (−1) 2 (x∧y)Jc ∧xI0 ∧yI00 = (−1) 2 cJc,I0,I00 .
c If we express a0,ϕ as a linear combination of cI,I0,I00 and choose all M ⊂ J with |M| = p, we can have
(|I0|+|I00|)(|I0|+|I00|−1)+|I| (q−|I|)(q−|I|−1)+|I| ∗s(ap,ϕ) = (−1) 2 an−p−q,ϕ = (−1) 2 an−p−q,ϕ
(q−|I|)(q−|I|−1)+|I| q(q−1) When |I| is even, we have (−1) 2 = (−1) 2 . Hence the second assertion is proved. 14 Assertion (2) can be summarized in the following diagram:
1 p−1 p p+1 p+2 n−q L : a0,ϕ −→ · · · −−→ ap−1,ϕ −→ ap,ϕ −−→ ap+1,ϕ −−→· · · −−→ an−q,ϕ n−q n−p−q+2 n−p−q+1 n−p−q n−p−q−1 1 Λ: a0,ϕ ←−−· · · ←−−−−−− ap−1,ϕ ←−−−−−− ap,ϕ ←−−−− ap+1,ϕ ←−−−−−−· · · ←− an−q,ϕ
Corollary 2.1.6. 1. (p + s)! (n − p − q)! ΛsLs(a ) = · (a ) (2.2) p,ϕ p! (n − p − q − s)! p,ϕ p! (n − p − q + s)! LaΛs(a ) = · (a ) (2.3) p,ϕ (p − s)! (n − p − q)! p,ϕ
2.
∗s∗s = 1. (2.4)
3. ∗sΛ = L∗s or Λ = ∗sL∗s.
From Lemma 2.1.5 and Corollary 2.1.6, we can see that the operators L, Λ and ∗s have simple formulas in B. If we can show that B generates V∗ V ∗, then we have given explicit formulas for these operators.
Lemma 2.1.7. V∗ V ∗is generated by B as a vector space.
Proof. It is enough to show that any cI,I0,I00 is a linear combination of elements in B. We will prove it by induction on |I|. Without loss of generality, we can assume I0 = I00 = ∅ and I = {1, 2, ··· , k}.
If I = ∅, we have c∅,I0,I00 = a0,I0,I00 . When I = {1},
n 1 X c = x ∧ y = (ω + (x ∧ y − x ∧ y )). {1},∅,∅ 1 1 n 1 1 i i i=2 Assume the statement is true for |I| ≤ k and
X c{2,3,··· ,k+1},∅,∅ = bp,ϕap,ϕ. p,ϕ
When I = {1, 2, ··· , k + 1},
X cI,∅,∅ = (x ∧ y)1 ∧ ( bp,ϕap,ϕ). p,ϕ 15 We will use the following equality
1 X (x ∧ y) = ( ((x ∧ y) − (x ∧ y) ) + ω − ω ) (2.5) 1 t 1 j ϕ j∈ /Iϕ where ω = P (x ∧ y) and t := n − |I |. Note that (2.5) holds whether 1 ∈ I or ϕ i∈Iϕ i ϕ ϕ not. It is easy to show that ϕ ∧ ωϕ = 0. If 1 ∈/ Iϕ, then 1 X ϕ ∧ (x ∧ y) = ϕ ∧ ( ((x ∧ y) − (x ∧ y) ) + ω − ω ) 1 t 1 j ϕ j∈ /Iϕ 1 X 1 = ϕ ∧ ((x ∧ y) − (x ∧ y) ) + a t 1 j t 1,ϕ j∈ /Iϕ and ϕ ∧ ((x ∧ y)1 − (x ∧ y)j) ∈ B. When 1 ∈ Iϕ and (x ∧ y)j − (x ∧ y)1 is a factor of ϕ, 0 we use the notation ϕ = ϕ ∧ ((x ∧ y)j − (x ∧ y)1). Then
0 ϕ ∧ (x ∧ y)1 = ϕ ∧ (x ∧ y)1 ∧ (x ∧ y)j.
We can assume j = 2. By the following equality
(x ∧ y)1 ∧ (x ∧ y)2 1 X = ((x ∧ y) − (x ∧ y) ) ∧ ((x ∧ y) − (x ∧ y) ) t(t + 1) 1 i 2 j i,j∈ /Iϕ,i6=j
1 2 1 − (ω − ω 0 ) + (ω − ω 0 ) ∧ ((x ∧ y) + (x ∧ y) ), t(t + 1) ϕ t ϕ 1 2 we have
0 ϕ ∧ (x ∧ y)1 ∧ (x ∧ y)2 1 X = ϕ0 ∧ ((x ∧ y) − (x ∧ y) ) ∧ ((x ∧ y) − (x ∧ y) ) t(t + 1) 1 i 2 j i,j∈ /Iϕ,i6=j 1 1 − ϕ0 ∧ ω2 + ϕ0 ∧ ω ∧ ((x ∧ y) + (x ∧ y) ). t(t + 1) t 1 2
Using induction assumption and the result for 1 ∈/ Iϕ, the assertion is proved.
If (M, ω) is a symplectic manifold with dimension 2n, we can set V = TxM and the k linear operators discussed in this section are defined on Ω (M, R). All lemmas above k hold for Ω (M, R) as well. 16 2.2 Lefschetz decomposition and primitive forms
In the previous section, we show that the set B generates V∗ V ∗. In this section, we will give a decomposition of V∗ V ∗ and show that B is compatible with this decomposition. Q V∗ ∗ Vq ∗ Q Let q : V → V be the projection and H = ⊕(n − q) q. L, Λ and H give a representation of the Lie algebra sl(2) acting on V∗ V ∗:
Lemma 2.2.1.
[Λ,L] = H, (2.6) [H, Λ] = 2Λ, (2.7) [H,L] = −2L (2.8)
Proof. The first assertion is proved in Lemma 2.1.4. The other two identities can be shown straightforwardly:
(HΛ − ΛH)cI,I0,I00 = (n − q)ΛcI,I0,I00 − (n − q + 2)ΛcI,I0,I00 = 2ΛcI,I0,I00
(HL − LH)cI,I0,I00 = (n − q − 2)LcI,I0,I00 − (n − q)LcI,I0,I00 = −2LcI,I0,I00
It gives the other way to decompose V∗ V ∗. Let Ik be the direct sum of irreducible V∗ ∗ k sl(2)-representation of dimension k + 1. Then V = ⊕I For a0,ϕ ∈ B, Lemma 2.1.5 and Corollary 2.1.6 show that
n−|ϕ|+1 n−|ϕ| n−|ϕ| 2 Λa0,ϕ = 0,L a0,ϕ = 0, Λ L a0,ϕ = ((n − |ϕ|)!) a0,ϕ
n−|ϕ| Hence a0,ϕ, a1,ϕ, ··· an−|ϕ|,ϕ ∈ I . Let
p,q L = Span{ap,ϕ||ϕ| = q}
Hence X Ik = Lp,k. 0≤p≤n−k The elements of the sl(2) representation with highest weight play an important role in our study of differential forms and their cohomologies. 17 Vq ∗ Definition 2.2.2. Bq ∈ V is call primitive if ΛBq = 0. Define
∗ ^ P := {ϕ ∈ V ∗|ϕ : primitive} and Pq = P ∩ Vq V ∗.
Proposition 2.2.3. (1) Pk = L0,k. In particular, Pk = 0 if k < 0 or k > n.
(2) Each α ∈ Vk V ∗ can be decomposed uniquely as
X 1 α = LrBα (2.9) r! k−2r r≥max{k−n,0}
α k−2r for some primitive elements Bk−2r ∈ P .
(3) a a (p + a)! (n − p − q)! Λ L | p,q = · Id L p! (n − p − q − a)!
a a p! (n − p − q + a)! L Λ | p,q = · Id L (p − a)! (n − p − q)!
p,q n−p−q,q 0,q (4) ∗s : L → L is an isomorphism and, for ϕ ∈ L ,
p q(q−1) p! n−p−q ∗ L ϕ = (−1) 2 L ϕ s (n − p − q)!
n−q (5) When p ≤ 2 , q(q−1) p! n−2p−q ∗ = (−1) 2 L . s (n − p − q)! n−q If p > 2 q(q−1) (n − p − q)! 2p−n+q ∗ = (−1) 2 Λ . s p! This proposition can be proved by Lemma 2.1.5, Corollary 2.1.6 and the theory of sl(2) representations. The decomposition (2.9) is called the Lefschetz decomposition of differential forms. 18
Ln,0 Ω2n(M) Ln−1,1 Ω2n−1(M) Ln−1,0 Ω2n−2(M) L2,n−2 L ↑ L1,n−1 . . L0,n L1,n−2 ··· . . Λ ↓ L0,n−1 L0,n−2 L0,1 Ω1(M) L0,0 Ω0(M)
I0 I1 I2 ···In−1 In
Since dimLp,q = dimLp0,q, 0 ≤ p, p0 ≤ n − q, we have
q q−2 ^ ^ 2n 2n dimLp,q = dimPq = dim V ∗ − dim V ∗ = − q q − 2
Now we want to generalize this theory to symplectic manifolds. For any point x in a symplectic manifold (M, ω), the Darboux theorem says that there is a neighborhood 2n U of x symplectomorphic to (R , ω0). Although the generators cI,I0,I00 and ap,α depend on the choice of local chart, the subspace Lp,q and the Lefschetz decomposition are well defined. Hence we can define the Lefschetz decomposition for any smooth forms on Ω∗(M) and all the above lemmas and propositions hold as well.
Λ 2.3 ∂+, ∂− and d operator
In this section, we first give a decomposition of the differential operator d which is com- patible with the Lefschetz decomposition. Then we compare the decomposed operators with the symplectic operator dΛ studied by Brylinski in [12]. Let (M, ω) be a symplectic manifold of dimension 2n. For any primitive form α ∈ 19 Pq(M), the (q + 1)-form dα has the Lefschetz decomposition (2.9) 1 dα = Bdα + LBdα + L2Bdα + ··· . q+1 q−1 2 q−2 The decomposition of d-exact forms has only the first two terms:
Proposition 2.3.1. Assume α ∈ Pq, then
dα dα dα = Bq+1 + ω ∧ Bq−1.
Moreover, for Lpα ∈ Lp,q,
p dα p+1 dα dα = L Bq+1 + L Bq−1.
Proof. It is enough to show it locally. Assume a canonical basis in a local chart is given ∞ and consider the set of generators B. If α = fa0,ϕ, then dα is a C -linear combination 2 of a0,ϕ0 and xi ∧ yi ∧ a0,ϕ0 . Applying Λ to each component, we can see that Λ dα = 0. By Corollary 2.1.6, the Lefschetz decomposition of dα has only the first two terms. The second assertion follows from the easy fact [d, L] = 0.
Definition 2.3.2. The maps
p,q p,q+1 p,q p,q−1 ∂+ : L → L , ∂− : L → P
p p dα p p dα are defined as ∂+(L α) = L Bq+1 and ∂−(L α) = L Bq−1.
It is clear that d = ∂+ + L∂− and [∂+,L] = [∂−,L] = 0. From the equation
2 0 = d = (∂+ + L∂−)(∂+ + L∂−) 2 = ∂+ + L∂−∂+ + ∂+L∂− + L∂−L∂− 2 2 2 = ∂+ + L(∂−∂+ + ∂+∂−) + L ∂−
2 2 we have ∂+ = ∂− = 0 and L(∂−∂+ + ∂+∂−) = 0.
Remark 2.3.3. The relation L(∂−∂+ +∂+∂−) = 0 can be simplified to ∂−∂+ +∂+∂− = 0 except the highest term p = n − q. 20 In [12], Brylinski defines the operator
dΛ := [d, Λ]. dΛ depends on the symplectic structure and is a differential operator of degree −1. Λ 2 ∗ Λ ∗ It is shown in [12] that (d ) = 0 and (Ω (M), d ) defines a cohomology HdΛ (M). Proposition 2.3.1 implies that
dΛ(Lp,q) ⊂ Lp−1,q+1 ⊕ Lp,q−1
Λ Λ Λ Λ Hence d is decomposed as d = ∂+ + ∂− with
Λ p,q p−1,q+1 Λ p,q p,q−1 ∂+ : L → L , ∂− : L → L
The decompositions of d and dΛ can be summarized in the following diagram:
Lp,q+1 Lp+1,q−1
∂+ -% L∂− Lp,q Λ Λ ∂+ .& ∂− Lp−1,q+1 Lp,q−1
Λ Actually, d can be described by ∂+, ∂−,L and Λ.
Proposition 2.3.4. On Lp,q,
(1)
Λ q d = (−1) ∗s d ∗s . (2.10)
(2) [dΛ,L] = d, [dΛ, Λ] = 0, [dΛ,H] = −dΛ.
(3) ( Λ 1 ∂+ = n−p−q Λ∂+ 0 ≤ p < n − q Λ p p−1 ∂+L ϕ = pL ∂+ϕ p = n − q and Λ ∂− = (p + q − n − 1)∂−. 21 Proof. (1) Assume ϕ is primitive and 0 < p ≤ n − q.
(dΛ − Λd)Lpϕ = dΛLpϕ − ΛdLpϕ p−1 p p+1 = p(n − p − q + 1)dL ϕ − ΛL ∂+ϕ − ΛL ∂−ϕ p−1 p = p(n − p − q + 1)L ∂+ϕ + p(n − p − q + 1)L ∂−ϕ p−1 p −p(n − p − q)L ∂+ϕ − (p + 1)(n − p − q + 1)L ∂−ϕ p−1 p = pL ∂+ϕ + (p + q − n − 1)L ∂−ϕ
On the other hand,
p q(q−1) p! n−p−q ∗ L ϕ = (−1) 2 L ϕ s (n − p − q)!
p q(q−1) p! n−p−q d ∗ L ϕ = (−1) 2 L dϕ s (n − p − q)!
q(q−1) p! n−p−q n−p−q+1 = (−1) 2 (L ∂ ϕ + L ∂ ϕ) (n − p − q)! + −
p q(q−1) p! (q+1)q (n − p − q)! p−1 ∗ d ∗ L ϕ = (−1) 2 ((−1) 2 L ∂ ϕ s s (n − p − q)! (p − 1)! +
(q−1)(q−2) (n − p − q + 1)! p +(−1) 2 L ∂ ϕ) p! − q2 p−1 (q−1)2 p = (−1) pL ∂+ϕ + (−1) (n − p − q + 1)L ∂−ϕ q p−1 p = (−1) (pL ∂+ϕ + (p + q − n − 1)L ∂−ϕ)
When p = 0, the assertion is given by ignoring all ∂+ terms in the above equalities.
(2)
[dΛ,L] = (dΛ − Λd)L − L(dΛL − Λd) = dΛL − ΛdL − LdΛ + LΛd = d(H + LΛ) − ΛLd − LdΛ + LΛd = dH + dLΛ − LdΛ + (LΛ − ΛL)d = dH − Hd = [d, H] = d 22 Λ q+1 q+1 q+1 [d , Λ] = (−1) ∗s d∗s ∗sL∗s −(−1) ∗s L∗s ∗sd∗s = (−1) ∗s (dL−Ld)∗s = 0
The third one is obvious.
(3) From the proof of part (1), we have
( p−1 p pL ∂+ϕ + (p + q − n − 1)L ∂−ϕ 0 < p ≤ n − q dΛLpϕ = (q − n − 1)∂−ϕ p = 0
Λ Λ So the assertion is true for ∂+ with p = 0, n − q and for ∂−. When 0 < p < n − q, by Proposition 2.2.3(3), 1 1 pLp−1∂ ϕ = p · ΛLLp−1∂ ϕ = Λ∂ Lpϕ + p(n − p − q) + (n − p − q) +
4 4 Example 2.3.5. Let M = R /Z be a 4-torus with standard symplectic structure ω = x1 ∧ y1 + x2 ∧ y2. The primitive forms of M are
0 1 P =< 1 >, P =< dx1, dy1, dx2, dy2 >,
2 P =< dx1 ∧ dx2, dx1 ∧ dy2, dy1 ∧ x2, dy1 ∧ dy2, σ >
∞ where σ = dx1 ∧ dy1 − dx2 ∧ dy2. For f ∈ C (M),
d(fdx1) = fy1 dy1 ∧ dx1 + fx2 dx2 ∧ dx1 + fy2 dy2 ∧ dx1 1 1 = −f dx ∧ dx − f dx ∧ dy − f ( ω + σ) x2 1 2 y2 1 2 y1 2 2 So 1 ∂ (fdx ) = −f dx ∧ dx − f dx ∧ dy − f σ, + 1 x2 1 2 y2 1 2 2 y1 1 ∂ (fdx ) = − f . − 1 2 y1
2.4 Primitive cohomologies
If we focus on primitive forms, we have two complexes
∂ ∂ ∂ ∂ 0 −→ P0 −→+ P1 −→+ · · · −→+ Pn−1 −→ Pn −→+ 0 23 ∂ ∂ ∂ ∂ 0 −→ Pn −→− Pn−1 −→− · · · −→− P1 −→ P0 −→− 0
The primitive cohomologies are defined as
k k k ker∂+ ∩ P k ker∂− ∩ P PH+(M) = k−1 , PH−(M) = k+1 ∂+P ∂−P for 0 ≤ k < n. PH+ and PH− has the following properties
Proposition 2.4.1. [48]
k 1. dimPH± ≤ ∞ for 0 ≤ k ≤ n − 1.
k ∼ k k ∼ k 2. For k = 0, 1, PH+(M) = H (M) and PH−(M) = HdΛ (M).
k ∼ k 3. There is a natural isomorphism PH+ = PH−, 0 ≤ k ≤ n.
2.5 Example: Kodaira-Thurston manifold
The Kodaira-Thurston manifold is the first example of symplectic manifold which is not K¨ahler.In this section, we will compute some primitive cohomology of this manifold. 4 The Kodaira-Thurston manifold M can be viewed as a quotient space M = R /Γ 4 4 where Γ is a discrete subgroup of Diff(R ). If (x1, y1, x2, y2) is the coordinate of R , Γ is the group generated by the diffeomorphisms:
(x1, y1, x2, y2) → (x1 + 1, y1, x2, y2 + λx2)
(x1, y1, x2, y2) → (x1, y1 + 1, x2, y2)
(x1, y1, x2, y2) → (x1, y1, x2 + 1, y2)
(x1, y1, x2, y2) → (x1, y1, x2, y2 + 1) for a fixed λ ∈ N. A smooth function f on M is equivalent to a Γ-invariant function on R4 and the differential forms on M correspond to the Γ-invariant differential forms on 4 R . Note that fx1 , fy1 , fx2 + λx1fy2 and fy2 are also Γ-invariant. It is easy to see that the forms
dx1, dy1, dx2, dy2 − λx1dx2 are invariant under Γ and their wedge products generate all differential forms on M. Assume M has a symplectic structure
ω = dx1 ∧ dy1 + dx2 ∧ (dy2 − λdx2) = dx1 ∧ dy1 + dx2 ∧ dy2 24 P2 is generated by
dx1 ∧ dx2, dx1 ∧ dy2, dy1 ∧ dx2, dy1 ∧ dy2, σ = dx1 ∧ dy1 − dx2 ∧ dy2
So the ∂± operator is 1 ∂ (P dx ) = −P dx ∧ dx − P dx ∧ dy − P σ + 1 x2 1 2 y2 1 2 2 y1 1 = −(P + λx P )dx ∧ dx − P dx ∧ (dy − λx dx ) − P σ x2 1 y2 1 2 y2 1 2 1 2 2 y1 1 ∂ (Qdy ) = −Q dy ∧ dx − Q dy ∧ dy + Q σ + 1 x2 1 2 y2 1 2 2 x1 1 = −(Q + λx Q )dy ∧ dx − Q dy ∧ (dy − λx dx ) + Q σ x2 1 y2 1 2 y2 1 2 1 2 2 x1 1 ∂ (Rdx ) = R dx ∧ dx + R dy ∧ dx + R σ + 2 x1 1 2 y1 1 2 2 y2
∂+(S(dy2 − λx1dx2)) = −λSdx1 ∧ dx2 + Sx1 dx1 ∧ (dy2 − λx1dx2) 1 + S dy ∧ (dy − λx dx ) − (S + λx S )σ y1 1 2 1 2 2 x2 1 y2
1 ∂ (P dx ) = − P − 1 2 y1 1 ∂ (Qdy ) = Q − 1 2 x1 1 ∂ (Rdx ) = − R − 2 2 y2 1 ∂ (S(dy − λx dx )) = (S + λx S ) − 2 1 2 2 x2 1 y2
∂−(Adx1 ∧ dx2) = Ay2 dx1 − Ay1 dx2
∂−(Bdx1 ∧ (dy2 − λx1dx2)) = −(Bx2 + λx1By2 )dx1 − By1 (dy2 − λx1dx2)
∂−(Cdy1 ∧ dx2) = Cy2 dy1 + Cx1 dx2
∂−(Ddy1 ∧ (dy2 − λx1dx2)) = −(Dx2 + λx1Dy2 )dy1 − λDdx2 + Dx1 (dy2 − λx1dx2)
∂−(Eσ) = −Ex1 dx1 − Ey1 dy1 + (Ex2 + λx1Ey2 )dx2 + Ey2 (dy2 − λx1dx2)
If Ψ = Adx1 ∧ dx2 + Bdx1 ∧ (dy2 − λx1dx2) + Cdy1 ∧ dx2 + Ddy1 ∧ (dy2 − λx1dx2) + Eσ, 25 then
∂−Ψ =(Ay2 − (Bx2 + λx1By2 ) − Ex1 )dx1 (2.11)
+ (Cy2 − (Dx2 + λx1Dy2 ) − Ey1 )dy1 (2.12)
+ (−Ay1 + Cx1 − λD + (Ex2 + λx1Ey2 ))dx2 (2.13)
+ (−By1 + Dx1 + Ey2 )(dy2 − λx1dx2) (2.14)
The computation of PH∗(M) is similar to the computation of de Rham cohomolo- gies. The main idea is to reduce the number of variables and to use induction. Let ∞ 4 f ∈ C (R ) be a Γ-invariant function. We define Z 1 f¯(x1, x2, y2) := f(x1, t, x2, y2)dt, 0 and Z y1 g(x1, y1, x2, y2) = f(x1, t, x2, y2) − f¯(x1, x2, y2)dt 0 ¯ ¯ Then gy1 = f−f. Moreover, g is Γ-invariant and f is independent of y1. So ∂−(−2gdx1) = 0 f −f¯and [f] = [f¯] ∈ PH (M). Hence we can assume f = f(x1, x2, y2) depends on x1, x2 and y2 only. Now we consider another description of M. If N is a smooth manifold and φ : N → N is a diffeomorphism, the mapping torus of N is the quotient space
Nφ := N × [0, 1]/(0, x) = (1, φ(x)).
3 So M can be viewed as a mapping torus of T with diffeomorphism φ(y1, x2, y2) =
(y1, x2, y2 + λx2). Let ρ : [0, 1] → [0, 1] be a smooth increasing function whose value is 0 near 0 and 1 near 1. Now we define Z 1 fˆ(x1, x2) := f(x1, x2, s)ds 0
Z y2 h1(x1, x2, y2) := f(x1, x2, t) − fˆ(x1, x2) dt 0 Z y2 h2(x1, x2, y2) := f(x1, x2, t) − fˆ(x1, x2) dt λx2 and
h(x1, x2, y2) = (1 − ρ(x1))h1(x1, x2, y2) + ρ(x1)h2(x1, x2, y2) 26 ˆ on M. It is easy to see that h1,y2 = h2,y2 = hy2 = f − f. So we can assume that f is independent of y2. Similar to the reduction of y1, we can eliminate x1, x2 and assume 0 ∼ that f = c is a constant. Hence PH−(M) = R.
Note that the elimination of x1 and x2 works when f is independent of y2. Let
P dx1 + Qdy1 + Rdx2 + S(dy2 − λx1dx2) be a ∂−-closed 1-form where P, Q, R and S are
Γ-invariant functions of x1, y1, x2, y2. So it satisfies
−Py1 + Qx1 − Ry2 + Sx2 + λx1Sy2 = 0 (2.15)
We want to reduce the variables of these functions. Here we will write down the variables to emphasize the dependence of variables and function. For example, P = P (x1, y2) means that P is independent of y1 and x2. We consider the formula (2.11), (2.12), (2.13), (2.14).
1. Use Ay2 ,Cy2 ,Ey2 to eliminate y2 in P, Q, S. By (2.15), Ry2 is independent of y2 and so is R.
2. Use By1 ,Dx1 to get S = S(x2).
3. Use Ey1 to get Q = Q(x1, x2).
4. Use Ay1 to get R = R(x1, x2). By (2.15), Py1 and P are independent of y1 and
P = P (x1, x2).
5. Use Ex1 to get P = P (x2).
6. Use Cx1 to get R = R(x2). By (2.15), Qx1 and Q are independent of x1 and
Q = Q(x2). So (2.15) becomes Sx2 = 0 and S =constant.
7. Use Bx2 to get P =constant.
8. Use Dx2 to get Q =constant.
9. Use Ex2 to get R =constant. Then use λD to eliminate R.
1 3 Hence PH−(M) = R . Chapter 3
Strong Lefschetz property
The Lefschetz operator L gives an isomorphism
Lk :Ωn−k(M) → Ωn+k(M) α 7→ α ∧ ωk
Moreover, the commutativity of d and L induces a map
Lk : Hn−k(M) → Hn+k(M) [α] 7→ [α ∧ ωk]
By Poincare duality, Hn−k(M) and Hn+k(M) have the same dimension. If Lk is an isomorphism for any k, we say that M satisfies the strong Lefschetz property. It is well know that this property holds for K¨ahlermanifolds:
Proposition 3.0.1 (Hard Lefschetz property). If M is a K¨ahlermanifold and ω is the K¨ahlerform, then the map
Lk : Hn−k(M) → Hn+k(M) [α] 7→ [α ∧ ωk] is an isomorphism for any 0 ≤ k ≤ n.
In general, the strong Lefschetz property does not hold for symplectic manifolds. Actually, it is one of the main tool to show that a symplectic manifold is not K¨ahler.
27 28 There are several concepts strongly relating to the strong Lefschetz property. The properties of symplectic and primitive harmonic forms are given in Section 3.1. Section Λ 3.2 discusses the dd -lemma and ∂+∂−-lemma. In Section 3.3, we reprove that the strong Lefschetz property has equivalent conditions given in the first two sections. In Section 3.4, we show that the existence of Lefschetz decomposition for de Rham cohomology is characterized by the strong Lefschetz property. In Section 3.5, we use the symplectic cohomologies to discuss the failure of the strong Lefschetz property.
3.1 Symplectic and primitive harmonic forms
In Hodge theory, one key ingredient is the existence of harmonic forms. In a symplectic manifold, a smooth form α is called symplectic harmonic if dα = dΛα = 0. A symplectic harmonic form can be characterized by its Lefschetz decomposition:
Proposition 3.1.1. For any α ∈ Ωk(M), the following conditions are equivalent:
(i) α is symplectic harmonic;
(ii) each primitive term of the Lefschetz decomposition of α is ∂+- and ∂−-closed.
Proof. (ii)⇒(i) is obvious. So we only need to show (i)⇒(ii). Assume α is symplectic harmonic and k−2r α = α0 + Lα1 + ··· , αr ∈ P is the Lefschetz decomposition of α. Note that α0 = α1 = ··· = αk−n−1 = 0 when k > n. The Lefschetz decomposition of dα is
2 dα = (∂+ + L∂−)(α0 + Lα1 + ··· ) = ∂+α0 + L(∂−α0 + ∂+α1) + L (∂−α1 + ∂+α2) + ···
α is d-closed implies that
∂+α0 = 0, ∂−αr−1 = −∂+αr, r = 1, 2, ··· .
On the other hand, dΛα = 0 implies that
Λ Λ r−1 Λ r ∂+α0 = 0, ∂−L αr−1 = −∂+L αr, r = 1, 2, ··· . 29 k−n Λ k−n When k > n and r = k − n, the first term of α is L αk−n and ∂+L αk−n = 0. It implies that ∂+αk−n = 0. In general,
Λ r−1 r−1 r−1 ∂−L αr−1 = ((r − 1) + (k − 2r + 2) − n − 1)∂−L αr−1 = (n − k + r)L ∂+αr and 1 −∂ΛLrα = − Λ∂ Lrα + r n − r − (k − 2r) + r (r − 1 + 1)(n − (r − 1) − (k − 2r + 1)) = − ∂ Lr−1α n − k + r + r r(n − k + r) = − ∂ Lr−1α n − k + r + r k−n So ∂−αr−1 = −∂+αr can be nonzero only when n − k + r = −r or r = 2 , which is impossible.
This equivalent condition play an important role in this chapter.
k Definition 3.1.2. α ∈ P (M) is called primitive harmonic if ∂+α = ∂−α = 0. In general, α ∈ Ωk(M) is primitive harmonic if each primitive term of its Lefschetz de- composition is primitive harmonic.
Λ 3.2 dd -lemma and ∂+∂−-lemma
In symplectic geometry, there is a property similar to the ∂∂¯-lemma in complex geom- etry.
Definition 3.2.1. M has the ddΛ-lemma if
Imd ∩ ker dΛ = ImdΛ ∩ ker d = ImddΛ
Similarly, we can define the primitive version of ddΛ-lemma in the space of primitive forms.
Definition 3.2.2. M has the ∂+∂−-lemma if
k k k Im∂+ ∩ ker ∂− ∩ P = Im∂− ∩ ∂+ ∩ P = Im∂+∂− ∩ P , 1 ≤ k ≤ n − 1 (3.1)
0 0 Im∂+ ∩ ker ∂− ∩ P = Im∂+∂− ∩ P (3.2) n n Im∂− ∩ ker ∂+ ∩ P = Im∂+∂− ∩ P (3.3) 30 These two lemmas hold simultaneously:
Proposition 3.2.3. Let (M, ω) be a symplectic manifold. Then M has the ddΛ-lemma if and only if M has the ∂+∂−-lemma.
Λ Proof. dd -lemma ⇒ ∂+∂−-lemma: k Assume α is a primitive harmonic primitive k-form, i.e. α ∈ P and ∂+α = ∂−α = 0. Then dα = dΛα = 0. 1 Λ Λ If α = ∂−γ = k−n−1 d γ, then α = dd c for some c = c0 + Lc1, c0, c1 ∈ P. ⇒ α = (q − n − 1)∂+∂−c0. Λ Λ Λ If α = ∂+β, then d dβ = d (α + L∂−β) = 0. So dβ = dd c for some c = c0 + Lc1 + 2 L c2 and ∂−∂+c0 = ∂+β = α. Λ ∂+∂−-lemma ⇒ dd -lemma: Λ If α = α0 + Lα1 + ··· and dα = d α = 0, then ∂+αi = ∂−αi = 0 for any i. Assume
γ = γ0 + Lγ1 + ··· and α = dγ = (∂−γ0 + ∂+γ1) + L(∂−γ1 + ∂+γ2) + ··· . Applying ∂+ to α0, we have
∂+α0 = ∂+∂−γ0 = 0
So ∂−γ0 ∈ Im∂− ∩ ker ∂+ and ∂−γ0 = ∂+∂−c0. Similarly, we can find c1, c2, ··· such that ∂−γi = ∂+∂−ci. Replacing ci by kci for some appropriate constant k, we have Λ d d(c0 + Lc1 + ··· ) = α. The argument is the same when α = dΛγ.
Remark 3.2.4. The condition (3.2) holds for any compact symplectic manifolds (c.f. proof of Theorem 3.3.2).
3.3 Strong Lefschetz property, primitive harmonic repre-
sentatives and ∂+∂−-lemma
In this section, we want to use primitive forms to describe the strong Lefschetz property. We have the following well known fact
Theorem 3.3.1. [13, 35, 39, 54] The following properties are equivalent for a symplectic manifold M: 31 1. M satisfies the strong Lefschetz property;
2. the ddΛ-lemma holds for M;
3. each cohomology class of H∗(M) has a symplectic harmonic representative.
According to Proposition 3.1.1 and 3.2.3, this theorem can be rephrased as
Theorem 3.3.2. The following properties are equivalent for a symplectic manifold M:
1. M satisfies the strong Lefschetz property (SLP).
2. The ∂+∂−-lemma holds for M.
3. Each cohomology class of H∗(M) has a primitive harmonic representative (PH).
In the proofs of Theorem 3.3.1, they use the language of quivers, superalgebras and generalized complex geometry. Here we use more elementary techniques to prove Theorem 3.3.2 directly.
Proof. ∂+∂−-lemma ⇒ PH
Assume α = α0 + Lα1 + ··· is closed:
2 0 = dα = ∂+α0 + L(∂−α0 + ∂+α1) + L (∂−α1 + ∂+α2) + ···
Let εi = ∂+αi = −∂−αi−1. By ∂+∂−-lemma, there exists γi such that ∂+∂−γi = εi. Let
γ = γ1 + Lγ2 + ··· andα ¯ = α − dγ. The Lefschetz decomposition ofα ¯ is
α¯ = α0 + Lα1 + · · · − d(γ1 + Lγ2 + ··· ) i = (α0 − ∂+γ1) + L(α1 − ∂−γ1 − ∂+γ2) + ··· + L (αi − ∂−γi − ∂+γi+1) + ···
Since
∂+(αi − ∂−γi − ∂+γi+1) = ∂+αi − ∂+∂−γi+1 = εi − εi = 0
∂−(αi − ∂−γi − ∂+γi+1) = ∂−αi − ∂−∂+γi = εi+1 − εi+1 = 0 α¯ is primitive harmonic and [¯α] = [α − dγ] = [α].
2 k ±Lε1 ±L ε2 ±L εk %-%-%- 2 k−1 k α0 Lα1 L α2 ··· L αk−1 L αk -%-%-% k−1 γ1 Lγ2 L γk 32 PH ⇒ SLP For any σ ∈ H2n−k(M), there exists a primitive harmonic representative
n−k n−k+1 k−2i α = L α0 + L α1 + ··· , αi ∈ P
n−k So α0 + Lα1 + ··· is closed and σ = L [α0 + Lα1 + ··· ].
SLP ⇒ ∂+∂−-lemma We want to prove it by induction at k. In a symplectic local chart with canonical basis 1 (x1, y1, ··· , xn, yn), the ∂− part of dΩ (M) is contributed by dxi ∧ dyi terms. Note that
ndxi ∧ dyi − ω = (dxi ∧ dyi − dx1 ∧ dy1) + ··· + (dxi ∧ dyi − dxn ∧ dyn) or 1 dx ∧ dy = (ω + (dx ∧ dy − dx ∧ dy ) + ··· + (dx ∧ dy − dx ∧ dy )) i i n i i 1 1 i i n n P If σ = (pidxi + qidyi), then
X ∂qi ∂pi dσ = ( − )dx ∧ dy + other terms ∂x ∂y i i i i i and X 1 ∂qi ∂pi ∂ σ = ( − ). − n ∂x ∂y i i i For any f ∈ P0(M) = Ω0(M), locally, ! X ∂f ∂f 1 X ∂2f ∂2f ∂ ∂ f = ∂ ( dx + dy ) = ( − ) = 0 − + − ∂x i ∂y i n ∂x ∂y ∂y ∂x i i i i i i i i
0 So ∂−∂+(P ) = 0. We also know that
0 ker ∂+ ∩ P = {f = c|c : constant}
0 Λ 0 0 2n Im∂− ∩ P = Imd ∩ P = Im ∗s d ∗s ∩P = ∗s(Imd ∩ Ω (M)) = {f 6= constant}
Hence 0 0 Im∂− ∩ ker ∂+ ∩ P = 0 = Im∂+∂− ∩ P
0 and the ∂+∂−-lemma holds for P (M). 33 0 0 Using ∂−∂+(P ) = 0 again, we have ∂+Ω (M) ⊂ ker ∂− and
1 0 1 Im∂+ ∩ ker ∂− ∩ P = ∂+Ω (M) = Im∂+∂− ∩ P
1 n−1 n−k−1 On the other hand, if σ ∈ Im∂−∩ker ∂+∩P and σ = ∂−η, then [L σ] = [dL η] = n−1 0 0 ∈ H (M) and SLP implies that [σ] = 0 and σ ∈ dΩ (M). So ∂+∂−-lemma holds for P1(M). i k Now assume ∂+∂−-lemma holds for P (M), i < k and ε ∈ P , ∂+ε = ∂−ε = 0.
If ε = ∂+γ, then [ε] = [ε − dγ] = [−L∂−γ] and ∂+∂−γ = −∂−∂+γ = −∂−ε = 0. By k−2 hypothesis, ∂−γ = ∂+∂−c for some c ∈ P . So γ + ∂+c satisfies
∂−(γ + ∂+c) = 0, ∂+(γ + ∂+c) = ε
n−k+1 n−k+1 n−k+1 and L (γ + ∂+c) is closed. SLP implies that [L (γ + ∂+c)] = L [β] for n−k+1 n−k some closed k-form β = β0 + Lβ1 + ··· . Hence L (γ + ∂+c − β) = dL ξ for some k k−2i ξ = ξ0 + Lξ1 + · · · ∈ Ω (M), ξi ∈ P . In particular, ∂−ξ0 + ∂+ξ1 = γ + ∂+c − β0 and
∂+∂−ξ0 = ∂+γ = ε. n−k n−k−1 If ε = ∂−η, then [L ε] = [dL η] = 0 and SLP implies that [ε] = 0 as well. So
ε = ∂+γ for some γ again.
3.4 Lefschetz decomposition for de-Rham cohomology
We are motivated by the Lefschetz decomposition of forms and ask whenever H∗(M) ∗ has a similar structure. The primitive cohomology PH±(M) seems to be a good fit to play the role of primitive forms in the Lefschetz decomposition. Unfortunately, the situation is not so simple for cohomology. It even does not have a natural map between ∗ ∗ H (M) and PH±(M). Let
k k Λ Ωh = {α ∈ Ω |dα = d α = 0} and k k Ωph = {α ∈ Ω |α is primitive harmonic}.
k k By Proposition 3.1.1, Ωh = Ωph. Consider the natural maps
k K φ :Ωh −→ H (M) 34 and k k φ± :Ωph −→ H±(M)
Theorem 3.3.2 implies that
SLP ⇐⇒ φ is surjective
Assume the strong Lefschetz property hold now. We define a map
i k−2i k Φ+ : ⊕iL PH+ (M) −→ H (M)
k by Φ+(σ) = φ(α) for some α ∈ Ωph such that φ+(α) = σ. Similar to Theorem 3.3.2, we k can show that SLP implies that φ+ is surjective. If α = α0 + Lα1 + · · · ∈ Ωph satisfies ∗ φ+(α) = 0, there exists γi ∈ P such that ∂+γi = αi. By ∂+∂−-lemma, there exists ∗ k ci ∈ P such that ∂+∂−ci = αi. Then α = d∂−(c0 + Lc1 + ··· ) and φ(α) = 0 ∈ H (M).
Hence Φ+ is well defined.
On the other hand, if φ(α) = 0, there exists γ = γ0 + Lγ1 + ··· such that dγ = α, i.e.
α0 = ∂+γ0, αi = ∂−γi−1 + ∂+γi.
∗ So ∂+∂−γi−1 = ∂+αi − ∂+∂+γi = 0 and [αi] = [∂−γi−1] ∈ PH+(M). By ∂+∂−-lemma, ∗ ∂−γi−1 ∈ Im∂+∂− and ∂−γi−1 = ∂+∂−ci for some ci ∈ P . So [αi] = [∂+∂−ci] = 0 ∈ ∗ PH+(M) and Φ is injective. ∗ This argument works for PH−(M) and we have shown:
Theorem 3.4.1. Assume a symplectic manifold (M, ω) satisfies the strong Lefschetz property. Then there are canonical isomorphisms
i k−2i k Φ+ : ⊕iL PH+ (M) −→ H (M) and i k−2i k Φ− : ⊕iL PH− (M) −→ H (M).
3.5 Symplectic cohomologies
There are other sympelctic cohomology given by the combination of d and dΛ. 35 Definition 3.5.1. [47]
ker(d + dΛ) ∩ Ωk(M) Hk (M) = d+dΛ ImddΛ ∩ Ωk(M)
ker ddΛ ∩ Ωk(M) Hk (M) = ddΛ (Imd + ImdΛ) ∩ Ωk(M) Λ k k ker(d + d ) ∩ Ω (M) H Λ (M) = d∩d dΩ˜ k−1(M) + dΛΩ˜ k+1(M) where Ω˜ ∗(M) is the space of ddΛ-closed forms.
In [47], it is showed that all the cohomologies HdΛ ,Hd+dΛ ,HddΛ ,Hd∩dΛ are finite dimensional. Furthermore, ∼ ∼ Proposition 3.5.2. Hd∩dΛ (M) = HdΛ (M) ∩ Hd(M) = Hd+dΛ (M) ∩ HddΛ (M)
Similarly, we can define the primitive cohomologies for other symplectic cohomolo- gies:
Definition 3.5.3. [47]
k k ker(∂+ + ∂−) ∩ P (M) PHd+dΛ (M) = k Im∂+∂− ∩ P (M)
k k ker ∂+∂− ∩ P (M) PHddΛ (M) = k−1 k+1 ∂+P (M) + ∂−P (M) ker(∂ + ∂ ) ∩ Pk(M) PHk (M) = + − d∩dΛ k−1 k+1 ∂+P˜ (M) + ∂−P˜ (M) ∗ where P˜ (M) is the space of ∂+∂−-closed forms.
Unlike PH+ and PH−, they are equal to the ”primitive” part of their corresponding cohomologies:
Proposition 3.5.4.
Λ k k ∼ ker(d + d ) ∩ P (M) PH Λ (M) = d+d ImddΛ ∩ Pk(M)
Λ k k ∼ ker dd ∩ P (M) PH Λ (M) = dd (Imd + ImdΛ) ∩ Pk(M) k ∼ k k PHd∩dΛ (M) = PHd+dΛ (M) ∩ PHddΛ (M) 36 Hence they have Lefschetz decomposition:
Theorem 3.5.5. [47] The cohomologies Hd+dΛ ,HddΛ and Hd∩dΛ have Lefschetz decom- position: k r k−2r H∗ (M) = ⊕rL PH∗ (M)
When we consider other symplectic cohomologies, Theorem 3.5.5 implies that
Theorem 3.5.6. [47] The cohomologies Hd+dΛ ,HddΛ and Hd∩dΛ satisfies the strong Lefschetz properties: k n−k ' n+k L : H∗ (M) → H∗ (M)
According to this theorem, it is natural to expect that these cohomologies can detect properties of the symplectic structure related to the strong Lefschetz property. If strong Lefschetz property does not hold, we have the following lemma
k n−k+1 Lemma 3.5.7. [α] ∈ ker L if and only if there exist ϕ ∈ ∂−P ∩ ker d such that [ϕ] = [α].
2 Proof. It is obvious for k = 0 and we assume k > 0. Let α = α0 +Lα1 +L α2 +··· , αi ∈ Pn−k−2i, be the Lefschetz decomposition of α. Lk([α]) = 0 implies that there exists 2 n−k+1−2i n−k−1 n−k η = η0 + Lη1 + L η2 + ··· , ηi ∈ P such that dL η = L α. We define 0 2 0 η := η1 + Lη2 + L η3 + ··· and ϕ = α − dη . Then [ϕ] = [α] and it is easy to show that
ϕ = α0 − ∂+η1 = ∂−η0. n−k+1 k k k−1 Conversely, if α = ∂−η for some η ∈ P , then L α = L ∂−η = dL η is exact.
The main result of this section is
k n−k Theorem 3.5.8. Let (M, ω) be a symplectic manifold. Then dimHd∩dΛ (M) ≤rankL .
Proof. Define
˜ k k H (M) := {α ∈ Hd (M)|α has a harmonic representative}
Then ker(d + dΛ) ∩ Ωk(M) H˜ k(M) = Imd ∩ ker(d + dΛ) ∩ Ωk(M) 37 If dα ∈ Imd ∩ ker(d + dΛ) ∩ Ωk(M), then 0 = (d + dΛ)dα = ddΛα and α ∈ Ω˜ k−1. So there is a surjective map ˜ k k p : H (M) −→ Hd∩dΛ (M). Moreover, Lemma 3.5.7 implies that ker Ln−k ⊂ H˜ k(M) and there is a projection
π : H˜ k(M) −→ H˜ k(M)/ ker Ln−k.
For α ∈ ker Ln−k, by Lemma 3.5.7, we can find ϕ ∈ Pk and η ∈ Pk+1 such that [ϕ] = α k+1 and ∂−η = ϕ. So η ∈ Ω˜ (M) and p(α) = 0. Hence there is a map
˜ k n−k k h : H (M)/ ker L −→ Hd∩dΛ (M) such that p = h ◦ π. p is surjective implies that h is also surjective and
k ˜ k n−k dimHd∩dΛ (M) ≤ dim(H (M)/ ker L )
So
k ˜ k n−k k n−k n−k dimHd∩dΛ (M) ≤ dim(H (M)/ ker L ) ≤ dim(H (M)/ ker L ) = rankL .
This lower bound is sharp for 4-manifolds:
k n−k Corollary 3.5.9. Let (M, ω) be a symplectic 4-manifold. Then dimHd∩dΛ (M) =rankL .
Proof. It is obvious for L2. When k = 1, ker d ∩ Ω1 = ker(d + dΛ) ∩ Ω1 implies that H1(M) = H˜ 1(M). It is clear that [dΩ˜ 0] = 0 in H˜ 1(M) and [dΛΩ˜ 2] ⊂ ker L1. So h is injective and
1 ˜ 1 1 1 1 1 dimHd∩dΛ (M) = dimH (M)/ ker L = dimH (M)/ ker L = rankL .
To prove it for L0, we first note that
Λ 2 2 ker(d + d ) ∩ Ω (M) = ∂−P ⊕ Rω and (ker d ∩ − ker(d + dΛ)) ∩ Ω2(M) = S where
2 0 S = {α0 + Lα1|α0 ∈ P , α1 ∈ P , ∂−α0 = −∂+α1 6= 0} 38 1 0 0 ˜ 1 ˜ 3 Recall that ∂−Ω = Ω − Rω and Rω = ker d ∩ Ω . So Ω = 0, Ω = 0. and
1 ∼ 2 Hd∩dΛ (M) = ∂−P ⊕ Rω.
In the composition of maps
p p ker(d + dΛ) ∩ Ω2 −→1 ker d ∩ Ω2 −→2 H2(M),
1 p1 is injective and p2 is surjective. Because dΩ ⊂ S, p2 ◦ p1 is injective. For any 1 α0 + Lα1 ∈ S, α1 = ∂−γ for some γ ∈ Ω . Then
2 [α0 + Lα1] = [α0 + Lα1 − dγ] = [α0 − ∂+γ], α0 − ∂+γ ∈ ker ∂−P .
1 1 Hence p2 ◦p1 is surjective and gives an isomorphism between Hd∩dΛ (M) and H (M). Chapter 4
Symplectic-de Rham spectral sequence
The main objective of this chapter is to focus on spectral sequences. Spectral sequences was first introduced by Leray in 40’s. Since then it has become an important tool in computing homology or cohomology of geometric or algebraic objects. In Section 4.1, we briefly review the theory of spectral sequences and some basic properties. In complex geometry, Fr¨olicher([19]) discovered a spectral sequence for any complex manifold which relates the Dolbeault cohomology of complex structures to the de Rham ∗,∗ cohomology of smooth manifolds. More precisely, the Fr¨olicher spectral sequence E∗ is created by the double complex (Ω∗(M), ∂, ∂¯) for a complex manifold M. It is well p,q p q ∗,∗ known that E1 is isomorphic to H (M, Ω ) and the spectral sequence Er converges ∗ ∗,∗ to H (M, R). The convergence of Er shows that the de Rham cohomology can be decomposed as the direct sum of the Dolbeault cohomology of a complex structure. ∗,∗ One special feature is the degeneration of Er in K¨ahlermanifolds. If M is K¨ahler, the Fr¨olicher spectral sequence degenerates at E1. This is proven by using the Hodge decomposition of differential forms. The other important property of K¨ahlermanifolds is the ∂∂¯-lemma. It states that a ∂-closed and ∂¯-exact form is also ∂∂¯-exact. We believe that the property can be described purely algebraically in the frame of spectral sequences, which is the main topic in Section 4.2. Using the operator d and dΛ, Brylinski defined the canonical spectral sequence for
39 40 any manifold with Poisson structures. He showed that this spectral sequence degener- ates at E1 when the manifold is symplectic. In Section 4.3, we construct a new spectral sequence for symplectic manifolds and show that it has more subtle degeneration prop- erties than the canonical spectral sequence. In Section 4.4, we analyze the spectral sequence of the Kodaira-Thurston manifold and a symplectic 6dimensional nilmanifold.
4.1 Spectral sequences and degeneration
In this section, we review the theory of spectral sequences and some of their properties.
Definition 4.1.1. A spectral sequence is a collection of differential bi-graded modules ∗,∗ ∗,∗ p,q ∗,∗ (Er , dr) such that dr is of degree (r, 1 − r) and Er+1 is isomorphic to H (E , dr). The most general approach to construct a spectral sequence is the method of exact couples. For our purpose, we will only consider the spectral sequences which arise from filtered complexes.
Definition 4.1.2. A filtered differential graded module is a N-graded module A = ∞ n ⊕n=0A over a field k, endowed with a filtration F and a linear map D : A → A satisfying
1. d is of degree 1: d(An) ⊂ An+1;
2. d ◦ d = 0;
3. The filtered structure is descending:
A = F 0A ⊇ F 1A ⊇ · · · ⊇ F pA ⊇ F p+1A ⊇ · · · ;
4. The map d preserves the filtered structure: d(F pA) ⊂ F pA for any p.
t(n) n ∗ Furthermore, if there exists a function t : N → N such that F A = 0, A is called a bounded filtered differential graded module.
∗,∗ Theorem 4.1.3. [36] (A, d, F ) defines a spectral sequence {Er , dr} with dr of bi-degree (r, 1 − r). Moreover, s,t ∼ s+t s s+1 E1 = H (F A/F A) s,t ∗ and Er converges to H (A, d) if (A, d, F ) is bounded. 41 More precisely, we have
s,t s s+t s+r s+t+1 Zr = {ϕ ∈ F A |dϕ ∈ F A } s,t s s+t s−r s+t−1 Br = F A ∩ dF A s,t s s+t Z∞ = F A ∩ ker d s,t s s+t B∞ = F A ∩ Imd s,t s,t Zr Er = s+1,t−1 s,t Zr−1 + Br−1 s+1,t−1 s+1 s+t s,t s+1 s+t+1 s,t s+r,t−r+1 (let Z−1 = F A and B−1 = dF A ) and dr : Er → Er is s,t s+r,t−r+1 induced by d : Zr → Zr .
s,t Example 4.1.4 (double complexes). Let A = ⊕A be a bi-graded C-(or R-)module k s,t and A = ⊕s+t=kA . A filtration on A is given as X F pAq = {ϕ = ϕs,t|ϕs,t = 0 for s < p} s+t=q Assume there are two maps d0 : As,t → As+1,t and d00 : As,t → As,t+1 satisfying d02 = d002 = 0 and d0d00 + d00d0 = 0. The double complex (A, d0, d00) defines a filtered complex 0 00 s,t (A, d + d ,F ) and the corresponding spectral sequence {Er }.
* d00 ↑ d0 α1 → * d00 ↑ d0 α2 → *
s,t 0 00 In the rest of this section, we assume {Er } is given by a double complex (A, d , d ). s,t We are interested in the degeneration of Er . In particular, we want to explore the s,t s,t relation between Er and Er+1. Consider the natural inclusions
s,t s,t s,t s,t s,t s,t B0 ⊂ B1 ⊂ · · · ⊂ B∞ ⊂ Z∞ ⊂ · · · ⊂ Z1 ⊂ Z0 , s,t s,t the inclusion map Zr+1 → Zr induces a homomorphism s,t s,t Zr+1 Zr αs,t,r : s+1,t−1 → s+1,t−1 Zr Zr−1 42
Lemma 4.1.5. (1) α = αs,t,r is injective.
(2) If α is not surjective, then there exists
s,t s+i,t−i ξ = ξ0 + ξ1 + ··· + ξr−1 ∈ Zr , ξi ∈ A
such that
(a) ξ0 6= 0;
(b) ξ0 is not the leading term of any d-closed form; s+r,t−r 00 0 (c) ξ can not extend to the right, i.e. @ξr ∈ A , d ξr = d ξr−1.
0 Remark 4.1.6. 1. (b) implies that d ξi 6= 0, i = 0, 1, ··· , r − 1.
2. Since r − 1 ≥ 0, we automatically assume r ≥ 1.
s s+t s+i,t−i Proof. Let ξ = ξ0 + ξ1 + · · · ∈ F A , ξi ∈ A .
s,t 0 s+r+1,t−r (1) Assume ξ ∈ Zr+1. So d(ξ0 + ··· + ξr) = d ξr ∈ A . If α([ξ]) = 0, then s+1,t−1 0 s+r,t−r+1 ξ ∈ Zr−1 . It means that ξ0 = 0 and d(ξ0 + ··· + ξr−1) = d ξr−1 ∈ A . s,t The second condition is true for any ξ ∈ Zr+1. The first condition implies that s+1,t−1 ξ ∈ Zr and [ξ] = 0.
(2) Consider the following decomposition of α:
s,t s,t s,t Zr+1 α1 Zr α2 Zr s+1,t−1 −→ s+1,t−1 −→ s+1,t−1 Zr Zr Zr−1
It is clear that α1 is injective and α2 is surjective. If α is not surjective, α1 is not s,t s+1,t−1 surjective and there exists ξ ∈ Zr such that [ξ] = (ξ mod Zr ), α2([ξ]) ∈/ s,t Zr Imα. Since [ξ] = [ξ0 + ··· + ξr] in s+1,t−1 , we can assume ξ = ξ0 + ··· + ξr. Zr First note that [ξ] 6= 0 and ξ0 6= 0. α2([ξ]) ∈/ Imα implies that [ξ] ∈/ Imα1, s,t which is equivalent to the condition that ξ∈ / Zr+1 and dξ has a nonzero term at As+r,t−r+1, i.e.
0 00 0 0 00 dξ = (d ξr−1 + d ξr) + d ξr, d ξr−1 + d ξr 6= 0
0 00 0 0 s+r,t−r If d ξr−1 = −d ξr for some ξr ∈ A , then
0 0 00 0 0 0 s+r+1,t−r d(ξ − ξr + ξr) = d ξr−1 + d ξr + d ξr = d ξr ∈ A 43 0 s,t 0 0 s,t s+1,t−1 and ξ − ξr + ξr ∈ Zr+1. So ξ = (ξ − ξr + ξr) + (ξr − ξr) ∈ Zr+1 + Zr−1 and
0 0 0 0 α2([ξ]) = α2([ξ − ξr + ξr]) + α2([ξr − ξr]) = α2([ξ − ξr + ξr]) = α([ξ − ξr + ξr])
which is a contradiction.
0 0 0 s s+t s+i,t−i 0 If there exists a d-closed form ξ = ξ0 + ξ1 · · · ∈ F A , ξi ∈ A with ξ0 = ξ0, 0 0 0 s+1.t−1 0 s,t then ξ = (ξ − ξ ) + ξ , ξ − ξ ∈ Zr−1 , ξ ∈ Zr+1. So
0 0 0 0 α2([ξ]) = α2([ξ − ξ ]) + α2([ξ ]) = α2([ξ ]) = α([ξ ]),
which is a contradiction again. Hence ξ satisfies conditions (a)-(c).
The converse of Lemma 4.1.5(2) may not be true, i.e. a homogeneous form ξ = ξ0 + s,t s+1,t−1 ···+ξr−1 ∈ Zr satisfying the conditions (a)-(c) does not imply that (ξ mod Zr−1 ) ∈/ 0 0 0 s,t 0 s+1,t−1 Imα. Suppose ξ = ξ0 + ξ1 + · · · ∈ Zr+1 and α([ξ ]) = (ξ mod Zr−1 ). Let ¯ 0 0 ξ = ξ − (ξ0 + ··· + ξr−1). Then
¯ 0 0 0 dξ = d ξr−1 − d ξr−1.
0 0 00 0 ¯ Since d ξr−1 = −d ξr, ξ can not extend to the right if and only if ξ can not extend to 0 s+1,t−1 0 the right. α([ξ ]) = (ξ mod Zr−1 ) implies that ξ0 = ξ0. So the first nonzero term ¯ 0 0 ¯ of ξ is ξi − ξi 6= 0 for some i > 0 (such term exists since ξr−1 − ξr−1 6= 0) and ξ can not extend to the right, i.e. ξ¯ satisfies (a)(c) (with different indexes s,t,r). If the leading term of ξ¯ is the leading term of some closed form ξ00, we can consider ξ¯− ξ00 and have a ¯ s+i,t−i larger index i. From (c), this process will stop at some i ≤ r −1. Note that ξ ∈ Zr−i . In summary,
s,t Lemma 4.1.7. If ξ = ξ0 + ··· + ξr−1 ∈ Zr satisfying the conditions in (2), then for some r − 1 ≥ i ≥ 0, the map
s+i,t−i s+i,t−i Zr−i+1 Zr−i αs+i,t−i,r−i : s+i+1,t−i−1 → s+i+1,t−i−1 Zr−i Zr−i−1 is not surjective.
Such forms also can be used to detect the surjectivity of the inclusion of boundaries: 44 Lemma 4.1.8. If
s,t s+i,t−i ξ = ξ0 + ξ1 + ··· + ξr−1 ∈ Zr , ξi ∈ A such that
(a) ξ0 6= 0;
(b) ξ is not the leading term of any d-closed form;
s+r,t−r+1 s+r,t−r+1 then the inclusion map βs+r,t−r+1,r : Br−1 → Br is not surjective.
0 s+r,t−r+1 s+r,t−r+1 0 s+r,t−r+1 Proof. dξ = d ξr−1 ∈ A ∩Br by definition. Assume d ξr−1 ∈ Br−1 , 0 0 0 0 s+i,t−i 0 0 0 then there exists ξ = ξ1 + ξ2 + ··· , ξi ∈ A such that dξ = d ξr−1. So ξ − ξ = 0 0 0 ξ0 + (ξ1 − ξ1) + ··· and d(ξ − ξ ) = 0. Hence ξ − ξ is a d-closed form whose leading term is the same as ξ, which is a contradiction.
p+q p+i,q−i Definition 4.1.9. Let η = η0 +η1 +η2 +···+ηr ∈ A , ηi ∈ A . η is call unstable indecomposable if
00 0 00 1. d η0 = 0, d ηi−1 = −d ηi 6= 0, 1 ≤ i ≤ r
0 p+r+1,q−r p,q p+r+1,q−r In other words, dη = d ηr ∈ A (so η ∈ Zr+1 and dη ∈ Br+1 ).
2. η0 is not the leading term of any d-closed form.
Moreover, η is called maximal if η + ηr+1 is not unstable indecomposable for any ηr+1 6= 0 ∈ Ap+r,q−r. 45
0 ↑ d0 ξ0 → ∗ ↑ d0 ξ1 → ∗ . .. ↑ d00 0 d 0 ξr−1 → d ξr−1
These lemmas show that the existence of unstable indecomposable elements can detect when the spectral sequence degenerates.
Theorem 4.1.10. The spectral sequence induced by a double complex (A, d0, d00) degen- erates at E1 if and only if there are no unstable indecomposable elements.
Proof. Assume there exists an unstable indecomposable element
s,t s+i,t−i ξ = ξ0 + ξ1 + ··· + ξr−1 ∈ Zr , ξi ∈ A
s+i,t−i By Lemma 4.1.7, αs+i,t−i,r−i is not surjective for some r − 1 ≥ i ≥ 0. Hence Er−i+1 s+i,t−i Er−i and 1 ≤ r − i ≤ r. Conversely, if Er Er+1, then either α or β is not surjective. If αs,t,r is not surjective, the existence of unstable indecomposable elements is proven by Lemma 4.1.5. s−r+i,t+(r−1)−i When βs,t,r is not surjective, there exists ξ = ξ0 + ξ1 + ··· , ξi ∈ A such s s+t s,t that dξ ∈ F A and dξ∈ / Br−1. ξ0 6= 0 by assumption. If ξ0 is the leading term 0 0 0 0 s,t of a closed form ξ = ξ0 + ξ1 + ··· , then d(ξ − ξ ) = dξ and dξ ∈ Br−1, which is a contradiction. So ξ is an unstable indecomposable element. 46 4.2 d0d00-lemma
In Chapter 2, we discuss the concepts of ∂+∂−-lemma and primitive harmonic forms for symplectic manifolds. They can be generalized to any double complex.
Definition 4.2.1. A double complex (A, d0, d00) satisfies the d0d00-lemma if
Imd0 ∩ ker d00 = Imd00 ∩ ker d0 = Imd0d00
Definition 4.2.2. An element ϕ ∈ Ap,q is called harmonic if d0ϕ = d00ϕ = 0.
The existence of harmonic forms can be characterized by the d0d00-lemma:
Lemma 4.2.3. If (A, d0, d00) satisfies d0d00-lemma and D = d0 + d00, then each class in
HD(A) has a harmonic representative.
Pl n i,n−i Proof. Assume α = i=k αi ∈ A , αi ∈ A is a D-closed form, i.e.
00 0 00 0 00 0 d αk = 0, d αk = −d αk+1, ··· , d αl−1 = −d αl, d αl = 0
0 00 i+1,n−i 0 00 Let εi = d αi = d αi+1 ∈ A , i = k, k + 1, ··· , l − 1. Then εi ∈ Imd ∩ ker d ∩ 00 0 0 00 0 00 i,n−i−1 0 Imd ∩ ker d . By d d -lemma, εi = d d γi for some γi ∈ A . Consider α = Pl−1 Pl−1 0 α − D i=k γi = i=k(αi − Dγi) + αl. It is clear that [α] = [α ] ∈ HD(A). Moreover,
0 0 0 00 d (αi − Dγi) = d αi − d d γi = εi − εi = 0 0 d αl = 0 00 d αk − Dγk = 0 + εk 00 00 00 0 d (αi − Dγi) = d αi − d d γi = −εi−1 + εi 00 d αl = −εl−1
So d0α0 = 0 and
00 0 d α = εk + (−εk + εk+1) + (−εk+1 + εk+2) + ··· + (−εl−2 + εl−1) − εl−1 = 0
Hence α0 is a harmonic form in the class [α].
Theorem 4.2.4. If A satisfies d0d00-lemma, then the spectral sequence of A degenerates at E1. 47 p,q 00 Proof. It is enough to show that d1 = 0. Assume α ∈ A is a d -closed element. We 0 00 0 0 00 0 0 00 want to show that d1[α] = [d α] = 0. Since d d α = −d d α = 0, d α ∈ Imd ∩ ker d . 0 00 0 0 00 00 By d d -lemma, d α = d d γ for some γ. So [α] = [α − d γ] ∈ Hd00 (A) and d1[α] = [d0(α − d00γ)] = 0.
4.3 Symplectic-de Rham spectral sequence
Using the Lefschetz decomposition, we can construct a spectral sequence similar to the Fr¨olicher spectral sequence for complex manifolds. Let As,t = Ls,t−s, i.e. s = p, t = p+q. Ω∗(M) is a graded algebra:
k s,t s,k−2s p,q k A = ⊕s+t=kA = ⊕s+t=kL = ⊕2p+q=kL = Ω (M) A filtration on Ω∗(M) is given as X F xA = {ϕ = ϕs,t|ϕs,t = 0 for s ≥ x}
s,t s,t+1 s+1,t The map d = ∂+ + L∂− : A → A ⊕ A is a differential operator with degree 1.
s,t Theorem 4.3.1. The spectral sequence defined by the double complex (A , L∂−, ∂+) satisfies: s,t s,t E0 = A Es,t = Hs,t−s(M) := Ls(PHt−s(M)) 1 ∂+ ∂+ In particular, 0,t ∼ 1,t+1 ∼ ∼ n−t,n E0 = E0 = ··· = E0 and 0,t ∼ 1,t+1 ∼ ∼ n−t−1,n−1 E1 = E1 = ··· = E1 Proof. This is an easy consequence of the definition.
L0,n L1,n L2,n ···Ln,0 L0,n−1 L1,n−1 ···Ln−1,0 L0,n−2 ··· . . L0,0 48 In [12], Brylinski constructed the canonical complexes for Poisson manifolds and showed that the induced spectral sequence degenerates at E1 for symplectic manifolds. By Theorem 4.2.4, we already see that the symplectic-de Rham spectral sequence have different properties from the canonical spectral sequence of Brylinski. When M is a 4-manifold, the degeneration is completely determined by the d0d00-lemma or the strong Lefschetz property.
Proposition 4.3.2. For a symplectic 4-manifold M, the strong Lefschetz property holds if and only if the symplectic-de Rham spectral sequence degenerates at E1.
Proof. If M satisfies strong Lefschetz property, the ∂+∂−-lemma also holds by Theorem
3.3.1, which implies that the symplectic-de Rham spectral sequence degenerates at E1 by Theorem 4.2.4. Conversely, assume the strong Lefschetz fails and there is a 1-form η ∈ P1 such 1 2 0 that [η] 6= 0 ∈ H (M) but [Lη] = 0 ∈ H3(M). So there exist ϕ2 ∈ P , ϕ0 ∈ P with
Lη = d(ϕ2 + Lϕ0) = L(∂−ϕ2 + ∂+ϕ0). We may replace η by η − ∂+ϕ0 and assume 0 ϕ = ∂−ϕ2. Since ∂+ϕ2 = 0 and η∈ / ∂+P , ϕ2 is an unstable indecomposable element.
So the spectral sequence does not degenerate at E1 by Theorem 4.1.10.
For higher dimensional symplectic manifolds, we hope that the ∂+∂−-lemma is also equivalent to the E1-degeneration of the symplectic-de Rham spectrl sequence.
4.4 Examples
In general, the computation of de Rham cohomology is very complicated. We usually compute other cohomology and use equivalent conditions or dualities to show that they are isomorphic to each other. For nilmanifolds, Nomizu’s theorem shows that their de Rham cohomology is isomorphic to the Chevalley-Eilenberg cohomology of their corresponding nilpotent Lie algebra. In this section, we consider two nilmanifolds and analyze some cohomologies and symplectic-de Rham spectral sequence given by the Chevalley-Eilenberg complex. In our cases, they are equivalent to the corresponding cohomologies and symplectic-de Rham spectral sequence of the manifolds. For general nilmanifolds, we need to find a Nomizu type theorem for these cohomologies. It will be discussed elsewhere. 49 4.4.1 Four dimensional nilpotent Lie algebra
Consider the nilpotent Lie algebra ch3 in [41]. It is given by the formulas
de1 = de2 = de4 = 0, de3 = e14
So de23 = −e124 and the symplectic-de Rham spectral sequence for ch3 is
2 2 e13, e14, e23, e24, σ Le1, Le2, Le3, Le4 L 1 = ω E0 : e1, e2, e3, e4 L1 = ω 1
2 2 e13, e23, e24, σ Le1, Le2, Le3, Le4 L 1 = ω E1 : e1, e2, e4 L1 = ω 1
2 2 e13, e24, σ Le1, Le2, Le3 L 1 = ω E2 : e1, e2, e4 L1 = ω 1
4.4.2 Six-dimensional nilmanifolds
The fine properties of spectral sequence can be seen in high dimension. We consider the same 6-dimensional example mentioned in [48]. g is the Lie algebra satisfying
de1 = de3 = de6 = 0
de4 = e15
de5 = e13
de2 = e14 + e35 + e36
P2 is generated by
e13, e14, e15, e16, e23, e24, e25, e26, e35, e36, e45, e46, σ1 = e12 − e34, σ2 = e34 − e56 50 P3 is generated by e135, e136, e145, e146, e235, e236, e245, e246, σ1∧e5, σ1∧e6, σ2∧e1, σ2∧e2, (σ1+σ2)∧e3, (σ1+σ2)∧e4
The differentials are
de13 = de14 = de15 = de16 = de35 = de36 = 0 1 1 de23 = −e134 = − 2 σ2 ∧ e1 − L( 2 e1) 1 1 de24 = e125 − e345 − e346 = σ1 ∧ (e5 + 2 e6) − L( 2 e6) de25 = e123 + e145 − e356 = e145 + (σ1 + σ2) ∧ e3 1 1 de26 = e146 + e356 = e146 − 2 (σ1 + σ2) ∧ e3 + L( 2 e3) 1 1 de45 = −e134 = − 2 σ2 ∧ e1 − L( 2 e1) 1 1 de46 = e156 = − 2 σ2 ∧ e1 + L( 2 e1) dσ1 = −2e135 − e136
dσ2 = e135 − e136
de135 = de136 = de145 = de146 = d((σ1 + σ2) ∧ e3) = d(σ2 ∧ e1) = 0
de235 = e1235 − e1345 = L(e35 − e15)
de236 = −e1346 = L(−e16)
de245 = −e1234 + e3456 = L(−σ1 − σ2)
de246 = e1256 − e3456 = L(σ1)
dσ1 ∧ e5 = e1356 = L(e13)
dσ1 ∧ e6 = −2e1356 = L(−2e13)
dσ2 ∧ e2 = e1235 − e1236 = L(e35 − e36)
d(σ1 + σ2) ∧ e4 = e1345 + 2e1346 = L(e15 + 2e16)
Hence we have 0,1 1 E1 = PH+ =< [e1], [e3], [e6] >
0,2 2 E1 = PH+ =< [e16], [e14 − e36], [e35 − e36], [e23 − e45], [e45 − e46] >
0,3 3 E1 = PH+ =< [e235], [e236], [e245], [e246], [σ1∧e5], [σ2∧e2], [(σ1+σ2)∧e3], [(σ1+σ2)∧e4] >
1 (Note that [σ1 ∧ e5] = − 2 [σ1 ∧ e6], [(σ1 + σ2) ∧ e3] = −[e145] = 2[e146].)
1,3 2 E1 = L(PH+) ⊕ L < [e23 + e46], [e24], [e25], [e26], [σ1], [σ2] > 51
For E2, we have
0,3 1,2 Z2 /Z1 =< e135, e136, e145, e146, (σ1+σ2)∧e3, σ2∧e1, σ1∧(2e5+e6), Le4−(σ1+σ2)∧e4−2e236 > 1 1 B0,3 =< e , e , Le , σ ∧e , σ ∧(2e +e )−Le , e +(σ +σ )∧e , e − (σ +σ )∧e +L( e ) > 1 135 136 1 2 1 1 5 6 6 145 1 2 3 146 2 1 2 3 2 3 0,3 ⇒ E2 = [e145, [σ1 ∧ (2e5 + e6)], [Le4 − (σ1 + σ2) ∧ e4 − 2e236] >
1,3 2,2 Z2 /Z1 = L < e13, e14, e15, e16, e25, e35, e36, e23 − e45, e45 + e46, σ1, σ2 >
1,3 B2 = L < e13, e15, e16, e35, e36, σ1, σ2 > 1,3 E2 = L < [e14], [e25], [e23 − e45], [e45 + e46] > 0,2 E2 =< e14, e16, e35, e36, e23 − e45 > /(e14 + e35 + e36)
2,3 2 E2 = L < [e2], [e4], [e5] >
The degrees of Er are:
14 14 6 1
E0 : 14 6 1 6 1 1
8 11 6 1
E1 : 5 3 1 3 1 1
3 4 3 1
E2 = E∞ : 4 3 1 3 1 1 52 If we consider the other nonsymplectomorphic symplectic structure, we can change the basis as (1, 2, 3, 4, 5, 6) → (1, 4, 3, 5, 6, 2) and have
de1 = de2 = de3 = 0
de4 = e15 − e23 + e36
de5 = e16
de6 = e13 The differentials are
de13 = de15 = de16 = de23 = de36 = 0 1 1 de14 = e123 − e136 = −e136 + 2 (σ1 + σ2) ∧ e3 + L( 2 e3) 1 1 de24 = e125 − e236 = −e236 + 2 σ1 ∧ e5 + L( 2 e5) 1 1 de25 = e126 = 2 σ1 ∧ e6 + L( 2 e6) 1 1 de26 = e123 = 2 (σ1 + σ2) ∧ e3 + L( 2 e3) de35 = e136 1 1 de = e − e − e = e − e + (σ + σ ) ∧ e − L( e ) 45 146 356 235 146 235 2 1 2 3 2 3
de46 = −e134 + e156 − e236 = −e236 − σ2 ∧ e1
dσ1 = −e135
dσ2 = 2e135
ϕ0de135 = de136 = de236 = d(σ2 ∧ e1) = d((σ1 + σ2) ∧ e3) = 0
de145 = e1235 + e1356 = L(e35 + e13)
de146 = e1236 = L(e36)
de235 = e1236 = L(e36)
de245 = e1246 − e2356 = L(e46 − e23)
de246 = e1256 − e1234 = L(−σ2)
dσ1 ∧ e5 = −e1346 = L(−e16)
dσ1 ∧ e6 = −e1356 = L(−e13)
dσ2 ∧ e2 = 2e1235 = L(2e35)
d(σ1 + σ2) ∧ e4 = −e1345 + e1236 + e2356 = L(−e15 + e36 + e23) Hence we have 0,1 1 E1 = PH+ =< [e1], [e2], [e3] > 53 0,2 2 E1 = PH+ =< [e15 + e23], [e23 + e36], [e14 + e35 − e26], [2σ1 + σ2] >
0,3 3 E1 = PH+ =< [e145], [e146], [e236], [e245], [e246], [σ2 ∧ e2], [(σ1 + σ2) ∧ e4] >
1 (Note that [e146] = [e235], [e236] = 2 [σ1 ∧ e5] = −[σ2 ∧ e1].)
1,3 2 E1 = L(PH+) ⊕ L < [e14], [e24], [e25], [e35], [e45], [e46], [σ1 − σ2] >
For E2, we have
0,3 1,2 Z2 /Z1 = < e135, e136, e236, e146 − e235, (σ1 + σ2) ∧ e3, σ2 ∧ e1, σ1 ∧ e5 + Le5,
σ1 ∧ e6 + Le6, σ2 ∧ e2 − 2e145 + Le6, (σ1 + σ2) ∧ e4 − 2e146 + Le4 > 1 1 1 1 B0,3 = < e , e , −e + σ ∧ e + L( e ), σ ∧ e + L( e ), 1 135 136 236 2 1 5 2 5 2 1 6 2 6 1 1 (σ + σ ) ∧ e + L( e ), e − e + (σ + σ ) ∧ e , −e − σ ∧ e > 2 1 2 3 2 3 146 235 1 2 3 236 2 1
0,3 ⇒ E2 =< [e236], [σ2 ∧ e2 − 2e145 + Le6], [(σ1 + σ2) ∧ e4 − 2e146 + Le4] >
1,3 2,2 Z2 /Z1 = L < e13, e14 − e26, e14 + e25, e15, e16, e23, e35, e36, e46, σ1, σ2 >
1,3 B2 = L < e13, e16, e35, e36, e46 − e23, e15 − e23, σ2 > 1,3 E2 = L < [e15 + e23 + e46], [e14 − e26], [e14 + e45], [σ1] > 0,2 E2 =< e15, e23, e36, e14 − e26 + e35, 2σ1 + σ2 > /(e15 − e23 + e36)
2,3 2 E2 = L < [e1], [e2], [e4] >
The degrees of Er are:
14 14 6 1
E0 : 14 6 1 6 1 1 54
7 11 6 1
E1 : 4 3 1 3 1 1
3 4 3 1
E2 = E∞ : 4 3 1 3 1 1 Using this example, we can observe that the information revealed by the symplectic- de Rham spectral sequence depends on the choice of symplectic structure. We believe that
Conjecture 4.4.1. If ω and ω0 are symplectic structures of M, then the symplectic-de Rham spectral sequence for (M, ω) and (M, ω0) will degenerate at the same step. Chapter 5
Luttinger surgery and Kodaira dimension
The main focus of this chapter is on Kodaira dimension and Luttinger surgery. The concept of symplectic Kodaira dimension is reviewed in Section 5.1. We also briefly discuss the classification problems for each Kodaira dimension. In Section 5.2, we define the Luttinger surgery and show some applications. In Section 5.3, we prove the main theorem of this chapter, which shows that Luttinger surgery preserves the symplectic Kodaira dimension. In Section 5.4, we use the classification of symplectic manifolds for each Kodaira dimension to discuss the effect of Luttinger surgery.
5.1 Symplectic Kodaira dimension
Let (M, ω) be a symplectic manifold. An almost complex structure on M is a linear map J ∈ TM ⊗ T ∗M such that J 2 = −1. An almost complex structure J on symplectic
4-manifold (M, ω) is tamed by ω if ω(v, Jv) > 0 for any x ∈ M and v 6= 0 ∈ TxM. J is compatible with ω if it is tamed by ω and satisfies ω(Ju, Jv) = ω(u, v) for any u, v ∈ TxM. Since the space J (M) of ω-compatible almost complex structures is nonempty and contractible, we can define the first Chern class c1(M, w) as c1(M,J) by choosing any J ∈ J (M). A codimension 2 smooth submanifold N of (M, ω) is a symplectic submanifold if (N, ω|N ) is also a symplectic manifold. A symplectic 4- manifold M is called symplectically minimal if there is no symplectic sphere N embedded
55 56 in M such that N 2 = −1. If M is not symplectically minimal and S is an embedded symplectic −1-sphere in M, it is shown in [37] that M can be blown down along S to get another symplectic manifold. Since the procedure of blowing down reduces b2 by 1, a minimal symplectic 4-manifold M 0 will be achieved in finite many steps. M 0 is called a symplectic minimal model of M. If (M, ω) is minimal, the Kodaira dimension of M is defined by
2 −∞ K < 0 or Kω · ω < 0 ω 2 0 K = 0 and Kω · ω = 0 κ(M, ω) = ω 2 1 K = 0 and Kω · ω > 0 ω 2 2 Kω > 0 and Kω · ω > 0 In general, the Kodaira dimension of a non-minimal symplectic manifold is defined as the Kodaira dimension of its minimal model.
2 Remark 5.1.1. 1. The case Kω > 0, Kω · ω = 0 is not contained in the list. From 2 Lemma 2.5 in [31], if (M, ω) is minimal with K · ω = 0 and Kω ≥ 0, then Kω is 2 a torsion class and hence Kω = 0. We can also observe that Kω is a torsion class 2 implies that Kω = 0 and Kω · ω = 0. Hence a symplectic 4-manifold M has κ = 0 exactly when its canonical class is a torsion class.
2. According to [31], the Kodaira dimension of minimal model is well defined. Ac- tually, the minimal model is uniquely determined up to diffeomorphism when κ(M, ω) ≥ 0. If M has minimal model diffeomorphic to rational or ruled sur- faces, the minimal model is not unique. But they still have the same κ.
3. It is also shown in [31] that κ(M, ω) is independent of choice of symplectic form ω.
4. If ω is a K¨ahlerform on M, the Kodaira dimension of M as a complex surface coincides with the Kodaira dimension κ(M, ω) of M for symplectic 4-manifolds.
5.1.1 κ = −∞
A rational symplectic 4-manifold is a symplectic 4-manifold whose underlying smooth 2 2 2 manifold is S × S or CP ]kCP2, k ≥ 0. A ruled sympelctic 4-manifold is a symplectic 57 4-manifold whose underlying smooth 4-manifold is either a sphere bundle over Riemann surface Σg, g ≥ 1 or blowups of such manifolds.
Theorem 5.1.2 ([33]). A symplectic 4-manifold (M, ω) has κ = −∞ if and only if it is rational or ruled.
There are some other characterization of such manifolds.
Theorem 5.1.3 ([37]). A symplectic 4-manifold (M, ω) is rational or ruled if and only if M has an embedded symplectic sphere with non-negative square.
5.1.2 κ = 0
As we have mentioned, a minimal symplectic 4-manifold (M, ω) has κ = 0 if and only if
Kω is a torsion class. There are some general properties of such symplectic manifolds.
Theorem 5.1.4 ([30]). 1. 2χ + 3σ = 0
2. M has even intersection form.
3. Kω is either trivial, or of order two, which only occurs when M is an integral homology Enriques surface.
4. M is spin except when M is an integral homology Enriques surface.
K¨ahlersurfaces with κ = 0, which also have (holomorphic) Kodaira dimension 0, have been classified: K3 surfaces, Enriques surfaces, hyperelliptic surfaces and torus T 4. The first non-K¨ahlersymplectic manifold, the Kodaira-Thurston manifold, also has κ = 0. It is an example of T 2 bundle over T 2. In fact, we know that
Lemma 5.1.5. Every oriented T 2 bundle over T 2 admits a symplectic form ω such that
Kω is a torsion class. Moreover, all such manifolds are minimal and with κ = 0.
Since M is a K(π, 1) manifold, there is no homotopy nontrivial sphere in M. Hence M is minimal. The existence of symplectic structures is given in [20]. If the fiber is homology essential, Thurston’s construction gives rise to a symplectic form such that M is a symplectic fibration. On the other hand, if the fundamental class of fibers is zero in 1 H2(M, Z), M is a total space of S bundle over a 3-manifold which in turn is a principal S1 bundle over T 2, and thus admits a symplectic structure by the construction in [15]. 58 No other minimal symplectic structure 4-manifolds are constructed so far. We be- lieve that the following conjecture is true.
Conjecture 5.1.6. The only minimal symplectic 4-manifolds with κ = 0 are K3 sur- faces, Enriques surfaces and T 2 bundles over T 2.
Although we can not answer the question now, some topological properties in our situation are found. Here is the table of symplectic surfaces with κ = 0 which we know so far:
+ − b b1 χ σ b manifolds 3 0 24 -16 19 K3 3 4 0 0 3 4-torus 2 3 0 0 2 Kodaira surface 1 0 12 -8 9 Enriques surface 1 2 0 0 1 hyperelliptic surface
We can observe that the Euler number of these manifolds are 0, 12 or 24 and satisfy the bound condition: + − (∗) b1 ≤ 4, b ≤ 3, b ≤ 19
It is conjectured in [31] that a minimal symplectic 4-manifold with κ = 0 satisfies (∗), which is proved in [32].
Theorem 5.1.7. If M is a minimal symplectic 4-manifold with κ = 0, then the rational homology of M is the same as that of K3 surface, Enriques surface, or a T 2 bundle over T 2. In particular, the Euler number of M is 0, 12, or 24, the signature of M is 0, -8, + − or -16, and the Betti numbers of M satisfy b1 ≤ 4, b ≤ 3 and b ≤ 19.
This theorem offers some evidence for the conjecture.
5.1.3 κ = 1
In [21], Gompf showed that any finitely presented group is the fundamental group of a symplectic 4-manifold with κ = 1 (and κ = 2). So it is impossible to classify symplectic 4-manifolds of such types. But we can ask what kind of invariants can be realized by 59 these symplectic manifolds. Gompf also asked whether a non-ruled symplectic four- manifold necessarily has non-negative Euler characteristic. When κ = 0, this statement is equivalent to σ ≤ 0. Since
2 + 0 = c1 = 2χ + 3σ = 4 + 4b − 4b1 + σ
We can conclude that σ σ b = 1 + b+ + ≥ 2 + (M is symplectic, so b+ ≥ 1.) 1 4 4 On the other hand, Noether’s formula implies that
+ χ + σ ≡ 0(mod 4) or b − b1 is odd
Hence σ = −2χ − 2σ ≡ 0(mod 8)
A pair of integer (a, b) corresponding to (σ, b1) is called admissible if a = 8k for some a non-positive integer k and b1 ≥max{0, 2 + 4 }. In simply connected case, σ = 8k with k < 0. The pairs (8k, 0) are realized by Dolgachev surfaces and elliptic surfaces E(−k) with k ≤ 2. 1 3 A symplectic 4-manifold (M, ω) induces a natural map ∪[ω]: H (M, R) → H (M, R). The rank of the kernel is called the degeneracy of (M, ω) and denoted as d(M, ω). A 3 triple (a, b, c) ∈ Z corresponding to (σ, b1, d) is called Lefschetz admissible if it satisfies
1. a = 8k where k is a non-positive integer,
a 2. b ≥ max{0, 2 + 4 },
3. 0 ≤ c ≤ b and b − c is even.
The symplectic realization of such triples are shown in [9]:
Theorem 5.1.8. For any Lefschetz admissible triple (a, b, c), there exists a minimal symplectic 4-manifold (M, ω) with κ = 1 and
(σ(M), b1(M), d(M, ω)) = (a, b, c) 60 1 The other similar problem is the nullity question. For any α ∈ H (M, R) and i i+1 i = 1, 2, consider the map iα = ∪α : H (M, R) → H (M, R). The dimension of 1 the linear space {α ∈ H (M, R)|iα = 0}, denoted by ni(M), is called the i-nullity of
M. By Poincar´eduality, n1(M) = n2(M). Hence we can just say nullity n(M) of M. It also follows from Poincar´eduality that nullity is a lower bound of degeneracy: n(M) ≤ d(M, ω). A triple (a, b, c) corresponding to (σ, b1, n) is called null admissible if it satisfies
1. a = 8k where k is a non-positive integer,
a 2. b ≥ max{0, 2 + 4 },
3. 0 ≤ c ≤ b and c 6= b − 1.
It is shown in [9] that many null admissible triples can be realized by symplectic 4-manifolds with κ = 0.
5.1.4 κ = 2
As we mention in previous section, it is also impossible to classify symplectic 4-manifolds with κ = 2. On the other hand, not many manifolds in this situation are known. But we are interested in whether these manifolds satisfy the Bogomolov-Miyaoka-Yau inequality χ ≥ 3σ. Fintushel and Stern conjectured that minimal symplectic 4-manifolds with κ = 2 satisfy the symplectic Noether inequality:
χ(M) + σ(M) K2 ≥ − (2 + c(K )) ω 4 ω
Here c(Kω) is the maximal number of connected components of an embedded sym- + plectic representative of Kω. When χ and σ are replaced by b and b1 as [χ(M) + + σ(M)]/4 = [b (M) − b1(M) + 1]/2, after dropping the b1 term, it induces a stronger inequality b+(M) − 3 K2 ≥ − c(K ) ω 2 ω We believe that any symplectic manifold with κ ≥ 0 satisfies this inequality. 61 5.2 Luttinger surgery
In this section, we describe the Luttinger surgery following [1]. Applications to La- grangian fibrations are also discussed. We assume all manifolds are oriented.
5.2.1 Construction
Topologically, Luttinger surgery is a framed torus surgery. We start with a general description of framed torus surgeries. Let X be a smooth 4-manifold and L ⊂ X an embedded 2-torus with trivial normal bundle. Then let U be a tubular neighborhood of L. If we assume Y = X − U is the complement of U, Z = ∂Y = ∂U and g : Z → Z is a diffeomorphism, a new manifold X˜ can be constructed by cutting U out of X and gluing it back to Y along Z via g:
X˜ = Y ∪g U. (5.1)
Such surgery is called a torus surgery. It is often more explicit to describe this process via a framing of L.
Definition 5.2.1. Let X, L, U and Z be given as above. A diffeomorphism ϕ : U → 2 2 −1 2 2 2 2 T × D is called a framing of L if ϕ (T × 0) = L. Let π1 : T × D → T be the 2 −1 projection. For any γ ⊂ L and z ∈ ∂D , the lift γϕ = ϕ (π1(γ) × z) of γ in Z is called a longitudinal curve of ϕ. Let
∂ϕ : Z → ∂(T 2 × D2) =∼ T 2 × S1
2 2 be the induced map. Two framings ϕ1, ϕ2 : U → T × D are smoothly isotopic to each other if the map −1 2 1 2 1 ∂ϕ2 ◦ (∂ϕ1) : T × S → T × S is homotopic to the identity map.
∂ϕ induces a S1-bundle structure on Z. A positive oriented fiber µ of Z is called a meridian of L. For X˜ in (5.1), we will use L˜ to denote the torus L ⊂ U ⊂ X˜. Notice that L˜ also inherits a framingϕ ˜ and its meridianµ ˜ ⊂ Z satisfies
[˜µ] = p[µ] + k[γϕ], 62 in H1(Z; Z). Here γϕ is a longitudinal curve of ϕ and p, k are coprime integers. The diffeomorphism type of X˜ only depends on the class [˜µ]. It is called a generalized logarithmic transform of X along (L, ϕ, γ) with multiplicity p and auxiliary multiplicity k, or of type (p, k) (see [22]), and denoted as X(L,ϕ,γ,p,k). For brevity, we will call it a (p, k)-surgery. If X is a symplectic 4-manifold, Weinstein’s theorem states that there is a canonical framing for any Lagrangian torus of X.
Definition 5.2.2. Let X be a symplectic 4-manifoldand and L a Lagrangian torus of X. A framing ϕ of L is called a Lagrangian framing if ϕ−1(T 2 × z) is a Lagrangian submanifold of X for any z ∈ D2.
Topologically, a Luttinger surgery is a (1, k)-surgery with respect to a Lagrangian framing. In order to deal with the sympelctic structure, it is more convenient to use a square neighborhood rather than the disk neighborhood of L as above. Express the cotangent bundle T ∗T 2 as
4 {(x1, x2, y1, y2) ∈ R }/(x1 = x1 + 1, x2 = x2 + 1) equipped with the canonical 2-form
ω0 = dx1 ∧ dy1 + dx2 ∧ dy2 and let ∗ 2 Ur = {(x1, x2, y1, y2) ∈ T T | − r < y1 < r, −r < y2 < r},
There exists a tubular neighborhood U of L and a symplectomorphism ϕ :(U, ω) →
(Ur, ω0) for small r which satisfies
ϕ(L) = T 2 × (0, 0).
In addition, given a simple closed curve γ on L, we can choose the coordinates x1, x2 of T 2 such that
ϕ(γ) = {(x1, 0, 0, 0) | x1 ∈ R/Z}.
Let As,t = Us − Ut (s > t) be an annular region and f :(−r, r) → [0, 1] be a smooth r r increasing function such that f(t) = 0 for t ≤ − 3 and f(t) = 1 for t ≥ 3 . For any 63 integer k, we can define a diffeomorphism hk:
A r → A r r, 2 r, 2 ( r (x1 + kf(y1), x2, y1, y2) y2 ≥ 2 (x1, x2, y1, y2) 7→ (x1, x2, y1, y2) otherwise
Observe that ∗ hk(ω0) = ω0, (5.2) which follows from the relation
0 (dx1 + kf (y1)dy1) ∧ dy1 + dx2 ∧ dy2 = ω0
r for y2 ≥ 2 . −1 −1 Let XL = X − ϕ (U r ) and define gk = ϕ ◦ hk ◦ ϕ via the following diagram 2
−1 gk −1 XL ⊃ U − ϕ (U r ) → U − ϕ (U r ) ⊂ U 2 2 ↓ ϕ ϕ ↓ ,
hk A r → A r r, 2 r, 2 then we can construct a new smooth manifold
˜ X := XL ∪gk U.
∗ ˜ Notice that, by (5.2), we have gk(ω) = ω. Thus X carries a symplectic formω ˜ induced by ω. This process is called a Luttinger surgery (along the Lagrangian torus L). We know that
Lemma 5.2.3. [17] Any two Lagrangian framings of a Lagrangian torus are smoothly isotopic to each other.
Hence the symplectomorphism type of (X,˜ ω˜) only depends on the Lagrangian iso- topy class of L, the isotopy class of γ in L, and the integer k. Therefore, X˜ is also denoted as X(L, γ, k). It is worth mentioning that a Luttinger surgery can be reversed. Let L,˜ γ˜ be the −1 2 −1 subsets ϕ (T × (0, 0)) and ϕ (R/Z × (0, 0, 0)) of X˜. We can apply the Luttinger surgery to X(L, γ, k), L,˜ γ˜ with coefficient −k to recover X. 64 5.2.2 Lagrangian fibrations
One natural source of Lagrangian tori is smooth fibers of Lagrangian fibrations.
Definition 5.2.4. Let (X, ω) be a symplectic 4-manifold, and let B be a 2-manifold (with boundary or vertices). A smooth map π : X → B is called a Lagrangian fibration −1 if there exists an open dense subset B0 ⊂ B such that π (b) is a compact Lagrangian submanifold of X for any b ∈ B0. X is called Lagrangian fibered if such a structure exists.
It is easy to see that any smooth fiber of a Lagrangian fibration must be a torus. Moreover, we have
Lemma 5.2.5. A Luttinger surgery along a Lagrangian fiber preserves the Lagrangian fibration structure.
Proof. Let π : X → B be a Lagrangian fibration and L = π−1(b) ⊂ X a generic fiber.
Using notations from section 2.1, it is shown in [40] that there is a neighborhood Br −1 of b and U = π (Br) with local charts ϕ :(U, ω) → (Ur, ω0) and ϕ0 : Br → Dr = (−r, r) × (−r, r) such that the diagram
ϕ U −→ Ur
π ↓ ↓ π0 ϕ0 Br −→ Dr commutes. Here π0 is the projection (x1, x2, y1, y2) 7→ (y1, y2). ˜ If X = XL ∪gk U is obtained by performing Luttinger surgery along L (indexed by ˜ γ ⊂ L and k ∈ Z), we can define a mapπ ˜ : X → B asπ ˜(˜x) = π(x). Since π0 ◦ hk = π0, we have
−1 −1 −1 π ◦ gk = π ◦ ϕ ◦ hk ◦ ϕ = ϕ0 ◦ π0 ◦ hk ◦ ϕ = ϕ0 ◦ π0 ◦ ϕ = π
Soπ ˜ is well-defined. It is clear thatπ ˜ is Lagrangian and X˜ also possesses a Lagrangian fibration structure.
Lagrangian fibrations appear widely in toric geometry, integral systems and mirror symmetry. We will discuss almost toric fibration introduced by Symington in some detail. 65 Definition 5.2.6. An almost toric fibration of a symplectic 4-manifold (X, ω) is a Lagrangian fibration π : X → B with the following properties: for any critical point x of π, there exists a local coordinate (x1, x2, y1, y2) near x such that x = (0, 0, 0, 0),
ω = dx1 ∧ dy1 + dx2 ∧ dy2, and π has one of the forms (x2 + y2, x2 + y2) 1 1 2 2 2 2 (x1, x2, y1, y2) → (x1 + y1, x2) 2 2 (x1 − y1, x2) An almost toric 4-manifold is a symplectic 4-manifold equipped with an almost toric fibration.
The base B of an almost toric fibration has an affine structure with boundary and vertices. Moreover, these three types of critical points project to vertices, edges and interior of B respectively. Almost toric fibrations are classified by Leung and Symington:
Theorem 5.2.7. [29] Let (X, ω) be a closed almost toric 4-manifold. There are seven types of almost toric fibrations according to the homeomorphism type of the base B.
2 2 2 1. CP ]nCP2 or S × S , B is (homeomorphic to) a disk;
2 2 2 2 2. (S × T )]nCP2 or (S ט T )]nCP2, B is a cylinder;
2 2 2 2 3. (S × T )]nCP2 or (S ט T )]nCP2, B is a M¨obiusband;
4. the K3 surface, B is a sphere;
2 5. the Enriques surface, B is RP ;
6. a torus bundle over torus with monodromy ( !) 1 m I, , m ∈ Z 0 1 B is a torus;
7. a torus bundle over the Klein bottle with monodromy ( ! !) 1 0 1 m , , m ∈ Z 0 −1 0 1 B is a Klein bottle. 66 An immediate consequence of this classification is the calculation of the symplectic Kodaira dimension.
Proposition 5.2.8. If (X, ω) → B is an almost toric fibration, then κ(X) ≤ 0. More- over, κ(X) = 0 if and only if the base B is closed.
The effect of Luttinger surgeries on almost toric fibrations is also easy to describe.
Proposition 5.2.9. Suppose (X, ω) → B is an almost toric fibration and (X,˜ ω˜) is obtained from (X, ω) by performing a Luttinger surgery along a smooth fiber L, then (X,˜ ω˜) retains an almost toric fibration structure with the same base. Moreover, X˜ is diffeomorphic to X if χ(B) > 0.
Proof. The first statement is given by Lemma 5.2.5. If χ(B) > 0, X and X˜ are in one of the types (1)-(5) in Theorem 5.2.7. In each of them, the list of manifolds are distinguished by the type of intersection forms and Euler numbers. So the second result for types (1)-(3) follows from Proposition 5.4.4 and the fact that homology classes of Lagrangian tori in manifolds with b+ = 1 are torsion. It is clear for (4) and (5) from the classification.
Propositions 5.2.8 and 5.2.9 provide examples of Luttinger surgeries preserving the symplectic Kodaira dimension. In the next section, we will show that it is true for any Luttinger surgery.
5.3 Preservation of Kodaira dimension
In this section, we prove Theorem 1.0.9. To proceed, we must first prove the invariance of minimality under Luttinger surgery.
5.3.1 Minimality
A symplectic (smooth) −1 class is a degree 2 homology class represented by an embedded symplectic (smooth) sphere with self-intersection −1. A symplectic 4-manifold is called symplectically (smoothly) minimal if it does not have any symplectic (smooth) −1 class. The symplectic minimality is actually equivalent to smooth minimality. 67 Proposition 5.3.1. The Luttinger surgery preserves the minimality.
Proof. Since a Luttinger surgery can be reversed and the reverse operation is also a Luttinger surgery, it suffices to show that, if we start with a non-minimal symplectic 4-manifold, then after a Luttinger surgery, the resulting symplectic manifold is still non-minimal. But this is a direct consequence of the following fact in [53]:
Theorem 5.3.2. Given a Lagrangian torus L and a symplectic −1 class, there is an embedded symplectic −1 sphere in that class which is disjoint from L.
5.3.2 Kodaira dimension
Now, we analyze the effect of Luttinger surgery on the symplectic canonical class Kω and the symplectic class [ω]. Recall that XL is an open submanifold of both X and X˜, and let ν : XL → X andν ˜ : XL → X˜ be the inclusions. To prepare for the following lemma, we use the notations from Section 5.2.1. For the sake of simplicity, we will identify any object in X with their image of ϕ and 0 0 0 0 (x1, x2, y1, y2), (x , x , y , y ) will denote the coordinates of A r on XL and U respec- 1 2 1 2 r, 2 tively.
Lemma 5.3.3. There exists a 2-dimensional submanifold S ⊂ XL such that ν∗([S]) =
PD(Kω) ∈ H2(X) and ν˜∗([S]) = PD(Kω˜ ) ∈ H2(X˜).
Proof. Let J be a ω-tamed almost complex structure in X which induces a complex structure on T ∗U as
J(dx1) = −dy1,J(dx2) = −dy2
Assume ρ :(−r, r) → [0, 1] is a continuous increasing function satisfying ( 0 t ≤ 0 ρ(t) = r 1 t > 3 . Another almost complex structure J 0 in T ∗U is defined as
0 0 0 0 2 2 0 0 0 0 J (dx1) = −kf (y1)ρ(y2)dx1 − (k f (y1)ρ(y2) + 1)dy1,J (dx2) = −dy2
0 0 It is easy to check that J is ω-tamed and (gk)∗(J) = J in XL ∩ U. 68 Let π : L → X andπ ˜ : L˜ → X˜ be the canonical bundles of X and X˜, respectively, and let s : X → L ands ˜ : X˜ → L˜ denote the corresponding embeddings of zero sections. 2 Since L is trivial on U, we can find a global section σ of L and a Thom class Φ ∈ Hcv(L) such that σ = (dx1 +idy1)∧(dx2 +idy2) in XL ∩U and Φ = 0 in s(U). Another nonzero (2, 0)-form in U is constructed as
0 0 0 0 0 0 0 σ = (dx1 + iJ (dx1)) ∧ (dx2 + iJ (dx2))
In XL ∩ U, we have
∗ 0 gk(σ ) ∗ 0 0 0 0 0 0 = gk((dx1 + iJ (dx1)) ∧ (dx2 + iJ (dx2))) ∗ 0 0 0 2 02 0 0 0 = gk((dx1 + i(−kf (y1)ρ(y2)dx1 + (k f (y1)ρ(y2) + 1)dy1) ∧ (dx2 + idy2)) 0 0 = (dx1 + kf (y1)dy1 + i(−kf (y1)dx1 + dy1)) ∧ (dx2 + idy2) 0 = (1 − ikf (y1))(dx1 + idy1) ∧ (dx2 + idy2) 0 = (1 − ikf (y1))σ
0 −1 σ and σ give two local trivializations ofπ ˜ (XL ∩ U) with transition function θ = 0 π π −1 1 − ikf (y1). Since − 2 < arg(θ) < 2 , we can normalize the frame ofπ ˜ (U) such that
−1 ˜ θ = 1. Hence Φ |π (XL) can be extended to L via constant function and form a Thom class Φ˜ satisfying
˜ ∼ ˜ −1 1. Φ = Φ in L |XL = L |XL (= π (XL)).
˜ 0 0 0 0 −1 ˜ 2. Φ is independent of the coordinates (x1, x2, y1, y2) inπ ˜ (U). In particular, Φ = 0 ins ˜(U).
∗ ∗ It is clear that these 2-forms e = s (Φ) ande ˜ =s ˜ (Φ)˜ are equivalent in XL and vanish in U ⊂ X and U ⊂ X˜ respectively. Using these representatives, we can find a
2-submanifold S ⊂ supp(e) ⊂ XL which is Poincar´edual to Kω in X, and dual to Kω˜ in X˜.
The main theorem can be proved now.
Proof of Theorem 1.0.9. Suppose X˜ is obtained from X by applying a Luttinger surgery along L. Let us first consider the case in which X is minimal. By Proposition 5.3.1, X˜ 69 is also minimal. Let Kω and Kω˜ denote the canonical classes of X and X˜ respectively.
By Lemma 5.3.3, there exists a submanifold S ⊂ XL such that ν∗([S]) = PD(Kω) and
ν˜∗([S]) = PD(Kω˜ ). We also know that ω =ω ˜ in XL. So Z Z 2 2 Kω = Kω = Kω˜ = Kω˜ S S Z Z Kω · [ω] = ω = ω˜ = Kω˜ · [˜ω] S S Thus the Kodaira dimensions of X and X˜ coincide. If X is not minimal, we can blow down X along symplectic −1 spheres disjoint from
L to a minimal model. These spheres are contained in XL and the same procedure can be applied to X˜, so we can argue as above.
Theorem 1.0.9 can be used to distinguish non-diffeomorphic manifolds. In [2, 3, 4, 8, 18], several symplectic manifolds homeomorphic but not diffeomorphic to non-minimal rational surfaces are constructed. With κ = 2 for the building blocks, it also easily follows from Theorem 1.0.9 that they are exotic.
Remark 5.3.4. 1. The main theorem is proved based on the invariance of Kω · [ω] 2 2 and Kω. Actually, the class [ω] is also preserved since the volume is invariant under a Luttinger surgery. Theorem 1.0.9 is expected, in light of Auroux’s Ques- tion 2.6 in [6]:
2 Let X1,X2 be two integral compact symplectic 4-manifolds with the smae (K , χ, K· 2 [ω], [ω] ). Is it always possible to obtain X2 from X1 by a sequence of Luttinger surgeries?
2. It is well known that the Dolgachev surfaces S(p, q) obtained by performing two 2 logarithmic transforms with multiplicities p > 1, q > 1 to CP ]9CP2 have κ = 1. So a generalized logarithmic transform may not preserve κ (see a related discussion in [11]). 70 5.4 Manifolds with non-positive κ
In this section we apply Theorem 1.0.9 to study the effect of Luttinger surgeries on symplectic 4-manifolds with κ ≤ 0.
5.4.1 Torus surgery and homology
We start by analyzing how homology changes under a general torus surgery. Suppose X is a smooth 4-manifold and L ⊂ X is an embedded 2-torus with trivial normal bundle. Moreover, U, Y, Z, g, X˜ are defined as in Section 5.2.1.
To compare the homology of X and X˜, we need to compare both of them with the homology of Y . The inclusion i : Z → Y induces homomorphisms
Z ik : Hk(Z; Z) → Hk(Y ; Z) (5.3)
and
Q ik : Hk(Z; Q) → Hk(Y ; Q) (5.4)
Q in homology. We often use ik to denote ik and Hk(−) to denote Hk(−, Q). We also use r(A) to denote the dimension of any Q-vector space A. The following lemma is a well know fact, for which we offer a geometric argument.
Lemma 5.4.1. [µ] ∈ ker i1 if and only if [L] 6= 0 in H2(X).
Z Proof. Suppose i1[µ] = 0 in H1(Y ), i.e. l i1 [µ] = 0 in H1(Y ; Z) for some positive integer l. Thus l copies of µ bounds an oriented surface A in Y . Extend A by l normal disks inside the tubular neighborhood to obtain a closed oriented surface A0 intersecting with
L at l points with the same sign. This implies in particular that [L] 6= 0 in H2(X).
Conversely, suppose [L] 6= 0 in H2(X), then there exists a closed oriented surface B in X intersecting L with nonzero algebraic intersection numbers, say l. We may assume that the intersection is transverse with l + b positive intersection points and b 71 negative intersection points. We can further assume that B intersects the closure of U at l + 2b normal disks, l + b of those having positive orientations, the remaining b disks having negative orientations. This implies that the complement of those disks in B is an oriented surface in Y , whose boundary is homologous to lµ, and thus i1[µ] is zero.
When we consider the integral homology, Lemma 5.4.1 immediately implies
Z Corollary 5.4.2. If [L] = 0 in H2(X; Z), then i1 [µ] is a non-torsion class in H1(Y ; Z).
Consider the Mayer-Vietoris sequence
∂k+1 ρk νk ∂k ρk−1 ··· −→ Hk(Z) −→ Hk(Y ) ⊕ Hk(U) −→ Hk(X) −→ Hk−1(Z) −→ · · · (5.5)
0 00 0 00 where ρk = (ik, jk) and νk = νk ⊕ (−νk ) with ik, jk, νk, νk induced by inclusions.
0 Lemma 5.4.3. 1. ∂1 = 0 and ν1 : H1(Y ) → H1(X) is surjective.
2. ( 3 if i1[µ] 6= 0 r(Imρ1) = 2 if i1[µ] = 0
and ρ1 is injective if and only if [µ] ∈/ ker i1.
3. ∼ Z H1(X; Z) = H1(Y ; Z)/ < i1 [µ] > (5.6)
0 4. If [L] = 0 ∈ H2(X), then ν2 : H2(Y ) → H2(X) is surjective.
Proof. 1. It is clear that any class a in H1(X; Z) can be represented by a 1-cycle C disjoint from L. C is also disjoint from Z if the neighborhood U is small enough. 0 So ∂1a = [C ∩ Z] = 0. C ⊂ Y implies that ν1 is surjective.
2. We know ker ρ1 = ker i1 ∩ ker j1 ⊂ ker j1. Since ker j1 =< [µ] >, ( 0 if i1[µ] 6= 0 ker ρ1 = < [µ] > if i1[µ] = 0
and ρ1 is injective if and only if i1[µ] 6= 0. The rank of Imρ is given from r(Imρ1) =
r(H1(Z)) − r(ker ρ1). 72 3. The sequence (5.5) induces a short exact sequence
0 → H1(Y ; Z) ⊕ H1(U; Z)/ ker ν1 → H1(X; Z) → Im∂1 → 0
∂1 = 0 implies that
∼ H1(Y ; Z) ⊕ H1(U; Z)/ ker ν1 = H1(X; Z)
0 Because ν1 is surjective, we also have
0 H1(X; Z) = H1(Y ; Z)/ ker ν1
If {[µ], γ1, γ2} is a basis of H1(Z; Z), then Imρ1 =< ([µ], 0), (γ1, γ1), (γ2, γ2) > 0 and γ1, γ2 6= 0 ∈ H1(U; Z). For a ∈ H1(Y ; Z), a ∈ ker ν1 if and only if (a, 0) ∈ Z 0 Z ker ν1 =Imρ1, or a = ki1 [µ] for some k ∈ Z. So ker ν1 =< i1 [µ] > and
Z H1(X; Z) = H1(Y ; Z)/ < i1 [µ] >
4. [L] = 0 ∈ H2(X) ⇔ [µ] 6= 0 ∈ H1(Y ) (by Lemma 5.4.1)
⇔ ρ1 injective (by part(2))
⇔ ∂2 = 0 (exactness)
⇔ ν2 surjective (exactness) 00 0 Since [L] = 0 also implies ν2 = 0, ν2 has to be surjective.
All the results hold if we replace X, L, µ by X,˜ L˜ andµ ˜. Now, we are ready to compare X and X˜.
Comparing H∗(X) and H∗(X˜)
Lemma 5.4.3, applied to torus surgeries, gives
Proposition 5.4.4. If X˜ is obtained from X via a torus surgery, then
1. χ(X˜) = χ(X), σ(X˜) = σ(X). 73 2. 0 if i1[µ] = 0 = i1[˜µ] or i1[µ] 6= 0 6= i1[˜µ] ˜ b1(X) − b1(X) = −1 if i1[µ] = 0 and i1[˜µ] 6= 0 1 if i1[µ] 6= 0 and i1[˜µ] = 0
3. |b1(X˜) − b1(X)| ≤ 1 and |b2(X˜) − b2(X)| ≤ 2.
Proof. 1. Obvious.
2. Since ∂1 = 0, we can conclude that ( b1(Y ) − 1 if i1[µ] 6= 0 b1(X) = b1(Y ) + 2 − r(Imρ1) = b1(Y ) if i1[µ] = 0.
The same is true for b1(X˜) with i1[µ] replaced by i1[˜µ]. The proof is finished by
comparing b1(X) and b1(X˜).
3. The first inequality is given by part (2). The second inequality follows from part (1) and the first inequality.
The next result concerns with the intersection forms.
Proposition 5.4.5. Suppose X and X˜ are defined as above. If [L] is a torsion class ˜ in H2(X; Z) and the intersection form Q(X) is odd, then Q(X) is odd as well. In particular, if both [L] and [L˜] are torsion, then X˜ and X have the same intersection form.
Proof. Since Q(X) is odd, there exists a closed oriented surface S in X such that S · S 0 is odd. By Lemma 5.4.3(4), [S] ∈ Imν2 and S can be chosen such that S ⊂ Y . Thus, S is contained in X˜, and hence, Q(X˜) is also odd.
5.4.2 κ = −∞
By Proposition 5.2.9, if a symplectic manifold (X, ω) with κ(X) = −∞ has an almost toric structure π : X → B and if we apply a Luttinger surgery along a smooth fiber of π, the new manifold (X,˜ ω˜) is diffeomorphic to (X, ω). Such phenomenon is still true for any 4-manifold with κ = −∞. Moreover, we have the stronger Theorem 1.0.10. 74 Proof of Theorem 1.0.10. Proposition 5.3.1 allows us to reduce to the case where (X, ω) is minimal. We first show that X˜ is diffeomorphic to X. Observe that the diffeomorphism types of minimal manifolds with κ = −∞ are distinguished by their Euler numbers and intersection forms. Since such manifolds have b+ = 1, the homology classes of Lagrangian tori are torsion. Thus, both quantities are preserved by Proposition 5.4.4 and 5.4.5. To show further that (X,˜ ω˜) and (X, ω) are symplectomorphic to each other, it is 2 enough to show that ω is cohomologous toω ˜ ([36]). If X is diffeomorpic to CP , the symplectic structure is determined by the volume [ω]2, which is preserved by Remark 5.3.4(1). 2 When X is ruled, H (X) is either generated by Kω and the Poincar´edual to the 2 homology class of a fiber F = S , or by Kω and [ω]. Hence the class of ω is determined 2 by Kω · [ω], [ω] and [ω](F ). As mentioned above, the first two quantities are preserved. By [53] the fiber sphere can be chosen to be disjoint from L, so it follows that the last quantity is also preserved.
5.4.3 Luttinger surgery as a symplectic CY surgery
A symplectic CY surface is a symplectic 4-manifold with torsion canonical class, or equivalently, a minimal symplectic 4-manifold with κ = 0. By Theorem 1.0.9 and Proposition 5.3.1, we have
Proposition 5.4.6. A Luttinger surgery is a symplectic CY surgery in dimension four.
There is a homological classification of symplectic CY surfaces in [32] and [12].
Theorem 5.4.7. A symplectic CY surface is an integral homology K3, an integral homology Enriques surface or a rational homology torus bundle over torus.
The table in p.71 lists possible rational homological invariants of symplectic CY surfaces.
Proof of Theorem 1.0.11. It follows from Propositions 5.4.4, 5.4.6, Theorem 5.4.7 and the table above. 75 It is also speculated that a symplectic CY surface is diffeomorphic to the K3 surface, the Enriques surface or a torus bundle over torus. Thus we make the following
Conjecture 5.4.8. If X is a K3 surface, or an Enriques surface, then under a Luttinger surgery along any embedded Lagrangian torus, X˜ is diffeomorphic to X.
As for torus bundles over torus, we have
Conjecture 5.4.9. Any smooth oriented torus bundle X over torus possesses a sym- plectic structure ω such that (X, ω) can be obtained by applying Luttinger surgeries to 4 (T , ωstd).
In the list of torus bundles over torus in [20], any manifold in classes (a), (b) and (d) has a Lagrangian bundle structure. For any such manifold, it is not hard to verify Conjecture 5.4.9 via Luttinger surgery along Lagrangian fibers.
Remark 5.4.10. 1. Conjectures 5.4.8 and 5.4.9 are clearly related to Question 2.6 in [6].
2. There is another symplectic CY surgery in dimension six, the symplectic conifold 6 transition. If (M , ω) with Kω = 0 contains disjoint Lagrangian spheres S1, ..., Sn Pn with homology relations generated by i=1 λi[Si] = 0 with all λi 6= 0, Smith, Thomas, Yau ([45]) construct from (M 6, ω) a new symplectic manifold (M 0, ω0)
with Kω0 = 0 and smaller b3.
5.4.4 Luttinger surgery in high dimension
In this section, we construct a symplectic surgery in high dimension, which can be viewed as a parametrized Luttinger surgery. In the following, the definition of Ur, ω0,As,t and hk are given in Section 5.2.1. Let (X, ω) be a symplectic manifold of dimension 2n and L ⊂ X a coisotropic submanifold of codimension two. Moreover, we assume that L possesses a local product structure. This means that there is a tubular neighborhood U of L, a symplectic manifold (Y, ωY ) of dimension 2n − 4 and a symplectomorphism
ϕ :(U, ω) → (Y × Ur, ωY + ω0) 76 2 −1 such that ϕ(L) = Y × T × (0, 0). Let XL = X − ϕ (Y × U r ) and define gk = 2 −1 ϕ ◦ (id × hk) ◦ ϕ via the following diagram
−1 gk −1 XL ⊃ U − ϕ (Y × U r ) → U − ϕ (Y × U r ) ⊂ U 2 2 ↓ ϕ ϕ ↓ ,
id×hk Y × A r → Y × A r r, 2 r, 2 then we can construct a new smooth manifold
˜ X := XL ∪gk U.
∗ ˜ Similar to the Luttinger surgery, the construction satisfies gk(ω) = ω and X is a sym- k n−k k 0 plectic manifold. We want to compare the value of Kω · Ω and Kω0 · ω . But the computation is more subtle and we will discuss it elsewhere. Chapter 6
Torus surgery and framings
In this chapter, we discuss some topological properties of embedded tori in 4−manifolds. In Section 5.4.9, the torus surgery for an 2-torus with trivial normal bundle embedded in a smooth 4-manifold is defined. This is the topological generalization of Luttinger surgery. We also discuss the effect of torus surgery in homology. Section 6.1.2 defines several local framings for torus surgery and their constraints. The proofs of some lemmas are postponed to Section ??.
6.1 Topological preferred framing and Lagrangian framing
In this section, we will introduce topological preferred framings and compare them with the Lagrangian framing for Lagrangian tori in κ ≤ 0 symplectic 4-manifolds. Suppose X is a smooth 4-manifold and L ⊂ X is an embedded 2-torus with trivial normal bundle. Recall that a framing is a diffeomorphism ϕ : U → T 2 ×D2 for a tubular neighborhood U of L such that ϕ−1(T 2 × 0) = L and a longitudinal curve of ϕ is a lift
γϕ of some simple closed curve γ ⊂ L in Z. Let
H1,ϕ :=< [γϕ]|γϕ : longitudinal curve of ϕ >
H1,ϕ is a subgroup of H1(Z; Z) and it induces a decomposition of H1(Z; Z):
H1(Z; Z) =< [µ] > ⊕H1,ϕ.
Conversely, any rank 2 subgroup V of H1(Z; Z) such that [µ] and V generate H1(Z; Z) corresponds to a framing of L.
77 78 In [34], Luttinger introduced a version of topological preferred framings of La- 4 Z grangian tori in R . It requires that H1,ϕ is in the kernel of i1 . On the other hand, Fintushel and Stern ([17]) defined null-homologous framings for a null-homologous torus Z via i2 (seemingly, under the assumption that H1(X; Z) vanishes, though not explicitly mentioned). The following definition is essentially the same as in [17], but without assuming that
H1(X; Z) vanishes.
Definition 6.1.1. Suppose L is null-homologous, i.e., [L] = 0 in H2(X; Z). A framing Z ϕ is called a topological preferred framing if [Lϕ] ∈ ker i2 . Here, Lϕ ⊂ Z is a longitudinal torus of ϕ given by ϕ−1(T 2 × z), z ∈ ∂D2.
There is the following generalization when [L] is a torsion class in H2(X; Z).
Definition 6.1.2. Assume [L] is a torsion class in H2(X; Z). A framing ϕ is called a Q rational topological preferred framing if [Lϕ] ∈ ker i2 .
When L is null-homologous, it is clear that a topological preferred framing is also a rational topological preferred framing.
6.1.1 Comparing ker i1 and ker i2
In order to compare various preferred framings and the Lagrangian framing, we need to investigate the relation of the maps i1 and i2 given by (5.3) and (5.4). Let Y be a smooth oriented 4-manifold with boundary Z = T 3.
Lemma 6.1.3. The maps i1 and i2 satisfy the following properties
1. r(ker i1) + r(ker i2) = 3.
2. With the pairing ∼ H1(Z) × H2(Z) → H0(Z) = Q,
given by the cap product, ker i2 and ker i1 annihilate each other:
ker i2 = ann(ker i1) := {c ∈ H2(Z)|a · c = 0 ∈ H0(Z) for any a ∈ ker i1}
and ker i1 = ann(ker i2). 79
3. r(ker i1) > 0.
Proof. 1. Consider the exact sequence
∂2 i2 δ2 ∂1 i1 δ1 ··· −→ H2(Z) −→ H2(Y ) −→ H2(Y,Z) −→ H1(Z) −→ H1(Y ) −→ · · ·
It induces a short exact sequence
ν2 ∂1 0 −→ H2(Y )/Imi2 −→ H2(Y,Z) −→ Im∂1 = ker i1 −→ 0 (6.1)
By Lefschetz duality and universal coefficient theorem,
∼ 2 ∼ H2(Y,Z) = H (Y ) = H2(Y )
and r(H2(Y,Z)) = r(H2(Y )). So (6.1) implies that
r(ker i1) = r(Imi2) = r(H2(Z)) − r(ker i2)
which is (1).
2. Consider the dual pairing
1 H1(Y ) × H (Y ) → H0(Y ) ∼ ↑ i1 ↓ j ↑= 1 H1(Z) × H (Z) → H0(Z)
Because the maps i1 and j are induced by embedding and restriction, this pairing 1 is natural, i.e., for a ∈ H1(Z) and α ∈ H (Y ),
< i1(a), α >=< a, j(α) >
There is an isomorphism of long exact sequences induced naturally by Lefschetz and Poincar´edualities:
j · · · −→ H1(Y,Z) −→ H1(Y ) −→ H1(Z) −→ H2(Y,Z) −→ · · · ↓ ∩[Y ] ↓ ∩[Y ] ↓ ∩[Z] ↓ ∩[Y ]
i3 δ3 ∂2 i2 δ2 ··· −→ H3(Y ) −→ H3(Y,Z) −→ H2(Z) −→ H2(Y ) −→ · · · 80 Using this diagram, the dual pairing induces the intersection pairing:
↓
H3(Y ) ↓
H1(Y ) × H3(Y,Z) → H0(Y ) ∼ ↑ i1 ↓ ∂ = ∂2 ↑=
H1(Z) × H2(Z) → H0(Z)
↓ i2
H2(Y ) ↓
Given z2 ∈ ker i2, there exists z3 ∈ H3(Y,Z) such that ∂z3 = z2. Let β be the 1 Lefschetz dual of z3 in H (Y ). For any z1 ∈ ker i1,
z1 · z2 = z1 · ∂z3 = i1(z1) · z3 =< i1(z1), β >= 0
It shows that ker i2 ⊂ ann(ker i1). From part (1),
r(ann(ker i1)) = r(H2(Z)) − r(ker i1) = r(ker i2).
So ker i2 = ann(ker i1). Similar argument shows that ker i1 = ann(ker i2).
3. Suppose r(ker i1) = 0, then r(ker i2) = 3 and i2 is the zero map by part (1). Let
T1,T2 be two nonisotopic embedded tori in Z intersecting in a curve C transversely. 3 Since Z = T ,[T1] ∩ [T2] = [C] 6= 0 in H1(Z). Meanwhile, each Ti bounds a 3-
manifold Wi in Y . W1 ∩ W2 is a 2-cycle whose boundary is C and [C] is in the
kernel of i1, which contradicts the assumption that i1 is injective.
Here is a geometric interpretation of this lemma. Assume z2 is an integral class of ker i2 and C is a closed curve in Z such that [C] · z2 6= 0 in Z. There exists a relative 3-cycle W in (Y,Z) such that [∂W ] = z2. In particular, we can assume that W intersects Z transversely and ∂W intersects C transversely at a1, ··· , ap and b1, ··· , bn in Z with positive and negative intersections respectively. Furthermore, we can give a collar structure V =∼ Z × [0, ) near Z and assume W ∩ V = ∂W × [0, ). If we 81 0 0 push C to C = C × 2 in the interior of Y , then C and W intersect transversely at a1 × 2 , ··· , ap × 2 and b1 × 2 , ··· , bn × 2 with positive and negative intersections 0 respectively. Hence [C ] = i1([C]) and
0 [C ] · [W ] = [C] · [∂W ] = p − n = [C] · z2 6= 0
So [C] can not be in ker i1.
Remark 6.1.4. 1. Part (1) of Lemma 6.1.3 is still true in arbitrary dimension. If
Y is a (n+1)-dimensional manifold with connected boundary Z and ik : Hk(Z) →
Hk(Y ) denotes the homomorphism induced by the inclusion Z → Y , then
r(ker ik−1) + r(ker ik) = r(Hk(Z))
for 2 ≤ k ≤ n − 1.
2. Part (3) of Lemma 6.1.3 is pointed out by Robert Gompf.
In the following, we give examples to illustrate Lemma 6.1.3 according to r(ker i1).
3 1 3 1. Let K0 be the trivial knot in S and X = S × S . The complement of the torus 1 L = S × K0 is 1 3 ∼ 1 1 2 Y = S × (S − K0) = S × (S × D )
1 2 If t, m denote the isotopy classes of these two S and l = ∂D , then H1(Z) =<
[t], [m], [l] > and ker i1 has rank 1 which is generated by [l]. On the other hand,
ker i2 is generated by [l × t], [l × m] and has rank 2. In general, if K is any knot in S3 and S is a Seifert surface with boundary K, we can define t and m as above and choose l as the push-off of K in S. Then 2 2 Y and T × D have isomorphic homology groups and ker i1 is still generated by
l, which bounds the surface S. Similarly, ker i2 has rank 2 and is generated by [l × t], [l × m]. They bound S1 × S and {pt} × (S3 − K) respectively. In [16], Fintushel and Stern use these manifolds as building blocks to define knot surgery in 4-manifolds.
In the next example, the results of Lemma 6.1.3 are not obvious. Let π : X → Σg
be a ruled surface and the loop γ ⊂ X be a lift of a loop in Σg. We can construct 82 a torus L in X as the product of γ and some circle b in the fiber. If µ ⊂ Z is
a meridian of L and π(γ) is nontrivial in π1(Σg), it is easy to show that ker i1 is 2 generated by a push-off of b. But ker i2 is not obvious even when X = S × Σg
is the trivial bundle. By Lemma 6.1.3, we know that ker i2 has rank 2 and is generated by [µ × b] and [µ × γ].
2 2. Let L = a × b be the Clifford torus embedded in the rational manifold X = CP .
The group H1(Z) is generated by [a], [b] and the meridian [µ]. It is easy to show
that ker i1 =< [a], [b] > has rank 2 and ker i2 =< [a × b] >. In general, if X is simply connected and L ⊂ X is a torus with trivial normal
bundle, then r(ker i1) = 2 if and only if [L] = 0 in H2(X).
3. If r(ker i1) = 3, it follows from Lemma 5.4.3 that such surgery will not change the homology for any torus surgery.
There are similar results for Lemma 6.1.3 over Z if we consider r(·) as the rank of abelian groups. In particular, the following lemma is the analogue of 6.1.3(2).
Lemma 6.1.5. With the pairing
∼ H1(Z; Z) × H2(Z; Z) → H0(Z; Z) = Z,
Z Z given by the cap product, ker i2 annihilates ker i1 :
Z Z Z ker i2 ⊂ annZ(ker i1 ) = {c ∈ H2(Z; Z)|a · c = 0 ∈ H0(Z; Z) for any a ∈ ker i1 }.
Q Q Proof. By Lemma 6.1.3(2), ker i1 = ann(ker i2 ). If H1(Z; Z) is considered as the Q Z Z Q Z integral elements of H1(Z; Q), then ker i1 = ker i1 ⊗ Q and ker i1 ⊂ ker i1 . So ker i1 ⊂ Q Z ann(ker i2 ) = ann(ker i2 ).
6.1.2 Preferred framings via ker i1
Now we characterize topological preferred framings via i1. We first consider the rational ones.
Proposition 6.1.6. Assume [L] = 0 in H2(X; Q) and ϕ is a framing of L. Then ϕ is Q a rational topological preferred framing if and only if ker i1 ⊂ H1,ϕ ⊗ Q. 83 Proof.
ϕ is a rational topological preferred framing
Q ⇔ [Lϕ] ∈ ker i2 Q ⇔ < [Lϕ] >⊂ ker i2 Q ⇔ ann(< [Lϕ] >) ⊃ ann(ker i2 ) Q ⇔ ker i1 ⊂ H1,ϕ ⊗ Q. (Lemma 6.1.3)
In the integral cases, we have
Proposition 6.1.7. Suppose L is null-homologous and ϕ is a framing of L. Then
1. L has topological preferred framings, and
Z 2. ker i1 ⊂ H1,ϕ if ϕ is a topological preferred framing.
Proof. 1. Since [L] = 0 in H2(X; Z), there exists a 3-chain W such that ∂W = L. We can assume that W intersects Z transversely. In fact, we can choose a framing 2 2 −1 2 2 ϕ : U → T ×D such that W ∩U = ϕ (T ×Sx), where Sx = {(x, 0) ∈ D |x ≥ 0}. Then W ∩Z = ϕ−1(T 2 ×(1, 0)) is a longitudinal torus of ϕ and W ∩Y is a relative Z 3-cycle of (Y,Z) with ∂(W ∩Y ) = W ∩Z. So [W ∩Z] ∈ ker i2 and ϕ is a topological preferred framing.
Z Z 2. [Lϕ] ∈ ker i2 implies that annZ([Lϕ]) ⊃ annZ(ker i2 ). It is easy to observe that
annZ(< [Lϕ] >) = H1,ϕ. By Lemma 6.1.5,
Z Z ker i1 ⊂ annZ(ker i2 ) ⊂ annZ([Lϕ]) = H1,ϕ.
If [L] is torsion in X, Proposition 6.1.7(1) may fail in two situations. First, there may Z Z exist a ∈ H1(Z; Z) such that a∈ / ker i1 but ka ∈ ker i1 for some nonzero integer k. So Z we can only define rational topological preferred framings. Second, [µ] and ker i1 might not generate the group H1(Z; Z). In this case, rational topological preferred framings also do not exist. 84 Remark 6.1.8. 1. In knot theory, the notion of preferred framings is similar to that of Definition 6.1.1. Let M be an integral homology 3-sphere and K ⊂ M be a knot. If V is a tubular neighborhood of K, a diffeomorphism h : S1 × D2 → V satisfying h(S1 × 0) = K is called a framing of K. Furthermore, h is called a preferred framing if h(S1 × a) is homologically trivial in M − V . For any knot K in M, preferred framings exist and are unique up to isotopy ([43]).
2. It is easy to see from Proposition 6.1.7 that Luttinger’s definition coincides with 4 Definition 6.1.1 when X = R .
3. In [17], an invariant λ(L) is defined when [L] = 0 and L has a unique topological
preferred framing ϕ0. Assume ϕLag is the Lagrangian preferred framing. Then
ϕLag = ϕ0 if and only if λ(L) = 0. Otherwise, λ(L) is the smallest positive
integer k such that k[µ] + [γϕ] ∈ H1,ϕLag for some [γϕ] ∈ H1,ϕ0 .
6.1.3 (1, k)-surgeries and topological preferred framings
The following proposition relates rational topological preferred framings and (1, k)- surgeries.
Proposition 6.1.9. Suppose X is a smooth 4-manifold and L ⊂ X is a torus with trivial normal bundle such that [L] = 0 in H2(X; Q) and ϕ is a framing of L. Let ˜ X = X(L,ϕ,γ,1,k) be constructed from X via (1, k)-surgery along (L, ϕ, γ).
1. If ϕ is a rational topological preferred framing of L, then X˜ satisfies
r(H1(X˜)) = r(H1(X))
for any γ and k.
˜ ∼ 2. If H1(X; Z) = H1(X; Z) for any γ and k, then ϕ is a rational topological preferred framing of L.
Proof. 1. By Lemma 5.4.3(3), we have ( r(H1(Y )) − 1 if i1[µ] 6= 0 r(H1(X)) = r(H1(Y )) if i1[µ] = 0. 85
Lemma 5.4.1 implies that i1[µ] 6= 0 in H2(Y ). Since ϕ is a rational topological
preferred framing, Proposition 6.1.6 implies that [µ]+k[γϕ] ∈/ ker i1 for any integer
k and [γϕ] ∈ H1,ϕ. So
r(H1(X˜)) = r(H1(Y )) − 1 = r(H1(X))
if X˜ is given via (1, k)-surgery.
Z 2. We first prove that any a ∈ ker i1 lies in H1,ϕ. Let a = s[µ]+t[γϕ] for some s, t ∈ Z ˜ ˜ and γ ⊂ L (Recall that γϕ is a lift of γ). For X = X(L,ϕ,γ,1,kt), the meridian of L in X˜ satisfies
[˜µ] = [µ] + kt[γϕ] = [µ] + k(a − s[µ]) = (1 − sk)[µ] + ka (6.2)
So ˜ Z H1(X; Z) = H1(Y ; Z)/ < i1 ((1 − sk)[µ] + ka) > (by (5.6)) Z Z = H1(Y ; Z)/ < i1 ((1 − sk)[µ]) > (a ∈ ker i1 ) ∼ Z = H1(Y ; Z)/ < i1 [µ] >
Because [µ] is essential in Y and k is arbitrary, s should be zero and a ∈ H1,ϕ.
Otherwise, H1(X; Z) has infinitely many torsion classes with different orders.
Tensoring with Q, we have
Z ker i1 = ker i1 ⊗ Q ⊂ H1,ϕ ⊗ Q
By Proposition 6.1.6, ϕ is a rational topological preferred framing.
For integral cases, we have
Proposition 6.1.10. Suppose H1(X; Z) has no torsion and L is null-homologous. Then ˜ ∼ a framing ϕ of L is a topological preferred framing if and only if H1(X; Z) = H1(X; Z) ˜ for any X = X(L,ϕ,γ,1,k) obtained from X via (1, k)-surgery.
Proof. Assume ϕ is a topological preferred framing. Consider the 3-chain W given in the proof of Proposition 6.1.7. It is clear that a meridian µ intersects W at one point. Z Z So i1 [µ] · [W ] = ±1 and i1 [µ] is a primitive class. Similarly, the simple closed curveµ ˜ 86 has class [µ] + k[γϕ] and is homotopic to a curve intersecting W at one point. Hence Z i1 [˜µ] is also a primitive class for any γ, k. Since H1(Y ; Z) is a free abelian group, we Z ∼ Z ∼ ˜ have H1(Y, Z)/ < i1 [µ] >= H1(Y, Z)/ < i1 [˜µ] >. So H1(X, Z) = H1(X; Z) by Lemma 5.4.3(3). ∼ ˜ ˜ Conversely, if H1(X, Z) = H1(X; Z) for any X, the proof of Proposition 6.1.9(2) shows that ϕ is a rational topological preferred framing. The assumption that H2(X; Z) has no torsion implies that ϕ is actually a topological preferred framing.
3 1 The knot surgery in [16] is an example of (1, k)-surgeries. Suppose X0 = S × S , K is a knot in S3 and L = K × S1. Let µ be the meridian of L, a the longitude of the 1 Z Z preferred framing of K and b = S . Then ker i1 =< [a] > and ker i2 =< [µ×a], [a×b] >.
Consider the framings ϕp of L where H1,ϕp is generated by p[µ] + [a] and [b]. It is clear that ϕp is a topological preferred framing of L exactly when p = 0. If K is the trivial knot and [γ] = p[µ] + [a], the resulting manifold of (1, k)-surgery is S3 ∼ L(1 + pk, k) × S1 (L,ϕp,γ,1,k) = and H (S3 ; ) =∼ ⊕ . If |p| > 1, H (S3 ; ) has rank 1 for any 1 (L,ϕp,γ,1,k) Z Z1+kp Z 1 (L,ϕp,γ,1,k) Z p, k, but the torsion subgroups vary. Propositions 6.1.10 and 6.1.9 show that ϕp is not a rational topological preferred framing. Actually, such framings do not exist for L.
6.1.4 Constraints for Lagrangian framings
Here we provide topological constraints on the isotopy classes of Lagrangian tori in many symplectic manifolds with non-positive Kodaira dimension. In particular, they imply that the invariant λ(L) of Fintushel and Stern (Remark 6.1.8(3)) is zero if the manifold has non-positive Kodaira dimension and vanishing integral H1. Recall that the Lagrangian framing for Lagrangian tori is defined in Section 5.2.1.
Proposition 6.1.11. Suppose L is a Lagrangian torus in (X, ω) and any Luttinger surgery along L preserves the integral homology.
1. If H2(X; Z) is torsion free and L is null-homologous, then the Lagrangian framing of L is a topological preferred framing. 87 2. If [L] is torsion then the Lagrangian framing of L is a rational topological preferred framing.
Proof. The result follows directly from Propositions 6.1.10 and 6.1.9.
In particular, we have
Corollary 6.1.12. If κ(X) = −∞ and L is a Lagrangian torus in X, then the La- grangian framing of L is a topological preferred framing.
+ Proof. Since b (X) = 1 and H2(X; Z) has no torsion, L is null-homologous. For any X˜ given from X by applying Luttinger surgery along L, X˜ is diffeomorphic to X by Theorem 1.0.10. Now the claim follows from Proposition 6.1.11.
In the case that X is a symplectic CY surface, it is convenient to introduce
Definition 6.1.13. An embedded 2-torus L of a 4-manifold X is called essential if Q [L] 6= 0 in H2(X). Moreover, L is called completely essential if r(ker i1 ) = 3.
Proposition 6.1.14. If κ(X) = 0 and X is an integral homology K3, then any La- grangian torus L satisfies one of the following conditions:
1. L is null-homologous and the Lagrangian framing is a topological preferred framing.
2. L is completely essential.
Proof. When L is null-homologous, the claim follows from Theorem 1.0.11 and Propo- sition 6.1.11. Q Q When L is essential, [µ] ∈ ker i1 by Lemma 5.4.1. If r(ker i1 ) 6= 3, there exists Q ˜ γ ⊂ L such that i1 ([γϕ]) 6= 0. In the manifold X = X(L, γ, 1), the class [˜µ] = [µ] + [γϕ] is nonzero in H1(Y ). By Proposition 5.4.4, b1(X) 6= b1(X˜), which contradicts Theorem Q 1.0.11. So r(ker i1 ) = 3 and L is completely essential.
Similarly we have
Proposition 6.1.15. If κ(X) = 0 and X is an integral homology Enriques surface, then any Lagrangian torus L satisfies one of the following conditions: 88 1. L is null-homologous and the Lagrangian framing is a topological preferred framing.
2. [L] is torsion and the Lagrangian framing is a rational topological preferred fram- ing.
Proof of Theorem 1.0.12. The first statement follows from Theorem 5.4.7, Corollary 6.1.12, Propositions 6.1.14(1) and 6.1.15(1). The last statement on λ(L) follows from Remark 6.1.8(3). References
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