Symplectic homology of lefschetz fibrations and floer. Is there an analogue of this theorem for symplectic lefschetz fibrations in higher dimensions. Lefschetz fibrations, fiber sums, symplectic 4manifolds with arbitrary. This volume gives an introduction to the surgery techniques adapted to these additional structures. Introduction a lefschetz pencil is a construction that comes from algebraic geometry, but it is closely related with symplectic geometry. We point out some applications to cutandpaste constructions of symplectic 4manifolds. The basic example is the open book in r2 whose binding is the origin and whose pages are the halflines.
The map f is said to have a lefschetz singularity at a point y. Lefschetz fibrations in symplectic geometry semantic scholar. We introduce symplectic manifolds and state some results about them. A publication of the european mathematical society ems. Lefschetz brations and symplectic structures janwillem tel abstract in this thesis, we study a relation between symplectic structures and lefschetz brations to shed some light on 4manifold theory. A study of hamiltonian symplectic geometry from a hodge theoretic point of view on free shipping on qualified orders. X f to a surface f such that all the critical points of f lie in the interior of x and for each critical. Morsetype singularities on stein domains from complex geometry and on. Two symplectic fibrations are never exactly the same. In particular, we will focus on planar contact manifolds. Moreover, we prove that there is a onetoone correspondence similar to the one above between \emphconvex stabilizations of an exact open book and \emphconvex stabilizations of. Fukaya a structures associated to lefschetz brations. Using theorems of eliashberg and mcduff, etnyre proved that the intersection form of a symplectic filling of a contact 3manifold supported by planar open book is negative definite. Columbia symplectic geometry and gauge theory seminar.
The last part discusses applications to lefschetz fibrations and contains many previously unpublished results. The first half of the book focuses on mcduffs characterization of symplectic rational and ruled surfaces, one of the classic early applications of holomorphic curve theory. Constructing lefschetztype fibrations on fourmanifolds. Planar open book decompositions and planar lefschetz. Donaldson, lefschetz fibrations in symplectic geometry, documenta math. We use this to show the existence of fixed points of certain symplectomorphisms. The reader is expected to have a certain background in symplectic geometry. Fukaya categories and picardlefschetz theory book, 2008. Buy equivariant symplectic hodge theory and strong lefschetz manifolds. Mark mclean submitted on 11 sep 2007, last revised 11 feb 2009 this version, v4. Submitted on 11 sep 2007 v1, last revised 11 feb 2009 this version, v4 abstract. In this first paper of a twopart series, we introduce the basic notions relating spinal open books to contact structures and symplectic or stein.
First is reflected in the properties of the lerayserre spectral sequence. Indeed, as shown by gompf and donaldson, a four dimensional manifold has the structure of a lefschetz pencil if and only if it admits a symplectic. We prove that lefschetz fibrations admitting a section of square cannot be decomposed as fiber sums. Simon donaldson has found a role for lefschetz pencils in symplectic topology, leading to more recent research interest in them. We then introduce lefschetz brations, which are a generalization of ber. Over the years, i have worked on symplectic fourmanifolds and lefschetz fibrations, hamiltonian dynamics, the geometry of symplectic diffeomorphisms and symplectic embeddings, and foundational issues related to filtered floer and morse homology. Symplectic fillings and lefschetz fibrations laura starkston december 20, 2015 there is a correspondence between stein llings and lefschetz brations which allows us to understand geometric structures complex symplectic through topological decompositions. Michael usher mathematician at the university of georgia.
Advanced topics in geometry imperial college london. Stabilizations via lefschetz fibrations and exact open books. Ams proceedings of the american mathematical society. In 45, it was pointed out that these categories always come with. Lefschetz fibration over d such that the induced open book at the boundary is isotopic to mzb n. Maksim maydanskiy exotic symplectic manifolds via lefschetz fibrations pdf, ical abstract.
Lefschetz fibrations of 4dimensional manifolds terry fuller department of mathematics california state university, northridge. An important special case is that of a lefschetz bration coming from an anticanonical lefschetz pencil a pencil of hypersurfaces representing the rst chern class on a closed symplectic manifold e. Citeseerx lefschetz fibrations in symplectic geometry. In particular, lefschetz fibrations on symplectic 4manifolds found by donaldson are indecomposable. In particular, x blown up some number of times is a lefschetz. This book will be particularly useful to graduate students and researchers who have basic literacy in symplectic geometry and algebraic topology, and would like to learn how to apply standard techniques from holomorphic curve theory without dwelling more than necessary on the analytical details. The proof presented here uses the language of lefschetz fibrations and pencils, thus it includes some background on these topics, in addition to a survey of the required analytical results on holomorphic curves. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Additionally, these all admit compatible stein structures. Inparticular,wewillfocusonplanarcontactmanifolds,those thatcorrespondtoanopenbookdecompositionwithgenuszeropages,andprovide. We also show that these surgered manifolds admit near symplectic structures, and prove more generally that achiral lefschetz fibrations with sections have near symplectic structures. Symplectic 4dimensional manifolds are known to be characterized as. Moreover the neighborhoods are lefschetz fibered with an easily described open book on the boundary supporting the induced contact structure.
As well, to terry fuller, whose paper in 2003 lefschetz fibrations of 4dimensional manifolds was a truly invaluable source for this project. A note on the signature of lefschetz fibrations with. Lefschetz fibrations, open book decompositions and handlebodies a lefschetz. Symplectic geometry and lefschetz fibrations 4 perelliptic lefschetz fibrations, which have certain symmetry on e, and showed that they can be realized as branched covers over s fibrations section 3. We construct a spectral sequence converging to symplectic homology of a lefschetz fibration whose e 1 page is related to floer homology of the monodromy symplectomorphism and its iterates. Moreover the neighborhoods are lefschetz fibered with an easilydescribed open book on the boundary supporting the induced contact structure. This observation also shows that symplectic lefschetz fibrations are not necessarily fiber sums of holomorphic ones. These results specifically concern 4manifolds that also happen to carry either a complex structure or a symplectic structure. A generalization of chakiris fibrations endo, hisaaki, 2008.
Part of the lecture notes in mathematics book series lnm, volume 1938. In this paper, we study compact convex lefschetz fibrations on compact convex symplectic manifolds i. The last part treats lefschetz fibrations and their fukaya categories and briefly illustrates the theory on the example of amtype milnor fibres. We show that for each k 3 there are infinitely many finite type stein manifolds diffeomorphic to euclidean space r2k which are pairwise distinct as symplectic manifolds. Symplectic lefschetz fibrations with arbitrary fundamental. Since the topology of kahler surfaces is fairly wellunderstood, in. Fibrations baharacu abstractthisnotesurveystwofundamentaltoolsthatunlockseveraltopological properties of contact and symplectic manifolds. The behavior of its differentials gives an information about the topology of symplectomorphism groups. The last part discusses applications to lefschetz fibrations, and contains many previously unpublished results. Pdf symplectic fillings of lens spaces as lefschetz. We define \emphexact open books as boundary open books of compatible exact lefschetz fibrations, and show that any exact open book carries a contact structure. Among other things, smith derived nodal invariants from symplectic geometry. A spinal open book decomposition on a contact manifold is a generalization of a supporting open book which exists naturally e. We show that under appropriate hypotheses, a plumbing of symplectic surfaces in a symplectic 4manifold admits strongly convex neighborhoods.
Surgery on contact 3manifolds and stein surfaces burak. In this paper, we prove a signature formula for allowable lefschetz fibrations over d 2 with planar fiber by computing maslov index appearing in walls nonadditivity formula. Lefschetz critical points constitute a 0dimensional submanifold c f of x, to wit a disjoint union of finitely many points. Symplectic fillings of lens spaces as lefschetz fibrations article pdf available in journal of the european mathematical society 187 july 20 with 66 reads how we measure reads. Holomorphic curves in low dimensions from symplectic.
Simplified broken lefschetz fibrations and trisections of. The necessary topological background on lefschetz fibrations and open book decompositions is developed in. Lefschetz pencils, branched covers and symplectic invariants. The book will be of interest to graduate students and researchers in symplectic geometry and mirror symmetry. Smith studied at the university of oxford, where he received his doctorate in 1999 under the supervision of simon donaldson with thesis symplectic geometry of lefschetz fibrations. Generally, the emphasis is on simplicity rather than generality. Lefschetz fibrations, open book decompositions and handlebodies.
Realizing 4manifolds as achiral lefschetz fibrations. On the boundary we see the giroux correspondence between contact structures and open book decompositions. Restrictions on symplectic fibrations sciencedirect. In other words, are symplectomorphism classes of symplectic lefschetz fibrations in bijective correspondence with factorizations of identity in the symplectic mapping class group of the general fiber. For example, the minimal symplectic fillings of integral planar spinal open books also admit planar lefschetz fibrations, though over more general compact surfaces the topology of which are determined by the spine of the open book. Glen bredon topology and geometry, robert gompf and andras stipsicz 4manifolds and kirby calculus, and alexandru scorpan the wild world of 4manifolds, amongst many others. Stein manifolds are known to symplectic geometers as liouville domains.
There is a correspondence between stein llings and lefschetz brations which allows us to understand geometric structures complex symplectic through topological decompositions. In this case more tools in algebraic geometry are available, at the price that only a proper subset of all lefschetz fibrations is considered. This paper is devoted to restrictions on symplectic fibrations coming from gromovwitten invariants. X, if there are orientationpreserving local coordinates around y and f y, with respect to which it conforms to the complex model z 1, z 2. To appear in advances in mathematical sciences, 2020, volume 21, journal version 4 an introduction to weinstein handlebodies for complements of smoothed toric divisors joint w o.
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