Integral symplectic 4-manifolds may be described in terms of Lefschetz fibrations. In this note we give a formula for the signature of any Lefschetz fibration in terms of the second cohomology of the moduli space of stable curves. As a consequence we see that the sphere in moduli space defined by any (not necessarily holomorphic) Lefschetz fibration has positive "symplectic volume"; it evaluates positively with the Kahler class. Some other applications of the signature formula and some more general results for genus two fibrations are discussed.
We (re)consider how the Fukaya category of a Lefschetz fibration is related to that of the fibre. The distinguishing feature of the approach here is a more direct identification of the bimodule homomorphism involved.
We consider Hamiltonian Floer cohomology groups associated to a Lefschetz fibration, and the structure of operations on them. As an application, we will (under an important additional assumption) equip those groups with connections, which differentiate with respect to the Novikov variable.
This is an announcement of results proved in [GGS1], [GGS2], [C], and [CG] where methods from Lie theory were used as new tools for the study of symplectic Lefschetz fibrations.
We consider the Fukaya category associated to a basis of vanishing cycles in a Lefschetz fibration. We show that each element of the Floer cohomology of the monodromy around infinity gives rise to a natural transformation from the Serre functor to the identity functor. This complements previously known constructions, in a way which fits in well with homological mirror symmetry.
We show that for each k > 3 there are infinitely many finite type Stein manifolds diffeomorphic to Euclidean space R^{2k} which are pairwise distinct as symplectic manifolds.
We prove that any genus-2 Lefschetz fibration without reducible fibers and with ``transitive monodromy'' is holomorphic. The latter condition comprises all cases where the number of singular fibers is not congruent to 0 modulo 40. An auxiliary statement of independent interest is the holomorphicity of symplectic surfaces in S^2-bundles over S^2, of relative degree up to 7 over the base, and of symplectic surfaces in CP^2 of degree up to 17.
We consider a version of the relative Fukaya category for anticanonical Lefschetz pencils. There are direct connections between the behaviour of this category and enumerative geometry: some of these are results announced here, others remain conjectural. The ultimate aim of this approach is to determine the Fukaya category of the Calabi-Yau hypersurfaces that constitute the pencil.
Floer cohomology groups are usually defined over a field of formal functions (a Novikov field). Under certain assumptions, one can equip them with connections, which means operations of differentiation with respect to the Novikov variable. This allows one to write differential equations for Floer cohomology classes. Here, we apply that idea to symplectic cohomology groups associated to Lefschetz fibrations, and obtain a relation with enumerative geometry.
Nadler introduced certain Lagrangian singularities indexed by trees, and determined their microlocal sheaves to be the category of modules over the corresponding tree quiver. Another family of spaces indexed by trees: the tree plumbings of spheres. The Fukaya-Seidel category of the Lefschetz fibration with this plumbing as fiber and all spheres as vanishing cycles is well known to also be modules over the tree quiver. Here we upgrade this matching of categories to a matching of geometry.
This (partially expository) paper discusses Lagrangian Floer cohomology in the context of Lefschetz fibrations, with emphasis on the algebraic structures encountered there. In addition to the well-known directed A_infinity algebras which appear in this situation, one has additional information encoded in a certain bimodule homomorphism. There are two approaches to constructing this homomorphism: in terms of the (noncompact) Lefschetz thimbles in the total space, or else in terms of vanishing cycle in the fibre. We prove a comparison result, which shows that (up to a certain remaining ambiguity…
Using the recent results of Siebert and Tian about the holomorphicity of genus 2 Lefschetz fibrations with irreducible singular fibers, we show that any genus 2 Lefschetz fibration becomes holomorphic after fiber sum with a holomorphic fibration.
In this article, we generalize the classification of genus one Lefschetz fibrations to genus one simplified broken Lefschetz fibrations, which have fibers of genera one and zero. We classify genus one Lefschetz fibrations over the 2-disk with certain non-trivial global monodromies using chart descriptions, and identify the 4-manifolds admitting genus one simplified broken Lefschetz fibrations.
We give a maximal set of disjoint $(-1)$-sections of the well-known Lefschetz fibration constructed by Matsumoto, Cadavid and Korkmaz. In fact, we obtain several such sets for a fixed genus, which implies that the Matsumoto-Cadavid-Korkmaz Lefschetz fibration has more than one supporting minimal Lefschetz pencils. We also determine the diffeomorphism types of the obtained supporting minimal Lefschetz pencils.
It was shown by Usher that any fiber sum of Lefschetz fibrations over $S^2$ is minimal, which was conjectured by Stipsicz. We prove that the converse does not hold by showing that there exists an indecomposable minimal genus-2 Lefschetz fibration.
We study Lagrangian cobordisms with the tools provided by Lagrangian quantum homology. In particular, we develop the theory for the setting of Lagrangian cobordisms or Lagrangians with cylindrical ends in a Lefschetz fibration, and put the different versions of the quantum homology groups into relation by a long exact sequence. We prove various practical relations of maps in this long exact sequence and we extract invariants that generalize the notion of discriminants to Lagrangian cobordisms in Lefschetz fibrations. We prove results on the relation of the discriminants of the ends of a cobord…
We give a short proof of a conjecture of Stipsicz on the minimality of fiber sums of Lefschetz fibrations, which was proved earlier by Usher. We then construct the first examples of genus g > 1 Lefschetz fibrations on minimal symplectic 4-manifolds which, up to diffeomorphisms of the summands, admit unique decompositions as fiber sums.
We show that any 4-manifold, after surgery on a curve, admits an achiral Lefschetz fibration. In particular, we show that the connected sum of any simply connected 4-manifold with a 2-sphere bundle over the 2-sphere will admit an achiral Lefschetz fibration. We also show these surgered manifolds admit near-symplectic structures and prove more generally that achiral Lefschetz fibrations with sections have near-symplectic structures. As a corollary to our proof we obtain an alternate proof of Gompf's result on the existence of symplectic structures on Lefschetz pencils.
In this paper we construct, in all odd complex dimensions, pairs of Liouville domains W_0 and W_1 which are diffeomorphic to the cotangent bundle of the sphere with one extra subcritical handle, but are not symplectomorphic. While W_0 is symplectically very similar to the cotangent bundle itself, W_1 is more unusual. We use Seidel's exact triangles for Floer cohomology to show that the wrapped Fukaya category of W_1 is trivial. As a corollary we obtain that W_1 contains no compact exact Lagrangian submanifolds.
We describe Lefschetz-Bott fibrations on complex line bundles over symplectic manifolds explicitly. As an application, we construct more than one strong symplectic filling of the link of the $A_{k}$-type singularity. In the appendix, we show that the total space of a Lefschetz-Bott fibration over the unit disk serves as a strong symplectic filling of a contact manifold compatible with an open book induced by the fibration.
We construct open symplectic manifolds which are convex at infinity ("Liouville manifolds") and which are diffeomorphic, but not symplectically isomorphic, to cotangent bundles T^*S^{n+1}, for any n+1 \geq 3. These manifolds are constructed as total spaces of Lefschetz fibrations, where the fibre and all but one of the vanishing cycles are fixed. We show that almost any choice of the last vanishing cycle leads to a nonstandard symplectic structure (those choices which yield standard T^*S^{n+1} can be exactly determined). The Corrigendum changes the statement and proof of Lemma 1.1 in the origi…
Questions of geography of various classes of $4$-manifolds have been a central motivating question in $4$-manifold topology. Baykur and Korkmaz asked which small, simply connected, minimal $4$-manifolds admit a genus $2$ Lefschetz fibration. They were able to classify all the possible homeomorphism types and realize all but one with the exception of a genus $2$ Lefschetz fibration on a symplectic $4$-manifold homeomorphic, but not diffeomorphic to $3 \mathbb{CP}^2 \# 11\overline{\mathbb{CP}}^2$. We give a positive factorization of type $(10,10)$ that corresponds to such a genus $2$ Lefschetz f…
The topology of broken Lefschetz fibrations is studied by means of handle decompositions. We consider a slight generalization of round handles, and describe the handle diagrams for all that appear in dimension four. We establish simplified handlebody and monodromy representations for a certain subclass of broken Lefschetz fibrations/pencils, while showing that all near-symplectic closed 4-manifolds can be supported by these a la Auroux, Donaldson, Katzarkov. Various constructions of broken Lefschetz fibrations and a generalization of the symplectic fiber sum operation to the near-symplectic se…
In this short note, we give an explicit construction of inequivalent Lefschetz pencils and fibrations of same genera on blow-ups of all rational and ruled surfaces. This complements our earlier results, concluding that every symplectic 4-manifold, after sufficiently many blow-ups, admits inequivalent Lefschetz pencils and fibrations, which cannot be obtained from one another even via any sequence of fibered Luttinger surgeries.
The purpose of this note is to explain a combinatorial description of closed smooth oriented 4-manifolds in terms of positive Dehn twist factorizations of surface mapping classes, and further explore these connections. This is obtained via monodromy representations of simplified broken Lefschetz fibrations on 4-manifolds, for which we provide an extension of Hurwitz moves that allows us to uniquely determine the isomorphism class of a broken Lefschetz fibration. We furthermore discuss broken Lefschetz fibrations whose monodromies are contained in special subgroups of the mapping class group; n…
In this article, we characterize isomorphism classes of Lefschetz fibrations with multisections via their monodromy factorizations. We prove that two Lefschetz fibrations with multisections are isomorphic if and only if their monodromy factorizations in the relevant mapping class groups are related to each other by a finite collection of modifications, which extend the well-known Hurwitz equivalence. This in particular characterizes isomorphism classes of Lefschetz pencils. We then show that, from simple relations in the mapping class groups, one can derive new (and old) examples of Lefschetz …
We develop techniques to construct explicit symplectic Lefschetz fibrations over the 2-sphere with any prescribed signature and any spin type when the signature is divisible by 16. This solves a long-standing conjecture on the existence of such fibrations with positive signature. As applications, we produce symplectic 4-manifolds that are homeomorphic but not diffeomorphic to connected sums of S^2 x S^2, with the smallest topology known to date, as well as larger examples as symplectic Lefschetz fibrations.
We show that every Stein or Weinstein domain may be presented (up to deformation) as a Lefschetz fibration over the disk. The proof is an application of Donaldson's quantitative transversality techniques.
For each g > 2 and h > 1, we explicitly construct (1) fiber sum indecomposable relatively minimal genus g Lefschetz fibrations over genus h surfaces whose monodromies lie in the Torelli group, (2) fiber sum indecomposable genus g surface bundles over genus h surfaces whose monodromies are in the Torelli group (provided g > 3), and (3) infinitely many genus g Lefschetz fibrations over genus h surfaces that are not fiber sums of holomorphic ones.
We investigate the bounded cohomology of Lefschetz fibrations. If a Lefschetz fibration has regular fiber of genus at least 2 and it has at least two distinct vanishing cycles, we show that its Euler class is not bounded. As a consequence, we exclude the existence of negatively curved metrics on Lefschetz fibrations with more than one singular fiber.
In this paper, we develop symplectic Hodge theory on transversely symplectic foliations. In particular, we establish the symplectic $dδ$-lemma for any such foliations with the (transverse) $s$-Lefschetz property. As transversely symplectic foliations include many geometric structures, such as contact manifolds, co-symplectic manifolds, symplectic orbifolds, and symplectic quasi-folds as special examples, our work provides a unifying treatment of symplectic Hodge theory in these geometries. As an application, we show that on compact $K$-contact manifolds, the $s$-Lefschetz property implies a ge…
In this article we find an upper and lower bound for the slope of genus g hyperelliptic Lefschetz fibrations, which is sharp when g = 2, and demonstrate the strong connection, in general, between the slope of hyperelliptic genus g Lefschetz fibrations and the number of separating vanishing cycles. Specifically, we show that the slope is greater than 4-4/g if and only if the fibration contains separating vanishing cycles. We also improve the existing bound on s/n, the ratio of number of separating vanishing cycles to the number of non-separating vanishing cycles, for hyperelliptic Lefschetz fib…
For every integer g greater than or equal to 2, there exist infinitely many pairwise nonhomeomorphic smooth 4-manifolds that admit genus-g Lefschetz fibrations over S^2 but do not carry any complex structure with either orientation. This extends a recent result of Ozbagci and Stipsicz.
We prove that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We then describe the topology of the regular and singular fibres, in particular calculating their middle Betti numbers. For the example of sl(2,C) we compute the Fukaya--Seidel category of Lagrangian vanishing cycles.
We prove that a positive allowable Lefschetz fibration, PALF in short, admits a structure of exact Lefschetz fibration in the sense of Seidel \cite{Se08}. If the two-fold first Chern class of the total space is zero, we obtain the Fukaya-Seidel category. We prove that the derived Fukaya-Seidel category of PALF is independent of the choice of the symplectic structure. At the end of this paper, we study examples and show that derived Fukaya-Seidel categories have more information than the Milnor lattices of PALFs.
We associate an exact Lefschetz fibration with a pair of a consistent dimer model and an internal perfect matching on it, whose Fukaya category is derived-equivalent to the category of representations of the directed quiver with relations associated with the pair. As a corollary, we obtain a version of homological mirror symmetry for two-dimensional toric Fano stacks.
We consider the suspension operation on Lefschetz fibrations, which takes p(x) to p(x)-y^2. This leaves the Fukaya category of the fibration invariant, and changes the category of the fibre (or more precisely, the subcategory consisting of a basis of vanishing cycles) in a specific way. As an application, we prove part of Homological Mirror Symmetry for the total spaces of canonical bundles over toric del Pezzo surfaces.
We show how to construct broken, achiral Lefschetz fibrations on arbitrary smooth, closed, oriented 4-manifolds. These are generalizations of Lefschetz fibrations over the 2-sphere, where we allow Lefschetz singularities with the non-standard orientation as well as circles of singularities corresponding to round 1-handles. We can also arrange that a given surface of square 0 is a fiber. The construction is easier and more explicit in the case of doubles of 4-manifolds without 3- and 4-handles, such as the homotopy 4-spheres arising from nontrivial balanced presentations of the trivial group.
In this note, we verify that the complex Kodaira dimension $κ^h$ equals the symplectic Kodaira dimension $κ^s$ for smooth 4-manifolds with complex and symplectic structures. We also calculate the Kodaira dimension for many Lefschetz fibrations.
We prove that any symplectic 4-manifold which is not a rational or ruled surface, after sufficiently many blow-ups, admits an arbitrary number of nonisomorphic Lefschetz fibrations of the same genus which cannot be obtained from one another via Luttinger surgeries. This generalizes results of Park and Yun who constructed pairs of nonisomorphic Lefschetz fibrations on knot surgered elliptic surfaces. In turn, we prove that there are monodromy factorizations of Lefschetz pencils which have the same characteristic numbers but cannot be obtained from each other via partial conjugations by Dehn twi…
We explicitly construct genus-2 Lefschetz fibrations whose total spaces are minimal symplectic 4-manifolds homeomorphic to complex rational surfaces CP^2 # p (-CP^2) for p=7, 8, 9, and to 3 CP^2 #q (-CP^2) for q =12,...,19. Complementarily, we prove that there are no minimal genus-2 Lefschetz fibrations whose total spaces are homeomorphic to any other simply-connected 4-manifold with b^+ at most 3, with one possible exception when b^+=3. Meanwhile, we produce positive Dehn twist factorizations for several new genus-2 Lefschetz fibrations with small number of critical points, including the smal…
We prove that every closed oriented smooth 4-manifold X admits a broken Lefschetz fibration (aka singular Lefschetz fibration) over the 2-sphere. Given any closed orientable surface F of square zero in X, we can choose the fibration so that F is a fiber. Moreover, we can arrange it so that there is only one Lefschetz critical point when the Euler characteristic e(X) is odd, and none when e(X) is even. We make use of topological modifications of smooth maps with fold and cusp singularities due to Saeki and Levine, and thus we get alternative proofs of previous existence results. Also shown is t…
In this article, we generalize the results discussed in [arXiv:1004.3762] by introducing a genus to generic fibers of Lefschetz fibrations. That is, we give families of relations in the mapping class groups of genus-1 surfaces with boundaries that represent rational homology disk smoothings of weighted homogeneous surface singularities whose resolution graphs are $3$-legged with a bad central vertex.
We construct examples of Lefschetz fibrations with prescribed singular fibers. By taking differences of pairs of such fibrations with the same singular fibers, we obtain new examples of surface bundles over surfaces with non-zero signature. From these we derive new upper bounds for the minimal genus of a surface representing a given element in the second homology of a mapping class group.
Let M be a smooth 4-manifold which admits a relatively minimal hyperelliptic genus h Lefschetz fibration over the 2-sphere. If all of the vanishing cycles for this fibration are nonseparating curves, then we show that M is a 2-fold cover of a 2-sphere bundle over the 2-sphere, branched over an embedded surface. If the collection of vanishing cycles for this fibration includes separating curves, we show that M is the relative minimalization of a Lefshetz fibration N, with N a 2-fold branched cover of a rational surface, branched over an embedded surface.
We show that under appropriate hypotheses, a plumbing of symplectic surfaces in a symplectic 4-manifold admits strongly convex neighborhoods. Moreover the neighborhoods are Lefschetz fibered with an easily-described open book on the boundary supporting the induced contact structure. We point out some applications to cut-and-paste constructions of symplectic 4-manifolds.
We consider product 4--manifolds S^1 X M, where M is a closed, connected and oriented 3-manifold. We prove that if S^1 X M admits a complex structure or a Lefschetz or Seifert fibration, then the following statement is true: S^1 X M admits a symplectic structure if and only if M fibers over S^1, under the additional assumption that M has no fake 3-cells. We also discuss the relationship between the geometry of M and complex structures and Seifert fibrations on S^1 X M.
We show that hyperelliptic symplectic Lefschetz fibrations are symplectically birational to two-fold covers of rational ruled surfaces, branched in a symplectically embedded surface. This reduces the classification of genus 2 fibrations to the classification of certain symplectic submanifolds in rational ruled surfaces.
We show that if a contact open book $(Σ,h)$ on a $(2n+1)$-manifold $M$ ($n\geq1$) is induced by a Lefschetz fibration $π:W \to D^2$, then there is a one-to-one correspondence between positive stabilizations of $(Σ,h)$ and \emph{positive stabilizations} of $π$. More precisely, any positive stabilization of $(Σ,h)$ is induced by the corresponding positive stabilization of $π$, and conversely any positive stabilization of $π$ induces the corresponding positive stabilization of $(Σ,h)$. We define \emph{exact open books} as boundary open books of compatible exact Lefschetz fibrations, and show that…
It is known that an arbitrary smooth, oriented 4-manifold admits the structure of what is called a broken Lefschetz fibration. Given a broken fibration, there are certain modifications, realized as homotopies of the fibration map, that enable one to construct infinitely many distinct fibrations of the same manifold. The aim of this paper is to prove that these modifications are sufficient to obtain every broken fibration in a given homotopy class of smooth maps. One notable application is that adding an additional "projection" move generates all broken fibrations, regardless of homotopy class.…
We construct two types of non-holomorphic Lefschetz fibrations over $S^2$ with $(-1)$-sections ---hence, they are fiber sum indecomposable--- by giving the corresponding positive relators. One type of the two does not satisfy the slope inequality (a necessary condition for a fibration to be holomorphic) and has a simply-connected total space, and the other has a total space that cannot admit any complex structure in the first place. These give an alternative existence proof for non-holomorphic Lefschetz pencils without Donaldson's theorem.
