LEFSCHETZ PENCILS Contents 1. Introduction 2 2. Classical ...

LEFSCHETZ PENCILS 

FRANCISCO PRESAS 

Abstract. Lectures over the use and applications of Lefschetz 

pencils in symplectic and contact geometry. The lectures are divided 

in three blocks: projective case, symplectic case and contact 

case. First, the classical notion of Lefschetz pencil is introduced 

and then the more modern setup in symplectic topology is outlined. 

Later, the analogous constructions in the contact category 

are briefly sketched. 

Contents 

1. Introduction 2 

2. Classical pencils 2 

2.1. Quick review of some basic complex geometry 2 

2.2. Positive bundles 4 

2.3. Lefschetz pencils 6 

2.4. Symplectic structures and monodromy 8 

2.5. Monodromy of a pencil 10 

2.6. Main equivalence theorem I: from pencil to symplectic 

structure 14 

3. Approximately holomorphic geometry 17 

3.1. Trivializations 19 

3.2. Local transversality 20 

3.3. Globalizing 20 

3.4. Main theorem II: from symplectic structure to pencil 22 

4. Approximately holomorphic techniques over contact 

manifolds. 23 

4.1. Almost contact manifolds 23 

4.2. Giroux’s theorems 23 

4.3. Contact pencils 23 

Date: May, 2012. 

1991 Mathematics Subject Classification. Primary: 53D10. Secondary: 53D15, 

57R17. 

Key words and phrases. contact structures, Lefschetz pencils. 

The author was supported by the Spanish National Research Project MTM2010- 

17389. 

1

2 FRANCISCO PRESAS 

4.4. Existence of contact structures 23 

References 23 

1. Introduction 

This series of lectures is intended to be an overview of the relation 

between Lefchetz pencils and symplectic and contact structures. We 

do not try to give precise proofs, but just a feeling of how the whole 

theory works. The following readings are recommended to formalize 

the outlines of the proofs provided in the lectures: 

(i) Complex geometry basics: first chapters of the book [We]. 

(ii) Basic Picard-Lefschetz theory: [La]. 

(iii) Symplectic pencils: Donaldson’s articles. The introductory one 

is [Do3]. Details provided in [Do2, Do1]. 

(iv) Basic introduction to open books in contact geometry: introductiry 

article [Gi]. 

(v) Contact pencils: basics [Pr1]. 

2. Classical pencils 

2.1. Quick review of some basic complex geometry. Let X be 

a smooth manifold with a fixed atlas {Vi, φi}i∈I. We say that the 

manifold is complex if φi : Vi → φi(Vi) ⊂ R 2n C n can be chosen to 

satisfy that φ −1 

j ◦ φi are bi-holomorphic diffeomorphisms. 

A complex vector bundle over a smooth manifold X is a pair (E, π : 

E → X) such that: 

(i) E is a smooth manifold, 

(ii) π : E → X is a surjective smooth submersion, 

(iii) There exists a covering by open subsets {Uj}j∈J, satisfying that 

there are diffeomorphisms ψj : π −1 (UJ) → Uj × C r making the 

following diagram commutative 

π −1 (Uj) 

ψj 

→ Uj × C r 

π ↓ π1 ↓ 

id 

Uj → Uj. 

(iv) The map ψ −1 

i ◦ ψj : Ui Uj → Diff(C r ) reduces to ψij = 

ψ −1 

i ◦ ψj 

 

: Ui Uj → GL(Cr ). 

The last condition implies that the fibers of the map π are canonically 

complex vector spaces. The dimension of those vector spaces, the 

constant r, is called the rank of the vector bundle.

LEFSCHETZ PENCILS 3 

A morphism of complex vector bundles Φ : E → F is a smooth map 

making the diagram commute 

E Φ → F 

πE ↓ πF ↓ 

X id → X, 

and such that the restriction to any fiber is a linear map. The morphism 

is called monomorphism whenever is injective, epimorphim whenever 

is surjective and isomorphism if it is bijective. 

A vector bundle E is completely determined, up to bundle isomorphism, 

by the maps {ψij} for any covering. In the other hand, any such 

collection of maps determines a vector bundle if it satisfies the cocycle 

condition 

ψjk ◦ ψij = ψik. 

These maps associated to the bundle are called the transition functions. 

With the help of them we can define various operations in the vector 

bundle category: 

• Let E have transition functions {ψij} and let F have transition 

functions {ρij}. We define the direct sum vector bundle as the 

bundle E F with transition functions 

 

ψj 0 

0 ρij 

• In the same spirit E ⊗ F is defined by the transition functions 

ψij ⊗ ρij. 

• The dual bundle E ∗ is defined by the transition functions ψ ∗ ij. 

• Any other conceivable operation with vector spaces has a translation 

to the category of vector bundles. 

Let X be a smooth complex manifold. We define a holomorphic 

vector bundle E over X to be a complex vector bundle E → X such 

that the maps ψij are holomorphic. This, in particular, implies that E 

is a complex manifold and π is a holomorphic submersion. 

The natural complex vector space structure of the fibers of a complex 

vector bundle E induces an isomorphism of the bundle J : E → E 

defined by the complex multiplication by i. It satisfies J 2 = −id. 

For any complex manifold, the tangent bundle T M → M has a natural 

structure of holomorphic vector bundle. Therefore, in particular 

a complex structure, i.e. an operator J : T M → T M. We say that a 

manifold is almost-complex if the tangent bundle possesses such operator. 

Realize that almost-complex manifold is a notion much weaker 

than that of complex manifold. 

 

.


The cotangent bundle T ∗M of an almost-complex manifold, therefore, 

has a complex structure as well. Define T ∗MC = T ∗M ⊗C, that it 

is naturally identified with the complex valued 1-forms. We have a natural 

division of that vector space between T ∗MC = T 1,0M T 0,1M, corresponding to the eigen-values i and −i of the J operator. We define 

E p,q = p 1,0 q 0,1 T M T M, its elements are called the (p, q)-forms. 

Moreover, for any fix complex vector bundle E we define Ep,q = E⊗E p,q 

whose elements are called the E-valued (p, q)-forms. Also, we have 

Em = E ⊗ m ∗ T MC. Therefore we obtain 

E m = 

E p,q . 

p+q=m 

Let E π → X be a complex vector bundle. We say that a map σ : 

X → E is a section of the bundle if π ◦ σ : X → X is the identity. The 

space of sections Γ(E) is a non-trivial vector space (it always possesses 

the zero section). For instance, the addition of elements is defined as 

(σ1 + σ2)(x) = σ1(x) + σ2(x), ∀x ∈ X, taking advantage of the vector 

space structure of the fibers. We can define sections over any open set 

U ⊂ X and we denote the vector space that they produce as Γ(U, E). 

In the same fashion we can define holomorphic sections as sections 

that satisfy the additional requirement of being holomorphic. Denote 

by H 0 (X, E) the vector space of holomorphic sections of a holomorphic 

vector bundle over a complex manifold. Some known facts: 

- If X is closed, then H 0 (X, E) is finite dimensional. 

- There exists always a choice of morphism ¯ ∂ : Γ(U, E) → Γ(U, E 0,1 ), 

such that ker ¯ ∂ = H 0 (U, E). It corresponds to take the anticomplex 

part of the differential of the map. 

- There is a formula relating its dimension to the topology of the 

space and of the bundle. It is called the Riemann-Roch formula. 

2.2. Positive bundles. An hermitian metric h for a complex bundle 

E is a section of the bundle E ∗ ⊗ E ∗ satisfying that h|Ep : Ep × Ep → C 

is a hermitian metric of the complex vector space Ep. 

A connection ∇E for the vector bundle is a C-linear mapping ∇E : 

Γ(U, E) → Γ(U, E 1 ) satisfying 

(1) ∇E(fs) = df ∧ s + f ∧ ∇E(s), 

for any function f : U → C and section s ∈ Γ(U, E). We have that 

an hermitian metric is compatible with ∇E if the following equation is 

satisfied 

dh(s1, s2) = h(∇Es1, s2) + h(s1, ∇Es2), 

for any pair of sections s1, s2.


Lemma 2.1. For any complex hermitian vector bundle E, there exists 

a compatible connection. Moreover, if the bundle is holomorphic and 

we decompose the connection as ∇E = ∂ + ¯ ∂, there exists a unique 

compatible connection for which ¯ ∂(s) = 0, for any holomorphic section. 

There is a natural way of extending a connection ∇E : Γ(U, E) → 

Γ(U, E 1 ) to a C-linear map 

∇E : Γ(U, E r ) → Γ(U, E r+1 ), 

satisfying equation (1). We can define, then, the operator ΘE := ∇E ◦ 

∇E : Γ(U, E) → Γ(U, E 2 ) that is called the curvature of the hermitian 

complex bundle. We have the following list of facts: 

- The operator is C ∞ -linear, i.e. Θ(fs) = fΘ(s), for any function 

f ∈ Γ(U, C) and section s ∈ Γ(U, E). Therefore, it is a section 

of (E ∗ ⊗ E ∗ ) 2 . 

- Moreover it is of type (E ∗ ⊗ E ∗ ) 1,1 . 

- dΘ = 0. 

- iΘ is a real form. (complexification of a real one) 

Let us restrict ourselves to the case of rank 1 holomorphic vector 

bundles. They are usually called line bundles. Then the curvature is 

a 2-form Θ that by the previous equations satisfies that is closed (i.e 

dΘ = 0). We have that 

Definition 2.2. A hermitian line bundle with connection is called positive 

if iΘ is a symplectic form 1 . 

Recall that the condition just imposes the non-degeneracy of the 

form, since the closedness is guaranteed by the previous results without 

any extra hypothesis. 

Fix a holomorphic line bundle L. Assume that H 0 (X, L) = 0. Fix 

a linear subspace L ⊂ H 0 (X, L), it is usually called a linear system. 

Define the base point set of L as 

BL = {x ∈ X : s(x) = 0, ∀s ∈ L}. 

Then there is a holomorphic map defined as 

φL : X \ BL → PL ∗ 

x → P({φ ∈ L ∗ : φ(s(x)) = 0, ∀s ∈ L}). 

The map defined by the whole H 0 (X, L) is called the Kodaira map. 

1 I will assume some basic knowledge of symplectic geometry, the level provided 

by the first chapters of [MS]


Theorem 2.3 (Kodaira’s embedding.). Let X be a complex manifold 

and L be an hermitian positive line bundle with connection over it. For 

any sufficiently large k > 0 positive integer, the Kodaira map associated 

to L ⊗k is an embedding. 

In fact the dimension of the target projective space is 

 

H 0 (X, L ⊗k ) = k n 

X Θn 

n! + O(kn−1 ), 

with n = dimCX. So, it roughly, increases with k n times the symplectic 

volume of X with respect to the symplectic form Θ. 

2.3. Lefschetz pencils. 

Definition 2.4. A Lefschetz pencil for the complex manifold X and for 

a positive line bundle L is a choice of a 2-dimensional L linear system 

in H 0 (X, L). 

Recall that it is equivalent to fix 2 non-zero sections s1, s2 (a basis of 

the subspace). The map φL with respect to that fixed basis becomes 

φL : X \ BL → CP 1 

x → [s1(x) : s2(x)]. 

We call Lefschetz pencil to this map slightly abusing notation. We can 

think of the map φL as the composition of the Kodaira map with the 

projection 

(2) 

π : CP n \ CP n−2 → CP 1 

[X0 : X1 : · · · : Xn] → [X0 : X1]. 

In other words, any Lefschetz pencil is the restriction of the standard 

projective pencil given by equation (2). 

We say that a complex manifold equipped with a bundle L which 

provides an embedding into PH 0 (X, L) is a polarized complex variety. 

Definition 2.5. The dual variety of a polarized complex manifold X ⊂ 

CP N is the set defined as 

X ∗ = {H ∈ (CP n ) ∗ : H is tangent to X}. 

Theorem 2.6 (Kodaira dual theorem). For a complex manifold polarized 

by a positive line bundle L ⊗k , for any k large enough, the dual 

variety is an irreducible algebraic variety of codimension 1 whose singularity 

locus is of positive codimension. 

A polarization with such a dual variety will be called extra ample. 

Thanks to this, we have the following definition

CP n-2 


Figure 1. Pencil in CP N and restriction to X. The red 

sub-manifolds are the fibers of the pencil. 

Definition 2.7. A pencil L for an extra ample polarization is called 

transverse if it intersects (as a line in PH 0 (X, L)) the dual variety 

transversely at smooth points of it. 

Then, we have 

Corollary 2.8. There are transverse pencils for any extra ample polarization. 

Proof. It is a direct consequence of the Bertini’s theorem since X ∗ 

and L have complementary dimensions and generically they intersect 

transversely (see Figure 2). 

The points of intersection of a Lefschetz pencil with the dual variety 

are singular values of the projection map φL, if the pencil is transverse, 

then the map has isolated critical points {pj} and for each of them 

pj ∈ X \ BL, there are holomorphic coordinates (z1, . . . , zn) around 

pj ∈ X and around φL(pj) ∈ CP 1 for which the map is written as 

φL(z1, . . . , zn) = z 2 1 + · · · z 2 n. 

X


L 

p 1 

Figure 2. Pencil in (CP N ) ∗ and intersection with X ∗ . 

The points of intersection are the critical fibers. 

2.4. Symplectic structures and monodromy. We are in the position 

of giving a purely topological definition of Lefschetz pencil. 

Definition 2.9. Let X be a smooth closed oriented 2n-dimensional 

manifold. An oriented smooth pencil on it is a triple (B, C, f) that 

conforms the following conditions: 

p 2 

(i) B is a codimension 4 submanifold, 

(ii) C is a set of points C = {pj} ∈ X \ B, 

(iii) f : X \ B → CP 1 is a smooth map that is a submersion away 

from C, 

(iv) for any point ∈ B, there are oriented coordinates (z1, . . . , zn) 

around it such that in those coordinates B = {z1 = z2 = 0} and 

moreover f(z1, . . . , zn) = z2 

z1 , 

(v) for any point p ∈ C, there are oriented coordinates (z1, . . . , zn) 

around it such that f(z1, . . . , zn) = z2 1 + · · · + z2 n. 

X *


A transverse Lefschetz pencil of a polarized manifold X is a smooth 

pencil over it. It has more structure because it also satisfies the following 

Definition 2.10. An oriented Lefschetz pencil over a symplectic manifold 

(M, ω) is compatible with the symplectic structure if 

(i) The sub-manifold B is symplectic, 

(ii) The fibers of the map f are symplectic sub-manifolds away from 

the critical points C. 

Away from the sets B and C the pencil is a fibration. We have the 

following general fact about symplectic fibrations 

Lemma 2.11. Let π : X → B a smooth fibration of a symplectic manifold 

(X, ω) with symplectic fibers. There is a natural connection H on 

X making the parallel transport 2 by symplectomorphims. Moreover, the 

parallel transport along contractible loops can be chosen to be through 

Hamiltonian diffeomorphisms. 

Proof. The connection is defined by the equation 

H(x) = {v ∈ T xX : ω(v, w) = 0, ∀w ∈ T xX such that dπx(w) = 0}, 

i.e. we are taking the symplectic orthogonal to the symplectic subspace 

ker dπx. As usual whenever we have a path γ : [0, 1] → B, there is a 

unique lift ˆγx0 provided by the three following conditions 

• ˆγx0(0) = x0, 

• π(ˆγx0(t)) = γ(t), ∀t ∈ [0, 1], 

• ˆγ ′ x0 (t) ∈ H(ˆγx0(t)), ∀t ∈ [0, 1]. 

This is true by the existence and uniqueness theorem for ODE’s. 

Recall that we need a compactness condition to guarantee the existence 

of the flow (not just the derivative of it), but we assume it for now. By 

using this construction, for any path γ : [0, 1] → B we can produce the 

diffeomorphism 

p 1 0 : π −1 (γ(0)) → π −1 (γ(1)) 

x → ˆγx(1). 

To check that it preserves the symplectic structure we have just to 

check it infinitesimally. Define P 1 0 = π −1 (γ{[0, 1]} that is an immersed 

sub-manifold with self-intersections at the self-intersection points of γ. 

Denote X(x) = ˆγx(0) ′ . We just have to check that 

(3) LXω|P = 0. 

2 There is an extra condition to be satisfied related with the completeness of the 

connection that we will have to check case by case


Using the Cartan formula we obtain 

LXω|P = diXω|P + iXdω|P = diXω|P . 

Now recall that X ∈ H and H is symplectically orthogonal to T Ft, so 

iXω|P (v) = 0, ∀v ∈ T Ft. Finally we have iXω(X) = 0 and therefore 

diXω|P = 0 as we wanted to show. So, being true formula (3), we 

obtain that m is a symplectomorphism. 

Check the details of the Hamiltonian case in [MS]. 

2.5. Monodromy of a pencil. The parallel transport of a Lefschetz 

pencil is called the monodromy of it. Let us study it. There is a 

standard way to transform a pencil into a fibration, it amounts to 

a surgery operation on X called a symplectic blow-up (respectively 

complex blow-up in the projective case). Let us describe it from a 

topological viewpoint skipping the details of the construction. 

Start by a codimension 2k (k > 1) symplectic sub-manifold B inside 

X. Denote by (ν(B), η) its symplectic normal bundle. We have that 

the symplectic structure in a small neighborhood of B is completely 

determined by the bundle ν(B). Choose a compatible almost complex 

structure J on ν(B) to make it a complex bundle. Now replace the zero 

section ZB (that is diffeomorphic to B) by the sub manifold Pν(B) → 

B, i.e. the projectivization of the bundle. This can be understood as 

reversing the radial coordinate 

r → 1 

r , 

and adding the infinity hyperplane (by the reversion it shows up at 

the origin). Moreover, to have an oriented manifold we place Pν(B) 

with the opposite complex orientation (see Figure 3). The new bundle 

is denoted ˜ν(B). The surgery takes place close to the zero section, 

therefore we can glue back to M \ B to produce a new manifold ˜ M. 

It possess a symplectic structure inherited by that of M. The submanifold 

E := Pν(B) is called the exceptional divisor and we have 

that 

˜M \ E → M \ B 

is a symplectic diffeomorphism. It is important to observe that there 

is a natural projection ˜ M π → M that restricts to the previous symplectomorphism 

away from E. 

There are multiple choices to be done in the blow-up surgery, in 

particular the identification of the normal bundle of B with the neighborhood 

of the sub-manifold. In the particular case of a Lefschetz 

pencil, we can be careful to ensure the following


ν(B) 0 

0 

∼ 

ν(B) 

P(ν(Β)) 

Figure 3. Blow-up scheme. The zero section gets replaced 

by P(ν(B)) 

Lemma 2.12. Let (f, B, C) a Lefschetz pencil, the blow-up of X along 

B has a natural induced Lefschetz pencil structure defined as (π ◦ 

f, π −1 (B), ∅). 

This just means that the Lefschetz pencil becomes a singular fibration, 

i.e. the projection map is defined everywhere. Realize that now 

the fibers are compact and the parallel transport, as previously defined, 

is global. For the following discussion we will be assuming that we have 

a Lefschetz fibration, i.e. B = ∅, maybe after blowing-up. 

Fix a point z0 ∈ CP 1 and also fix the set of critical values ∆ = 

{z1, . . . , zk}. Choose a smooth path γ : S 1 → CP 1 \ ∆, with γ(0) = z0, 

this induces a symplectomorphism on the fiber F0 = π −1 (z0) by means 

of Lemma 2.11. Define the symplectic mapping class group of the fiber 

as Map(F0) = Symp(F0)/Ham(F0). We have that there is an induced 

map 

m1 : π1(CP 1 \ ∆, z0) → Map(F0),


called the geometric monodromy representation. Fix a set of disjoint 

paths γi : [0, 1] → CP 1 satisfying the following conditions: 

(i) γi(0) = z0, 

(ii) γi(1) = zi, 

(iii) γi is an embedded path. 

This is usually called an arc system. We construct an associated set 

of paths of π1(CP 1 \ ∆, z0) by choosing small positive embedded circle 

paths ci : S1 → CP 1 \ ∆ around each zi. We, then, define li = γ −1 

i ◦ 

ci ◦ γi (see Figure 4). It is clear that {li} are a set of generators of 

π1(CP 1 \ ∆, z0). There is a unique relation among them expressed as 

(4) l1 ◦ · · · ◦ ln = 1 

Therefore, there is a special word in the mapping class group Map(F0) 

written as 

z 6 

z 5 

z 1 

z 0 

z 4 

Figure 4. Paths associated to an arc system. 

(5) m1(l1) ◦ m1(l2) ◦ · · · ◦ m1(lk) = id. 

z 2 

z 3


This is just the translation through the morphism m1 of the relation (4). 

We have to study in more detail the element m1(li) to geometrically 

understand what is happening. 

Definition 2.13. An n-dimensional sub-manifold i : N → M inside a 

2n-dimensional symplectic manifold (M, ω) is Lagrangian if i ∗ ω = 0. 

It is known (due to Weinstein) that a small tubular neighborhood U 

of a Lagrangian sub-manifold L is symplectomorphic to a small tubular 

neighborhood of the zero section of the cotangent bundle T ∗ L with its 

canonical symplectic structure. 

For any diffeomorphism g : M → N, there is a canonical lift g ∗ : 

T ∗ N → T ∗ M that is a symplectomorphism. 

Recall that a Hamiltonian diffeomorphism φ : M → M is completely 

characterized by its Hamiltonian function H : M → R, the Hamiltonian 

is recovered out of the function as the time 1 flow associated to the 

vector field XH defined by the equation 

iXHω = dH. 

Denote by e : Sn → Sn the antipodal map over the sphere, i.e. e(v) = 

−v. We can give the following 

Definition 2.14. The Dehn twist associated over T ∗ S n is the composition 

of the map e ∗ with the Hamiltonian H(v) = ||v|| (choosing an 

arbitrary metric and smoothing out at the origin) 

This map is compactly supported since the Hamiltonian defines the 

geodesic flow at time 1 and the composition e ∗ ◦φH is the identity away 

from an arbitrary small neighborhood of the origin. We can, therefore, 

use the canonical tubular neighborhood theorem for any Lagrangian 

sphere L to define a Dehn twist φL supported on it. Due to the various 

choices in the construction, it works up to Hamiltonian isotopy. 

Therefore, it defines an element [φL] ∈ Map(M). 

Given a path γi as previously defined we can construct the set 

Si = {x ∈ F0 : γx0(1) = pi}. 

Since, the parallel transport is not correctly defined at the end of the 

path, it actually happens that there is not unicity and the set Si is not 

a point. It is actually a Lagrangian sphere, that is called the vanishing 

sphere (or cycle) for the singular value zi. 

Lemma 2.15. The monodromy for the path li is provided by the Dehn 

twist along the vanishing sphere Si associated to the critical value zi. 

So, we have that the monodromy of any path is always the composition 

of a series of Dehn twists.


We may wonder about the uniqueness of the word provided by equation 

(5). There is an exhaustive action of the braid group Bk over the 

disk on the arc systems. Recall that the braid group is defined as the 

mapping class group of the puntured disk D 2 \{z1, . . . , zk} (the infinity 

being z0). The action is the obvious composition. This induces new 

arc systems with new associated words. (see Figure 5) 

z 1 

z 0 

= 

z 2 

z 3 

z 1 

+ 

Figure 5. Composition with a braid element of a path system. 

2.6. Main equivalence theorem I: from pencil to symplectic 

structure. We restrict ourselves to the 4-dimensional case in the following 

discussion. We want to state that the monodromy representation 

provided by the word in formula (5) recovers the symplectic type 

of the manifold up to deformation of the symplectic structure. There 

are a lot of subtleties that we will skip: 

- We consider fibrations instead of pencils. The case of a pencil 

amounts to the study of geometric monodromy in symplectic 

manifolds with boundary (we remove the base points B on 

each fiber producing fibers with boundary). The argument goes 

z 0 

z 1 

z 2 

z 3 

z 2 

z 3 

z 1 

z 2 

z 3


through but it is more technical. For a good summary see 

[AMP]. 

- We restrict ourselves to the dimension 4 case. The statement 

is partially true in higher dimensions, but a long discussion is 

required. See [Go] for the more general statements. 

First an auxiliary construction 

Lemma 2.16. For any genus g ≥ 1 surface Σg and any embedded loop 

L ∈ Σg, there exists a fibration f : X → D 2 with a unique singular 

point at the origin of the disk and such that: 

(i) it conforms the local model (v) in Definition 2.9, 

(ii) the vanishing cycle associated to the singularity is L. 

Proof. Start by a holomorphic Lefschetz pencil φ2 : CP 2 \ B4 → CP 1 

defined by the line bundle of conics L ⊗2 in CP 2 . Blow it up along the 

4 points of the base locus B4 to obtain a singular Lefschetz fibration 

ˆφ2 : CP 2 → CP 1 . The map has 3 singularities and the fibers are spheres 

(they are conics on CP 2 ). Choose the neighborhood of a singular value 

z0 and center a small chart around it U0. Define φ0 = ˆ φ2 restricted to 

ˆV0 = ˆ φ −1 

2 (U0). Therefore we have a Lefschetz fibration over the disk by 

spheres φ0 : ˆ V0 → U0 with vanishing cycle L0. 

Since the monodromy is supported at a tubular neighborhood of L0 

we can take L 1 0 and L 2 0 two copies of L0 at one fiber and transport 

them to produce a pair of 3 dimensional manifolds: ∂V 1 

0 the parallel 

transport of L0 and ∂V 2 

0 the parallel transport of L2. We take V0 

to be the domain inside ˆ V0 whose boundary is ∂V 1 2 

0 ∂V0 and that 

contains the singular point of the fibration. We therefore obtain a 

fibration φ0 : V0 → U0 by cylinders with monodromy generated by the 

Dehn twist around L0. Denote by ¯ φ0 : ∂V0 → U0 its restriction to the 

boundary of V0. 

Define ˆ Ψ0 : Σg × U0 → U0 to be the trivial Lefschetz fibration over 

the disk. Take U(L) a small neighborhood of L in Σg. Set Ψ0 : W0 = 

(Σg \ U(L)) × U0 → U0. Restricted to the boundary we get ¯ Ψ0 : 

∂U(L) × U0 → U0. Under the natural identification ∂U(L) = ∂V0 we 

have that ¯ Ψ0 = ¯ φ0. We can glue the two fibrations along this common 

boundary to obtain f = Ψ0#φ0 : W0#V0 → U0 that is a Lefschetz 

fibration satisfying all the requirements. 

Theorem 2.17. Let L1, . . . Lk be an ordered set of embedded loops in 

Σg such that φLk ◦ · · · ◦ φL1 = id. Assume that g > 1, then there exists 

a unique symplectic Lefschetz fibration up to deformation of the symplectic 

structure with an arc system providing the given set of vanishing 

cycles.


Remark that the symplectic mapping class group Map(Σg) is isomorphic 

to the topological mapping class group 

Mapt(Σg) = Diff + (Σg)/Diff + 0 (Σg). 

This is because the symplectomorphism group in the case of Riemann 

surfaces corresponds to the area preserving diffeomorphism group and 

it is well-known that the area preserving diffeomorphisms retract to 

the orientation preserving ones. 

Proof. First we want to recover a differentiable manifold out of the 

2 π0 2 combinatorial data. We start with X0 = Σg × D → D . We fix a 

genus g Lefschetz fibration as provided by Lemma 2.16 with vanishing 

cycle Li denoted as πk : Vk → Uk.We perform the fiber-connected sum 

of the fibrations π0, π1, . . . , πk that gives a fibration over the larger 

disk V = D 2 #U1# · · · #Uk. So, we clearly get a fibration Xk → V 

over a disk that has an arc system for which the associated word in the 

mapping class group is the required one. 

Now, it it left to glue another disk to close the infinity. The gluing 

morphism is an element λ : S 1 = ∂V → Diff + 0 (Σg). So, the diffeomor- 

phim type of the constructed manifold depends only on the homotopy 

class of the loop λ inside the space Diff + 

0 (Σg). But it is well-known that 

for g > 1, the fundamental group of the space Diff + 0 (Σg) is trivial. 

Therefore, any choice of λ leads to diffeomorphic smooth manifolds; we 

will denote the so built manifold as X. Moreover, there is a section 

e : CP 1 → X. This is because we can choose a point in the central 

fiber Σg that is away from all the support domains of the Dehn twists 

and so the parallel transport of that point provides the section. 

As for the symplectic structure we use an adaptation of a Thurston’s 

argument due to Gompf (and Donaldson). Represent the Poincaré dual 

of the sub-manifold e(CP 1 ) as a closed 2-form τ. Take a covering Ui of 

CP 1 by open contractible sets such that each one contains at most one 

critical value. We distinguish two cases: 

(i) Ui does not contain a critical point. Then we have that fUi : 

f −1 (Ui) (Σg × Ui) → Ui, and we can equip Vi = f −1 (Ui) with 

the standard symplectic product structure induced out of the 

pair of symplectic structures ωUi (restriction of the standard 

symplectic structure over CP 1 ) and ωΣg. The choice of ωΣg is 

such that the restriction of τ to the fiber is cohomologous to it. 

This implies just that 

 

f −1 

ωΣg = 

(pt) 

f −1 τ = 〈f 

(pt) 

−1 (pt), e(CP 1 )〉 = 1,


i.e it has to have volume 1. 

(ii) Ui contains a critical point. We place a symplectic structure 

on W0 as in the previous case. We have a canonical symplectic 

structure on it, because it is a holomorphic Lefschetz fibration 

V0. To check that they glue along the boundary we slightly 

thicken it, we have that in both cases is a band the thickening 

is a band inside T ∗ (S 1 × [−ɛ, ɛ]). Therefore, by Weinstein’s 

tubular neighborhood theorem, they are symplectomorphic. To 

setup the symplectomorphism we may need to slightly shrink 

Ui. We repeat the same trick for making the symplectic form 

ωi restricted to the regular fibers cohomologous to τ. 

Fix a partition of the unity {χi} subordinated to the covering Ui. We 

select 1-forms λi over each Ui such that 

Define 

ωi − τ = dλi. 

ˆτ = τ + Σid(χi ◦ f)λi. 

It is a closed 2-form that is symplectic when restricted to any vertical 

fiber. We define 

ωX,K = ˆτ + Kf ∗ (ω CP 1) 

that is symplectic for any large K > 0. 

In [Go], you can see why the construction is unique up to deformation 

of the sympplectic structure. The intuition is just that in 2dimensions 

the space of symplectic structures is contractible and a 

Lefschetz fibration is (up to standard singularities) a cartesian product 

of 2-dimensional symplectic spaces. 

3. Approximately holomorphic geometry 

The goal of this Section is to reproduce the results provided in the 

first pages of these notes in the projective setting, now for a general 

symplectic manifold. We can formally copy the setup just by defining 

an ”approximately holomorphic geometry”. Most of the Section is 

about creating an approximately holomorphic section. The last Subsection 

adapts the construction to create a approximately holomorphic 

pencil (complex line of sections). 

Let (M, ω) be a symplectic manifold of integer class, i.e. the cohomology 

class [ω] ∈ H 2 (M, R) admits an integer lift 3 . Therefore, there 

is a complex line bundle with connection L such that the curvature of 

3 The form is in the image of the map H 2 (M, Z) → H 2 (M, R).


the connection satisfies curv(∇L) = −iω. We fix a compatible almostcomplex 

structure J in the manifold M. We mean by this an endomorphism 

of the tangent bundle J : T M → T M such that J 2 = −Id and 

g(u, v) = ω(u, Jv) is a Riemannian metric. The space of such adapted 

almost-complex structures is non-empty and connected. 

We have a sequence of line bundles L k = L ⊗k , k ∈ Z + . They are 

naturally hermitian complex line bundles endowed with a connection 

∇k. Fix the sequence of Riemannian metrics gk(u, v) = kω(u, Jv). We 

may introduce the following definition 

Definition 3.1. We say that a sequence of sections sk : M → L k is 

asymptotically holomorphic if the following set of uniform estimates 

hold 

|sk| = O(1), | ¯ ∂sk|gk = O(k−1/2 ), |∇ r sk|gk = O(1), |∇r ¯ ∂sk|gk = O(k−1/2 ), 

for r = 1, 2. 

We also give the following 

Definition 3.2. A section s : M → L k is ɛ-transverse to zero over the 

domain U ⊂ M if at least one of the following conditions hold: 

(i) |s(x)| > ɛ, 

(ii) |∇ks(x)|ɛ. 

This condition sharpens the usual notion of transversality of a function. 

It gives a qualitative version of the usual transversality definition. 

Our goal is to use the following result: 

Proposition 3.3. Let sk : M → L k be an asymptotically holomorphic 

sequence of sections. Assume that for k large enough, the sections are 

ε-transverse to zero all over M. Then, the zero sets of the sections 

Z(sk) are smooth symplectic sub manifolds for k large. 

Proof. At x ∈ Z(sk), there is a unitary vector v ∈ TxM such that 

|∇vsk| > ɛ. At this point we have that 

| ¯ ∂vsk(x)| = O(k −1/2 ), |∂vsk(x)| > 3ɛ 

4 , |∂Jvsk(x)| > 3ɛ 

4 , 

and so, we obtain that |∇Jvsk(x)| > ɛ/2. Thus, we have that ∇sk(x) is 

surjective at x ∈ Z(sk) and so the set Z(sk) is a smooth sub manifold 

by the implicit function theorem. 

By the same argument we have that 

|∂vsk(x)| >> | ¯ ∂vsk(x)|, 

for all x ∈ Z(sk). A simple linear algebra argument shows that the subspace 

TxZ(sk) = ker ∇sk(x) ⊂ (TxM, ω(x), J(x)) is symplectic. The


reason being that it is close to be complex, i.e. TxZ(sk) is approximately 

J-invariant. 

So it is left to show that such kind of sections do exist. 

3.1. Trivializations. We want to trivialize in an approximately holomorphic 

way the manifold and the bundle L k , in order to compute 

things in the euclidean space instead of the manifold. This is our goal 

now. 

Lemma 3.4. For any point x ∈ (M, kω), there is a chart (Uk := 

Bgk (x, k1/2 ), φk : Uk → (C n , ω0)) that satisfies the following conditions: 

a) |(φk)∗J(v) − J0(v)| = ck −1/2 ||v||, 

b) φ ∗ k ω0 = kω. 

Proof. It is just a matter of selecting a standard Darboux chart ψk : 

Uk → C n and compose it with a linear map B ∈ Sp(2n, R), in such a 

way that B ◦ dφ(x) is complex-linear. This immediately provides all 

the bounds trivially. 

Now we go for the bundle 

Lemma 3.5. For any point x ∈ M, there is an asymptotically holomorphic 

sequence of sections σk,x : M → L k satisfying: 

(i) |σk,x(x)| = 1, 

(ii) |∇ r σk,x(y)| ≤ ce −d2 k (x,y)/5 ,, for r = 0, 1, 2, 3. 

(iii) |∇ r ¯ ∂σk,x(y)| ≤ ck −1/2 e −d2 k (x,y)/5 , for r = 0, 1, 2. 

Proof. Fix the Darboux chart φk : Uk → C n , obtained by Lemma 3.4. 

Recall that by means of that chart the bundle L k is pushed-forward to 

the hermitian complex bundle L0 over C n with curvature −iω0. Trivialize 

that bundle by parallel transport along radial directions starting at 

0 ∈ C n . We obtain that the connection in that trivialization becomes 

∇0 = d + 1 

4 (Σzid¯zi − ¯zidzi), 

and so we obtain 

¯∂0 = ¯ ∂ + 1 

4 (Σzid¯zi), 

It is an exercise to check that 

σ0 = e −|z|2 /4 

is a holomorphic section for this bundle. We, therefore, pull-back to 

obtain ˆσk,x = φ ∗ k σ0. Cutting-off by a suitable function, we obtain σk,x 

that satisfies all the required estimates.


3.2. Local transversality. To produce estimated transversality we 

need an estimated Sard Lemma. This is the content of the following 

result. Define 

Fp(δ) = δlog(δ −1 ) −p , δ > 0, p ∈ Z + . 

Theorem 3.6 (Theorem 20 in [Do1]). For σ > 0, let Hδ denote the 

set of functions f on B(0, 1) such that 

(i) ||f|| C 0 ≤ 1, 

(ii) ||∂f|| C 1 ≤ σ. 

Then there is an integer p. depending only on the dimension n, such 

that for any δ with 0 < δ < 1 

2 if σ ≤ Fp(δ), then for any f ∈ Hδ there 

is a w ∈ C with |w| ≤ δ such that f − w is Fp(δ)-trasnverse to zero 

over the region B(0, 1). 

3.3. Globalizing. We need to find an asymptotically holomorphic sequence 

of sections sk : M → Lk that are ɛ-transverse to zero all over 

M. The idea is that there are plenty of those sections and a clever 

choice makes the trick. Let us try an informal approach first. Take a 

finite number of big Darboux charts (Uj, φj), over each of them take 

the lattice L = (k−1/2Z) 2n ⊂ Cn ⊃ φj(Uj), the image of the lattice 

Lj = φ −1 

j (L) is a set of points {xij} on M. Recall that the number of 

points is O(kn ). We define the sequence 

sk = wijσk,xij , 

with wij ∈ C such that |wij| ≤ 1. It is trivial to check that it is asymptotically 

holomorphic. The conclusion is that the space of asymptotically 

holomorphic sequences is huge. The size of the space, informally 

speaking, grows as fast as in the holomorphic setting. 

Now, assume that there is a sequence sk fixed. We want to perturb 

it to make it ɛ-transverse to zero in a neighborhood of a point xij. We 

first trivialize the section by defining 

fij = sk 

, 

σk,xij 

that it is well-defined and with bounded derivatives in the ball Bgk (xij, 1). 

Now we use Lemma 3.4 to trivialize the manifold, we obtain ˆ 

fij = 

fij ◦ φk : B(0, 1) → C. It is a function that, for k large, satisfies the 

hypothesis of Theorem 3.6. We apply it and we obtain that ˆ fij − wij 

is transverse to zero. Going back we have that 

sk − wijσk,xij , 

is transverse to zero over Bgk (xij, 1). Now, there are two obvious ways 

to proceed:


All together method. 

Do the previous process for all the points of the lattices at the same 

time, the set of balls Bgk (xij, 1) cover the manifold. We just have 

to control the interference between different perturbations. The ɛtransversality 

is C 1 -stable, i.e. let f : U ⊂ C n → C be a function 

ɛ-transverse to 0 over U, and let g : U ⊂ C n → C be a function satisfying 

that |g| C 1 ≤ δ, then we have that f −g is (at least) (ɛ−δ)-transverse 

to zero over U. We have that over a fixed point xij the norm of the 

rest of the perturbations created by the other points (recall that they 

exponentially decay) is around 

(6) AδΣ ∞ n=1e (−1/5)n2 

, 

for a fixed A > 0. We need that number to be smaller than δ and it is 

not. 

One by one method. 

Do the process ball by ball. Call δ1 the allowed amount of the perturbation 

in the first ball. We obtain σ1(= Fp(δ1))- transversality in 

the neighborhood of that point. To not completely destroy the obtained 

transversality we impose the following condition for the second 

perturbation 

δ2 = 1 

2 σ1. 

So, denote σ2 = min{σ1/2, Fp(δ1)}, we obtain σ2 transversally in the 

first two balls. We go on to obtain at the end of the day σN-transversality 

all over the manifold. The problem is that N = O(k 2n ) and so, the 

transversality depends on k. Bad news! 

Mixed method. 

Mix the two methods. Construct D 2n sublattices (by taking (DZ) 2n ⊂ 

Z 2n ), D > 0 large but independent of k. We will perturb simultaneously 

over each sub lattice. The sub lattice satisfies the property that 

any two points x and y verify dk(x, y) ≥ D. Then we have that the 

stability equation (equivalent to equation 6) is 

(7) σj ≥ AδΣ ∞ n=1e −(D/5)n2 

, 

for all j = 1, . . . D 2n . So D has to be chosen in such a way that it 

holds for all the sublattices (index j moving). It is required a study of 

the iteration provided by equation (7). This strongly depends on the 

function Fp(δ). Just for fun: we have that 

cδ ≥ Fp(δ) ≥ δ 1+α , 

for any small positive constants c > 0 and α > 0 and for every sufficiently 

small value of δ. The linear function cδ is not achievable as a


result of a perturbation like te one performed in Theorem 3.6 (there 

are counter-examples). The function δ 1+α does not converge for the 

global iteration method just described. In other words, the function 

Fp(δ) is optimal in many ways. 

This concludes the proof of the existence of an asymptotically holomorphic 

sequence of sections, just proving that there are “approximately 

holomorphic” zero dimensional linear systems. 

3.4. Main theorem II: from symplectic structure to pencil. 

Now we will prove the converse of the Theorem 2.17. It is stated 

as follows 

Theorem 3.7. Let (M, ω) be a closed symplectic manifold of integer 

class, for k sufficiently large there exists a symplectic Lefschetz pencil 

(f, B, C) over M such that the fibers are Poincaré dual to the class [kω] 

The statement about the homology class of the fibers just tells that 

the fibers are zeroes of sections of the bundle L k . So we can copy the 

arguments of the previous subsection. We need to slightly generalize 

the setup. Let us introduce the following definitions 

Definition 3.8. A section s : M → E of a hermitian vector bundle is 

ɛ-trasnverse to zero over the domain U if at least one of the following 

two conditions hold for any point x ∈ U: 

(i) |s(x)| ≥ ɛ, 

(ii) ∇s(x) is surjective and it admits a right inverse Rx such that 

|Rx| ≤ ɛ. 

This generalizes the notion of estimated transversality of a section of 

a line bundle. Now, we have that given a bundle E, we can construct 

the sequence E ⊗ L k , then we have 

Definition 3.9. We say that a sequence of sections sk : M → E ⊗ L k 

is asymptotically holomorphic if the following set of uniform estimates 

hold 

|sk| = O(1), | ¯ ∂sk|gk = O(k−1/2 ), |∇ r sk|gk = O(1), |¯ ∂∇sk|gk = O(k−1/2 ), 

for r = 1, 2. 

We can produce sequences of sections sk : M → E ⊗ L k ɛ-transverse 

to zero over M by slightly adapting the arguments of the previous 

Subsection. 

Let us sketch the proof of Theorem 3.7. Let us take an asymptotically 

holomorphic sequence of sections sk,1 ⊕ sk,2 : M → L k L k . We 

perturb the sequence to obtain the following transversality conditions:


(i) sk,1 is ɛ-transverse to zero, 

(ii) sk,1 ⊕ sk,2 is s ɛ-transverse to zero, 

is ɛ-transverse to zero, away from a neighborhood of the 

zero set Z(s,1). 

(iii) ∂ sk,2 

sk,1 

The second condition is the one that ensures the good picture around 

the base point set B = Z(sk,1 ⊕ sk,2). The first one guarantees that the 

zero fiber is symplectic and the critical values of the pencil are away 

from that fiber. The third one makes sure that the singularities follow 

the “approximately holomorphic” local model, and that the fibers are 

symplectic. This last statement is not completely true and in fact the 

map fk = [sk,1 ⊕ sk,2] : M − B → CP 1 needs a perturbation in order to 

satisfy what we claim. The main reason is that 

4. Approximately holomorphic techniques over contact 

manifolds. 

4.1. Almost contact manifolds. 

4.2. Giroux’s theorems. 

4.3. Contact pencils. 

4.4. Existence of contact structures. 

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Cabrera, 13-15, 28049, Madrid, Spain 

E-mail address: fpresas@icmat.es

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