Some of these relations will aid us in defining the symplectic surgeries along certain plumbed 4manifolds. Lefschetz fibrations in symplectic geometry semantic scholar. We now have many interesting examples of lefschetz fibrations coming from algebraic geometry, mirror symmetry, and elsewhere, and we are only beginnning to understand how to study them. Also, the homotopy 4spheres arising from nontrivial presentations of the trivial group see kirby25, problem. Using our methods, one can effectively capture various interesting symplectic surfaces in symplectic 4manifolds as multisections, such as seibergwitten basic classes and exceptional classes, or branched loci of compact stein surfaces as branched. We prove upper bounds for the number of critical points in semistable symplectic lefschetz brations. Let x be a compact, connected, oriented, smooth 4manifold. Singular fibers in lefschetz fibrations on manifolds with b2. 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. Advanced topics in geometry imperial college london. We use this to show the existence of fixed points of certain symplectomorphisms.
We will also apply these relations to known lefschetz fibrations to construct new lefschetz fibrations. We develop a technique for gluing relative trisection diagrams of 4manifolds with nonempty connected boundary to obtain trisection diagrams for closed 4manifolds. Simplified broken lefschetz fibrations and trisections of. Morse theoretic methods in nonlinear analysis and in symplectic topology, p biran, o cornea, f lalonde, editors, nato sci. In this paper, we study compact convex lefschetz fibrations on compact convex symplectic manifolds i. Also, the homotopy 4spheres arising from nontrivial presentations of the trivial group see kirby25, problem 5.
In this case more tools in algebraic geometry are available, at the price that only a proper subset of all lefschetz fibrations is considered. Using our methods, one can effectively capture various interesting symplectic surfaces in symplectic 4manifolds as multisections, such as seibergwitten basic classes and exceptional classes, or branched loci of compact stein. For example bertinis theorem is a holomorphic version. The second aspect concerns symplectic geometry, particularly symplectic floer homology. 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 also obtain a new lower bound for the number of nonseparating vanishing cycles in lefschetz pencils and reprove. 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 form. A note on the signature of lefschetz fibrations with.
We then introduce lefschetz brations, which are a generalization of ber. Hyperelliptic szpiro inequality bogomolov, fedor, katzarkov, ludmil, and pantev, tony, journal of differential. Lecture notes geometry of manifolds mathematics mit. Indeed, as shown by gompf and donaldson, a four dimensional manifold has the structure of a lefschetz pencil if and only if it admits. Introduction a lefschetz pencil is a construction that comes from algebraic geometry, but it is closely related with symplectic geometry. Symplectic structures from lefschetz pencils in high. Plan morse theory complex morse theory i on riemann surfaces branched covers i on 4manifolds picard lefschetz theory pencils relation to symplectic geometry lagrangian submanifolds jonny evans eth zurich symplectic picard lefschetz theory 17th april 2012 3.
Nov 21, 2012 in the case of fibers of genus 4, it is shown that after a 2. Slides of a talk given at the gesta 2006 workshop on symplectic and contact topology madrid, august 2006 homological mirror symmetry for blowups of cp 2. X f to a surface f such that all the critical points of f lie in the interior of x and for each critical. Lefschetz fibr ations and exotic symplectic structures on cotangent bundles of spheres the mit faculty has made this article openly available.
We initiate a study of positive multisections of lefschetz fibrations via positive factorizations in framed mapping class groups of surfaces. These fibrations were first discovered on complex surfaces by lefschetz, who used them as a tool in the study of their topology. Existence of broken lefschetz fibrations international. Jonny evans eth zurich symplectic picard lefschetz theory 17th april 2012 11 122 branched covers let t 2 be the 2torus embedded in r 3 and poke an axis through it. 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. 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. This class of maps was first introduced by auroux, donaldson, and katzarkov in ref. In other words, are symplectomorphism classes of symplectic lefschetz fibrations in bijective correspondence with factorizations of identity in the symplectic mapping. Stein manifolds are known to symplectic geometers as liouville domains. 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. Symplectic 4manifolds lefschetz fibrations hurwitz spaces trigonal coverings research originally supported by edge, research training network hprnct200000101, supported by the european human potential programme, and by the deutsche forschungsgemeinschaft. 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. This means that in the case of a complex algebraic variety v, a lefschetz pencil is something like a fibration over the riemann sphere.
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. Broken fibrations form one part of what may be termed near symplectic geometry. Ana cannas da silva, lectures on symplectic geometry available online here. I am interested particularly in near symplectic geometry and in maps from 4manifolds to surfaces lefschetz fibrations and their generalisations. Using our methods, one can effectively capture various interesting symplectic surfaces in symplectic 4manifolds as multisections, such as seibergwitten basic classes and exceptional classes, or branched loci. Lefschetz fibr ations and exotic symplectic structures on. Lefschetz fibrations in symplectic geometry 311 the classical theory sketched in section 1 involves, beyond the existence of a plentiful supply of holomorphic sections, holomorphic versions of various familiar transversality statements. The last section of this work is devoted to discussing in greater detail the case of lefschetz. Symplectic lefschetz fibrations of curves as gonal coverings. Nov 18, 2011 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. Symplectic surgeries along certain singularities and new. A smooth 4manifold x is symplectic if it admits a closed nondegenerate 2form. Lefschetz fibrations and symplectic structures universiteit utrecht.
Keywords symplectic 4manifolds lefschetz fibrations hurwitz spaces trigonal coverings. Gompf had proven that chiral lefschetz fibrations carry symplectic structure. In the case of lefschetz fibrations over a disc, the monodromy homo morphism determines the total space of the fibration up to symplectic deformation. Multisections of lefschetz fibrations and topology of. It concerns the seibergwitten theory of broken lefschetz fibrations and, more particularly, their lagrangian matching invariants gadgets developed by the p. As well, to terry fuller, whose paper in 2003 lefschetz fibrations of 4dimensional manifolds was a truly invaluable source for this project. A pencil is a particular kind of linear system of divisors on v, namely a oneparameter family, parametrised by the projective line. Symplectic lefschetz fibrations with arbitrary fundamental. Theorem 25 concludes our general discussion on how to turn a lefschetz. The ratio of these sections is a meromorphic function on x, i. That construction works in the lowest dimension n 2, but is. Lefschetz brations in this case arise from preliminary structures called lefschetz pencils.
Find materials for this course in the pages linked along the left. Lefschetz fibrations and symplectic homology nasaads. Maksim maydanskiy exotic symplectic manifolds via lefschetz fibrations pdf, ical abstract. Fukaya a structures associated to lefschetz brations. Surfacefibrations, fourmanifolds, and symplectic floer. We introduce symplectic manifolds and state some results about them. We will investigate quantitative symplectic geometry, focusing on the problem of when a given symplectic manifold embeds inside another, and how the answer to this question may depend on quantitative parameters of the spaces in question. Exotic symplectic manifolds from lefschetz fibrations.
We also show that these surgered manifolds admit nearsymplectic structures, and prove more generally that achiral lefschetz fibrations with sections have nearsymplectic structures. Symplectic lefschetz fibrations of curves as gonal. These results specifically concern 4manifolds that also happen to carry either a complex structure or a symplectic structure. Extending the study of lefschetz fibrations, we detect certain constraints on the symplectic fiber sum operation to result in a 4manifold with nontrivial seibergwitten invariant, as well as the selfintersection numbers that sections of broken lefschetz fibrations can acquire. Constructing lefschetztype fibrations on fourmanifolds. A abbondandolo, m schwarz, note on floer homology and loop space homology, from. Citeseerx lefschetz fibrations in symplectic geometry. Columbia symplectic geometry and gauge theory seminar. This connection means that results concerning lefschetz. Recent results in symplectic topology show that lefschetz fibrations pro. We consider a slight generalization of round handles, and describe the handle diagrams for all that appear in dimension four. Michael usher mathematician at the university of georgia. Lefschetz fibrations, open book decompositions and handlebodies a lefschetz.
Lefschetz fibrations and exotic symplectic structures on cotangent bundles of spheres. Lefschetz fibrations, fiber sums, symplectic 4manifolds with arbitrary. Symplectic homology of lefschetz fibrations and floer. Surfacefibrations, fourmanifolds, and symplectic floer homology. In this paper we classify symplectic lefschetz fibrations with empty base locus on a fourmanifold which is the product of a threemanifold with a circle. Among other things, smith derived nodal invariants from symplectic geometry.
By continuing to use our website, you are agreeing to our use of cookies. Slides of a talk given at the summer institute in algebraic geometry seattle, july 2005 mapping class group factorizations and fiber sums of lefschetz fibrations. We begin by explaining our results for stein domains. Oct 23, 2018 both yield decompositions of the ambient 4manifold into much simpler pieces, such as symplectic fibrations or thickened handlebodies, allowing one to bring a hefty combination of ideas and techniques from complex and symplectic geometry, classical 3manifold topology, and geometric group theory. Broken fibrations form one part of what may be termed nearsymplectic geometry. Lefschetz fibrations on complex algebraic varieties 1. 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. Realizing 4manifolds as achiral lefschetz fibrations. Is there an analogue of this theorem for symplectic lefschetz fibrations in higher dimensions. In section 4, we turn our attention to understanding how far certain facts regarding symplectic 4manifolds and lefschetz.
Simon donaldson has found a role for lefschetz pencils in symplectic topology, leading to more recent research interest in them. The goal of this workshop is to bring participants up to the forefront of this subject and discuss the future directions of the field. Gompfs result on the existence of symplectic structures on lefschetz pencils. In that category, however, 4manifolds after perhaps some blowup present themselves as lefschetz fibrations due to a magni cent theorem of s. In sm, smith gave an elegant signature formula using the geometry of. There are several books on symplectic geometry, but i still took the trouble of writing up lecture notes.
We prove that every closed oriented smooth 4manifold x admits a broken lefschetz fibration aka singular lefschetz fibration over the 2sphere. Simplified broken lefschetz fibrations and trisections of 4. 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. These are lecture notes for a course on symplectic geometry in the dutch mastermath program. Exotic symplectic manifolds from lefschetz fibrations by maksim maydanskiy bachelor of arts, university of california, berkeley, june 2004 submitted to the department of mathematics in partial ful. In what follows, we will consider a family of relations in the mapping class group which will be necessary for our main purpose. Broken lefschetz fibrations and smooth structures on 4. It involves moduli spaces of pseudoholomorphic sections of relative hilbert schemes of points on the fibres, with lagrangian boundary conditions which arise by a vanishingcycle construction.