LEFSCHETZ PENCILS FRANCISCO PRESAS

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1 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 Quick review of some basic complex geometry Positive bundles Lefschetz pencils Symplectic structures and monodromy Monodromy of a pencil Main equivalence theorem I: from pencil to symplectic structure Approximately holomorphic geometry Trivializations Local transversality Globalizing Main theorem II: from symplectic structure to pencil Approximately holomorphic techniques over contact manifolds Almost contact manifolds Giroux s theorems Contact pencils 23 Date: May, 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 MTM

2 2 FRANCISCO PRESAS 4.4. Existence of contact structures 23 References 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 {V i, φ i } i I. We say that the manifold is complex if φ i : V i φ i (V i ) R 2n C n can be chosen to satisfy that φ 1 j φ i 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 {U j } j J, satisfying that there are diffeomorphisms ψ j : π 1 (U J ) U j C r making the following diagram commutative π 1 (U j ) π U j ψ j Uj C r π 1 id U j. (iv) The map ψ 1 i ψ j : U i Uj Diff(C r ) reduces to ψ ij = ψ 1 i ψ j : U i Uj GL(C r ). 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.

3 LEFSCHETZ PENCILS 3 A morphism of complex vector bundles Φ : E F is a smooth map making the diagram commute π E E Φ F π F 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 {ψ ij } for any covering. In the other hand, any such collection of maps determines a vector bundle if it satisfies the cocycle condition ψ jk ψ ij = ψ ik. 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 {ψ ij } and let F have transition functions {ρ ij }. We define the direct sum vector bundle as the bundle E F with transition functions ( ) ψj 0. 0 ρ ij In the same spirit E F is defined by the transition functions ψ ij ρ ij. The dual bundle E is defined by the transition functions ψij. 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 ψ ij 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.

4 4 FRANCISCO PRESAS The cotangent bundle T M of an almost-complex manifold, therefore, has a complex structure as well. Define T M C = 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 M C = T 1,0 M T 0,1 M, corresponding to the eigen-values i and i of the J operator. We define E p,q = p T 1,0 M q T 0,1 M, its elements are called the (p, q)-forms. Moreover, for any fix complex vector bundle E we define E p,q = E E p,q whose elements are called the E-valued (p, q)-forms. Also, we have E m = E m T M C. 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 (σ 1 + σ 2 )(x) = σ 1 (x) + σ 2 (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 Positive bundles. An hermitian metric h for a complex bundle E is a section of the bundle E E satisfying that h Ep : E p E p C is a hermitian metric of the complex vector space E p. 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(s 1, s 2 ) = h( E s 1, s 2 ) + h(s 1, E s 2 ), for any pair of sections s 1, s 2.

5 LEFSCHETZ PENCILS 5 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 Θ E := 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 B L = {x X : s(x) = 0, s L}. Then there is a holomorphic map defined as φ L : X \ B L 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]

6 6 FRANCISCO PRESAS 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 + O(k n 1 ), n! with n = dim C X. So, it roughly, increases with k n times the symplectic volume of X with respect to the symplectic form Θ 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 s 1, s 2 (a basis of the subspace). The map φ L with respect to that fixed basis becomes φ L : X \ B L CP 1 x [s 1 (x) : s 2 (x)]. We call Lefschetz pencil to this map slightly abusing notation. We can think of the map φ L as the composition of the Kodaira map with the projection (2) π : CP n \ CP n 2 CP 1 [X 0 : X 1 : : X n ] [X 0 : X 1 ]. 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

7 LEFSCHETZ PENCILS 7 X 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 φ L, if the pencil is transverse, then the map has isolated critical points {p j } and for each of them p j X \ B L, there are holomorphic coordinates (z 1,..., z n ) around p j X and around φ L (p j ) CP 1 for which the map is written as φ L (z 1,..., z n ) = z1 2 + zn. 2

8 8 FRANCISCO PRESAS p 2 L p 1 X * Figure 2. Pencil in (CP N ) and intersection with X. The points of intersection are the critical fibers 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: (i) B is a codimension 4 submanifold, (ii) C is a set of points C = {p j } 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 (z 1,..., z n ) around it such that in those coordinates B = {z 1 = z 2 = 0} and moreover f(z 1,..., z n ) = z 2 z 1, (v) for any point p C, there are oriented coordinates (z 1,..., z n ) around it such that f(z 1,..., z n ) = z zn. 2

9 LEFSCHETZ PENCILS 9 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 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 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 dπ x (w) = 0}, i.e. we are taking the symplectic orthogonal to the symplectic subspace ker dπ x. As usual whenever we have a path γ : [0, 1] B, there is a unique lift ˆγ x0 provided by the three following conditions ˆγ x0 (0) = x 0, π(ˆγ x0 (t)) = γ(t), t [0, 1], ˆγ x 0 (t) H(ˆγ x0 (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 ˆγ x (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) = ˆγ x (0). We just have to check that (3) L X ω 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

10 10 FRANCISCO PRESAS Using the Cartan formula we obtain L X ω P = di X ω P + i X dω P = di X ω P. Now recall that X H and H is symplectically orthogonal to T F t, so i X ω P (v) = 0, v T F t. Finally we have i X ω(x) = 0 and therefore di X ω 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] 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 Z B (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

11 LEFSCHETZ PENCILS 11 ν(b) ν(b) 0 0 P(ν(Β)) Figure 3. Blow-up scheme. The zero section gets replaced by P(ν(B)) Lemma 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 z 0 CP 1 and also fix the set of critical values = {z 1,..., z k }. Choose a smooth path γ : S 1 CP 1 \, with γ(0) = z 0, this induces a symplectomorphism on the fiber F 0 = π 1 (z 0 ) by means of Lemma Define the symplectic mapping class group of the fiber as Map(F 0 ) = Symp(F 0 )/Ham(F 0 ). We have that there is an induced map m 1 : π 1 (CP 1 \, z 0 ) Map(F 0 ),

12 12 FRANCISCO PRESAS called the geometric monodromy representation. Fix a set of disjoint paths γ i : [0, 1] CP 1 satisfying the following conditions: (i) γ i (0) = z 0, (ii) γ i (1) = z i, (iii) γ i is an embedded path. This is usually called an arc system. We construct an associated set of paths of π 1 (CP 1 \, z 0 ) by choosing small positive embedded circle paths c i : S 1 CP 1 \ around each z i. We, then, define l i = γ 1 i c i γ i (see Figure 4). It is clear that {l i } are a set of generators of π 1 (CP 1 \, z 0 ). There is a unique relation among them expressed as (4) l 1 l n = 1 Therefore, there is a special word in the mapping class group Map(F 0 ) z 1 z 2 z 6 z 0 z 3 z 5 z 4 Figure 4. Paths associated to an arc system. written as (5) m 1 (l 1 ) m 1 (l 2 ) m 1 (l k ) = id.

13 LEFSCHETZ PENCILS 13 This is just the translation through the morphism m 1 of the relation (4). We have to study in more detail the element m 1 (l i ) to geometrically understand what is happening. Definition 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 X H defined by the equation i XH ω = dh. Denote by e : S n S n the antipodal map over the sphere, i.e. e(v) = v. We can give the following Definition 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 φ H 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 φ L supported on it. Due to the various choices in the construction, it works up to Hamiltonian isotopy. Therefore, it defines an element [φ L ] Map(M). Given a path γ i as previously defined we can construct the set S i = {x F 0 : γ x0 (1) = p i }. 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 S i is not a point. It is actually a Lagrangian sphere, that is called the vanishing sphere (or cycle) for the singular value z i. Lemma The monodromy for the path l i is provided by the Dehn twist along the vanishing sphere S i associated to the critical value z i. So, we have that the monodromy of any path is always the composition of a series of Dehn twists.

14 14 FRANCISCO PRESAS We may wonder about the uniqueness of the word provided by equation (5). There is an exhaustive action of the braid group B k 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 \{z 1,..., z k } (the infinity being z 0 ). The action is the obvious composition. This induces new arc systems with new associated words. (see Figure 5) z 1 z 2 + z 1 z 2 z 1 z 2 z 0 z 3 z3 z 3 z 1 z 2 = z 0 z 3 Figure 5. Composition with a braid element of a path system 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

15 LEFSCHETZ PENCILS 15 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 For any genus g 1 surface Σ g and any embedded loop L Σ g, 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 φ 2 : CP 2 \ B 4 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 B 4 to obtain a singular Lefschetz fibration ˆφ 2 : 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 z 0 and center a small chart around it U 0. Define φ 0 = ˆφ 2 restricted to 1 ˆV 0 = ˆφ 2 (U 0 ). Therefore we have a Lefschetz fibration over the disk by spheres φ 0 : ˆV 0 U 0 with vanishing cycle L 0. Since the monodromy is supported at a tubular neighborhood of L 0 we can take L 1 0 and L 2 0 two copies of L 0 at one fiber and transport them to produce a pair of 3 dimensional manifolds: V0 1 the parallel transport of L 0 and V0 2 the parallel transport of L 2. We take V 0 to be the domain inside ˆV 0 whose boundary is V0 1 V 2 0 and that contains the singular point of the fibration. We therefore obtain a fibration φ 0 : V 0 U 0 by cylinders with monodromy generated by the Dehn twist around L 0. Denote by φ 0 : V 0 U 0 its restriction to the boundary of V 0. Define ˆΨ 0 : Σ g U 0 U 0 to be the trivial Lefschetz fibration over the disk. Take U(L) a small neighborhood of L in Σ g. Set Ψ 0 : W 0 = (Σ g \ U(L)) U 0 U 0. Restricted to the boundary we get Ψ 0 : U(L) U 0 U 0. Under the natural identification U(L) = V 0 we have that Ψ 0 = φ 0. We can glue the two fibrations along this common boundary to obtain f = Ψ 0 #φ 0 : W 0 #V 0 U 0 that is a Lefschetz fibration satisfying all the requirements. Theorem Let L 1,... L k be an ordered set of embedded loops in Σ g such that φ Lk φ L1 = 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.

16 16 FRANCISCO PRESAS Remark that the symplectic mapping class group Map(Σ g ) is isomorphic to the topological mapping class group Map t (Σ g ) = Diff + (Σ g )/Diff + 0 (Σ g ). 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 combinatorial data. We start with X 0 = Σ g D 2 π 0 D 2. We fix a genus g Lefschetz fibration as provided by Lemma 2.16 with vanishing cycle L i denoted as π k : V k U k.we perform the fiber-connected sum of the fibrations π 0, π 1,..., π k that gives a fibration over the larger disk V = D 2 #U 1 # #U k. So, we clearly get a fibration X k 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 + (Σ g ). So, the diffeomorphim type of the constructed manifold depends only on the homotopy class of the loop λ inside the space Diff + 0 (Σ g ). But it is well-known that for g > 1, the fundamental group of the space Diff 0 + (Σ g ) 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 Σ g 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 U i of CP 1 by open contractible sets such that each one contains at most one critical value. We distinguish two cases: (i) U i does not contain a critical point. Then we have that f Ui : f 1 (U i ) (Σ g U i ) U i, and we can equip V i = f 1 (U i ) with the standard symplectic product structure induced out of the pair of symplectic structures ω Ui (restriction of the standard symplectic structure over CP 1 ) and ω Σg. The choice of ω Σg is such that the restriction of τ to the fiber is cohomologous to it. This implies just that ω Σg = τ = f 1 (pt), e(cp 1 ) = 1, f 1 (pt) f 1 (pt)

17 LEFSCHETZ PENCILS 17 i.e it has to have volume 1. (ii) U i contains a critical point. We place a symplectic structure on W 0 as in the previous case. We have a canonical symplectic structure on it, because it is a holomorphic Lefschetz fibration V 0. 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 U i. We repeat the same trick for making the symplectic form ω i restricted to the regular fibers cohomologous to τ. Fix a partition of the unity {χ i } subordinated to the covering U i. We select 1-forms λ i over each U i such that Define ω i τ = dλ i. ˆτ = τ + Σ i d(χ i f)λ i. It is a closed 2-form that is symplectic when restricted to any vertical fiber. We define ω X,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 2- dimensions 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).

18 18 FRANCISCO PRESAS 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 g k (u, v) = kω(u, Jv). We may introduce the following definition Definition 3.1. We say that a sequence of sections s k : M L k is asymptotically holomorphic if the following set of uniform estimates hold s k = O(1), s k gk = O(k 1/2 ), r s k 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) k s(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 s k : 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(s k ) are smooth symplectic sub manifolds for k large. Proof. At x Z(s k ), there is a unitary vector v T x M such that v s k > ɛ. At this point we have that v s k (x) = O(k 1/2 ), v s k (x) > 3ɛ 4, Jvs k (x) > 3ɛ 4, and so, we obtain that Jv s k (x) > ɛ/2. Thus, we have that s k (x) is surjective at x Z(s k ) and so the set Z(s k ) is a smooth sub manifold by the implicit function theorem. By the same argument we have that v s k (x) >> v s k (x), for all x Z(s k ). A simple linear algebra argument shows that the subspace T x Z(s k ) = ker s k (x) (T x M, ω(x), J(x)) is symplectic. The

19 LEFSCHETZ PENCILS 19 reason being that it is close to be complex, i.e. T x Z(s k ) is approximately J-invariant. So it is left to show that such kind of sections do exist 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 (U k := B gk (x, k 1/2 ), φ k : U k (C n, ω 0 )) that satisfies the following conditions: a) (φ k ) J(v) J 0 (v) = ck 1/2 v, b) φ k ω 0 = kω. Proof. It is just a matter of selecting a standard Darboux chart ψ k : U k 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 σ k,x : M L k satisfying: (i) σ k,x (x) = 1, (ii) r σ k,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 φ k : U k 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 L 0 over C n with curvature iω 0. 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 (Σz id z i z i dz i ), and so we obtain 0 = (Σz id z i ), It is an exercise to check that σ 0 = e z 2 /4 is a holomorphic section for this bundle. We, therefore, pull-back to obtain ˆσ k,x = φ k σ 0. Cutting-off by a suitable function, we obtain σ k,x that satisfies all the required estimates.

20 20 FRANCISCO PRESAS 3.2. Local transversality. To produce estimated transversality we need an estimated Sard Lemma. This is the content of the following result. Define F p (δ) = δlog(δ 1 ) p, δ > 0, p Z +. Theorem 3.6 (Theorem 20 in [Do1]). For σ > 0, let H δ 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 F p(δ), then for any f H δ there is a w C with w δ such that f w is F p (δ)-trasnverse to zero over the region B(0, 1) Globalizing. We need to find an asymptotically holomorphic sequence of sections s k : M L k 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 (U j, φ j ), over each of them take the lattice L = (k 1/2 Z) 2n C n φ j (U j ), the image of the lattice L j = φ 1 j (L) is a set of points {x ij } on M. Recall that the number of points is O(k n ). We define the sequence s k = w ij σ k,xij, with w ij C such that w ij 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 s k fixed. We want to perturb it to make it ɛ-transverse to zero in a neighborhood of a point x ij. We first trivialize the section by defining f ij = s k, σ k,xij that it is well-defined and with bounded derivatives in the ball B gk (x ij, 1). Now we use Lemma 3.4 to trivialize the manifold, we obtain f ˆ ij = f ij φ k : 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 ˆf ij w ij is transverse to zero. Going back we have that s k w ij σ k,xij, is transverse to zero over B gk (x ij, 1). Now, there are two obvious ways to proceed:

21 LEFSCHETZ PENCILS 21 All together method. Do the previous process for all the points of the lattices at the same time, the set of balls B gk (x ij, 1) cover the manifold. We just have to control the interference between different perturbations. The ɛ- transversality 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 x ij 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 δ 1 the allowed amount of the perturbation in the first ball. We obtain σ 1 (= F p (δ 1 ))- transversality in the neighborhood of that point. To not completely destroy the obtained transversality we impose the following condition for the second perturbation δ 2 = 1 2 σ 1. So, denote σ 2 = min{σ 1 /2, F p (δ 1 )}, we obtain σ 2 transversally in the first two balls. We go on to obtain at the end of the day σ N -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 d k (x, y) D. Then we have that the stability equation (equivalent to equation 6) is (7) σ j 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 F p (δ). Just for fun: we have that cδ F p (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

22 22 FRANCISCO PRESAS 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 F p (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 Main theorem II: from symplectic structure to pencil. Now we will prove the converse of the Theorem 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 R x such that R x 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 s k : M E L k is asymptotically holomorphic if the following set of uniform estimates hold s k = O(1), s k gk = O(k 1/2 ), r s k gk = O(1), s k gk = O(k 1/2 ), for r = 1, 2. We can produce sequences of sections s k : 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 s k,1 s k,2 : M L k L k. We perturb the sequence to obtain the following transversality conditions:

23 LEFSCHETZ PENCILS 23 (i) s k,1 is ɛ-transverse to zero, (ii) s k,1 s k,2 is s ɛ-transverse to zero, (iii) s k,2 s k,1 is ɛ-transverse to zero, away from a neighborhood of the zero set Z(s,1 ). The second condition is the one that ensures the good picture around the base point set B = Z(s k,1 s k,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 f k = [s k,1 s k,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 Almost contact manifolds Giroux s theorems Contact pencils Existence of contact structures. References [AMP] D. Auroux, V. Muñoz and F. Presas, Lagrangian submanifolds and Lefschetz pencils, J. Symplectic Geom. 3(2) (2005), [BW] W. Boothby and H. Wang, On contact manifolds, Ann. of Math. 68 (1958), [CPP] R. Casals, D. Pancholi, F. Presas, Almost contact implies contact in dimension 5, preprint. [Do1] Donaldson, S. K. Symplectic submanifolds and almost-complex geometry. J. Differential Geom. 44 (1996), no. 4, [Do2] S. Donaldson, Lefschetz pencils on symplectic manifolds. J. Differential Geom. 53 (1999), no. 2, [Do3] S. Donaldson, Lefschetz fibrations in symplectic geometry. Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. 1998, Extra Vol. II, [El] Y. Eliashberg, Classification of overtwisted contact structures on 3 [Ge] manifolds, Invent. math. 98 (1989), H. Geiges, An Introduction to Contact Topology, Cambridge studies in Advanced Mathematics 109. Cambridge University Press (2008). [Ge1] H. Geiges, Contact structures on 1 connected 5 manifolds, Mathematica 38 (1991),

24 24 FRANCISCO PRESAS [Gi] [Go] E. Giroux, Gomtrie de contact: de la dimension trois vers les dimensions suprieures. (French) Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), , Higher Ed. Press, Beijing, R. Gompf, Toward a topological characterization of symplectic manifolds. (English summary) J. Symplectic Geom. 2 (2004), no. 2, [Ha] A. Hatcher, Algebraic Topology. Cambridge University Press (2002). [IM2] A. Ibort, D. Martínez Torres, Lefschetz pencil structures for 2 calibrated manifolds, C. R. Acad. Sci. Paris, Ser. I. 339 (2004), [IMP] A. Ibort, D. Martínez Torres, F. Presas, On the construction of contact submanifolds with prescribed topology, Journal of Differential Geometry 56 (2000), 2, [La] K. Lamotke, The topology of complex projective varieties after S. Lefschetz. Topology 20 (1981), no. 1, [MMP] D. Martínez Torres, V. Muñoz, F.Presas, Open book decompositions for almost contact manifolds, Proceedings of the XI Fall Workshop on Geometry and Physics, Oviedo, 2002, Publicaciones de la RSME. [Ma] J. Martinet, Formes de contact sur les varieétés de dimension 3, in Proc. Liverpool Singularity Sympos. II, Lecture notes in Math. 209 (1971), [MS] D. Mcduff, D. Salamon, Introduction to symplectic topology, Oxford Mathematical Monographs (1998). [Pr1] F. Presas, Lefschetz type pencils on contact manifolds, Asian J. Math. 6 [We] (2002), 2, R. O. Wells, Differential analysis on complex manifolds. Third edition. Graduate Texts in Mathematics, 65. Springer, New York, xiv+299 pp Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, C. Nicolás Cabrera, 13-15, 28049, Madrid, Spain address: fpresas@icmat.es