arXiv subjects

Chris Wendl

Publications and source records attributed to Chris Wendl.

At least 19 recordsLinked to original sources

arXiv2022

Transversality and super-rigidity for multiply covered holomorphic curves

We develop new techniques to study regularity questions for moduli spaces of pseudoholomorphic curves that are multiply covered. Among the main results, we show that unbranched multiple covers of closed holomorphic curves are generically regular, and simple index zero curves in dimensions greater than four are generically super-rigid, implying e.g. that the Gromov-Witten invariants of Calabi-Yau 3-folds reduce to sums of local invariants for finite sets of embedded curves. We also establish partial results on super-rigidity in dimension four and regularity of branched covers, and briefly discuss the outlook for bifurcation analysis. The proofs are based on a general stratification result for moduli spaces of multiple covers, framed in terms of a representation-theoretic splitting of Cauchy-Riemann operators with symmetries.

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arXiv2020

On symplectic fillings of spinal open book decompositions II: Holomorphic curves and classification

In this second paper of a two-part series, we prove that whenever a contact 3-manifold admits a uniform spinal open book decomposition with planar pages, its (weak, strong and/or exact) symplectic and Stein fillings can be classified up to deformation equivalence in terms of diffeomorphism classes of Lefschetz fibrations. This extends previous results of the third author to a much wider class of contact manifolds, which we illustrate here by classifying the strong and Stein fillings of all oriented circle bundles with non-tangential $S^1$-invariant contact structures. Further results include new vanishing criteria for the ECH contact invariant and algebraic torsion in SFT, classification of fillings for certain non-orientable circle bundles, and a general "symplectic quasiflexibility" result about deformation classes of Stein structures in real dimension four.

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arXiv2019

Contact 3-manifolds, holomorphic curves and intersection theory

This is a revision of some expository lecture notes written originally for a 5-hour minicourse on the intersection theory of punctured holomorphic curves and its applications in 3-dimensional contact topology. The main lectures are aimed primarily at students and require only a minimal background in holomorphic curve theory, as the emphasis is on topological rather than analytical issues. Some of the gaps in the analysis are then filled in by the appendices, which include self-contained proofs of the similarity principle and positivity of intersections, and conclude with a "quick reference" for the benefit of researchers, detailing the basic facts of Siefring's intersection theory.

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arXiv2019

Unknotted Reeb orbits and nicely embedded holomorphic curves

We exhibit a distinctly low-dimensional dynamical obstruction to the existence of Liouville cobordisms: for any contact 3-manifold admitting an exact symplectic cobordism to the tight 3-sphere, every nondegenerate contact form admits an embedded Reeb orbit that is unknotted and has self-linking number -1. The same is true moreover for any contact structure on a closed 3-manifold that is reducible. Our results generalize an earlier theorem of Hofer-Wysocki-Zehnder for the 3-sphere, but use somewhat newer techniques: the main idea is to exploit the intersection theory of punctured holomorphic curves in order to understand the compactification of the space of so-called "nicely embedded" curves in symplectic cobordisms. In the process, we prove a local adjunction formula for holomorphic annuli breaking along a Reeb orbit, which may be of independent interest.

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arXiv2019

Spine removal surgery and the geography of symplectic fillings

We prove that for any contact 3-manifold supported by a spinal open book decomposition with planar pages, there is a universal bound on the Euler characteristic and signature of its minimal symplectic fillings. The proof is an application of the spine removal surgery operation recently introduced in joint work of the authors with Van Horn-Morris.

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arXiv2018

On symplectic fillings of spinal open book decompositions I: Geometric constructions

A spinal open book decomposition on a contact manifold is a generalization of a supporting open book which exists naturally e.g. on the boundary of a symplectic filling with a Lefschetz fibration over any compact oriented surface with boundary. In this first paper of a two-part series, we introduce the basic notions relating spinal open books to contact structures and symplectic or Stein structures on Lefschetz fibrations, leading to the definition of a new symplectic cobordism construction called spine removal surgery, which generalizes previous constructions due to Eliashberg, Gay-Stipsicz and the third author. As an application, spine removal yields a large class of new examples of contact manifolds that are not strongly (and sometimes not weakly) symplectically fillable. This paper also lays the geometric groundwork for a theorem to be proved in part II, where holomorphic curves are used to classify the symplectic and Stein fillings of contact 3-manifolds admitting a spinal open book with a planar page.

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arXiv2016

Lectures on Symplectic Field Theory

This is the preliminary manuscript of a book on symplectic field theory based on a lecture course for PhD students given in 2015-16. It covers the essentials of the analytical theory of punctured pseudoholomorphic curves, taking the opportunity to fill in gaps in the existing literature where necessary, and then gives detailed explanations of a few of the standard applications in contact topology such as distinguishing contact structures up to contactomorphism and proving symplectic non-fillability.

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arXiv2016

Generic transversality for unbranched covers of closed pseudoholomorphic curves

We prove that in closed almost complex manifolds of any dimension, generic perturbations of the almost complex structure suffice to achieve transversality for all unbranched multiple covers of simple pseudoholomorphic curves with deformation index zero. A corollary is that the Gromov-Witten invariants (without descendants) of symplectic 4-manifolds can always be computed as a signed and weighted count of honest J-holomorphic curves for generic tame J: in particular, each such invariant is an integer divided by a weighting factor that depends only on the divisibility of the corresponding homology class. The transversality proof is based on an analytic perturbation technique, originally due to Taubes.

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arXiv2016

Subcritical contact surgeries and the topology of symplectic fillings

By a result of Eliashberg, every symplectic filling of a three-dimensional contact connected sum is obtained by performing a boundary connected sum on another symplectic filling. We prove a partial generalization of this result for subcritical contact surgeries in higher dimensions: given any contact manifold that arises from another contact manifold by subcritical surgery, its belt sphere is null-bordant in the oriented bordism group $ΩSO^*(W)$ of any symplectically aspherical filling $W$, and in dimension five, it will even be nullhomotopic. More generally, if the filling is not aspherical but is semipositive, then the belt sphere will be trivial in $H^*(W)$. Using the same methods, we show that the contact connected sum decomposition for tight contact structures in dimension three does not extend to higher dimensions: in particular, we exhibit connected sums of manifolds of dimension at least five with Stein fillable contact structures that do not arise as contact connected sums. The proofs are based on holomorphic disk-filling techniques, with families of Legendrian open books (so-called "Lobs") as boundary conditions.

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arXiv2014

Lectures on Holomorphic Curves in Symplectic and Contact Geometry

This is a set of expository lecture notes created originally for a graduate course on holomorphic curves taught at ETH Zurich and the Humboldt University Berlin in 2009/2010. The notes are still incomplete, but due to recent requests from readers, I've decided to make a presentable half-finished version available here. Further chapters will be added in future updates.

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arXiv2013

Contact Hypersurfaces in Uniruled Symplectic Manifolds Always Separate

We observe that nonzero Gromov-Witten invariants with marked point constraints in a closed symplectic manifold imply restrictions on the homology classes that can be represented by contact hypersurfaces. As a special case, contact hypersurfaces must always separate if the symplectic manifold is uniruled. This removes a superfluous assumption in a result of G. Lu, thus implying that all contact manifolds that embed as contact type hypersurfaces into uniruled symplectic manifolds satisfy the Weinstein conjecture. We prove the main result using the Cieliebak-Mohnke approach to defining Gromov-Witten invariants via Donaldson hypersurfaces, thus no semipositivity or virtual moduli cycles are required.

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arXiv2013

Non-Exact Symplectic Cobordisms Between Contact 3-Manifolds

We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably less rigid than its counterparts for exact or Stein cobordisms: in particular, we exhibit large new classes of contact 3-manifolds which are symplectically cobordant to something overtwisted, or to the tight 3-sphere, or which admit symplectic caps containing symplectically embedded spheres with vanishing self-intersection. These constructions imply new and simplified proofs of several recent results involving fillability, planarity and non-separating contact type embeddings. The cobordisms are built from generalized symplectic handles which have cores that are arbitrary symplectic surfaces with boundary and co-cores that are symplectic disks or annuli; these can be attached to contact 3-manifolds along sufficiently large neighborhoods of transverse links or pre-Lagrangian tori. We also sketch a construction of J-holomorphic foliations in these cobordisms and formulate a conjecture regarding maps induced on Embedded Contact Homology with twisted coefficients.

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arXiv2013

A Hierarchy of Local Symplectic Filling Obstructions for Contact 3-Manifolds

We generalize the familiar notions of overtwistedness and Giroux torsion in 3-dimensional contact manifolds, defining an infinite hierarchy of local filling obstructions called planar torsion, whose integer-valued order $k \ge 0$ can be interpreted as measuring a gradation in "degrees of tightness" of contact manifolds. We show in particular that any contact manifold with planar torsion admits no contact type embeddings into any closed symplectic 4-manifold, and has vanishing contact invariant in Embedded Contact Homology, and we give examples of contact manifolds that have planar k-torsion for any $k \ge 2$ but no Giroux torsion. We also show that the complement of the binding of a supporting open book never has planar torsion. The unifying idea in the background is a decomposition of contact manifolds in terms of contact fiber sums of open books along their binding. As the technical basis of these results, we establish existence, uniqueness and compactness theorems for certain classes of J-holomorphic curves in blown up summed open books; these also imply algebraic obstructions to planarity and embeddings of partially planar domains. The results are applied further in followup papers on weak symplectic fillings (arXiv:1003.3923, joint with K. Niederkrueger), non-exact symplectic cobordisms (arXiv:1008.2456) and Symplectic Field Theory (arXiv:1009.3262, joint with J. Latschev).

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arXiv2012

Weak and strong fillability of higher dimensional contact manifolds

For contact manifolds in dimension three, the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent. We extend these facts to higher dimensions: in particular, we define a natural generalization of weak fillings and prove that it is indeed weaker (at least in dimension five),while also being obstructed by all known manifestations of "overtwistedness". We also find the first examples of contact manifolds in all dimensions that are not symplectically fillable but also cannot be called overtwisted in any reasonable sense. These depend on a higher-dimensional analogue of Giroux torsion, which we define via the existence in all dimensions of exact symplectic manifolds with disconnected contact boundary.

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arXiv2012

Algebraic Torsion in Contact Manifolds

We extract a nonnegative integer-valued invariant, which we call the "order of algebraic torsion", from the Symplectic Field Theory of a closed contact manifold, and show that its finiteness gives obstructions to the existence of symplectic fillings and exact symplectic cobordisms. A contact manifold has algebraic torsion of order zero if and only if it is algebraically overtwisted (i.e. has trivial contact homology), and any contact 3-manifold with positive Giroux torsion has algebraic torsion of order one (though the converse is not true). We also construct examples for each nonnegative k of contact 3-manifolds that have algebraic torsion of order k but not k - 1, and derive consequences for contact surgeries on such manifolds. The appendix by Michael Hutchings gives an alternative proof of our cobordism obstructions in dimension three using a refinement of the contact invariant in Embedded Contact Homology.

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arXiv2010

Weak Symplectic Fillings and Holomorphic Curves

We prove several results on weak symplectic fillings of contact 3-manifolds, including: (1) Every weak filling of any planar contact manifold can be deformed to a blow up of a Stein filling. (2) Contact manifolds that have fully separating planar torsion are not weakly fillable - this gives many new examples of contact manifolds without Giroux torsion that have no weak fillings. (3) Weak fillability is preserved under splicing of contact manifolds along symplectic pre-Lagrangian tori - this gives many new examples of contact manifolds without Giroux torsion that are weakly but not strongly fillable. We establish the obstructions to weak fillings via two parallel approaches using holomorphic curves. In the first approach, we generalize the original Gromov-Eliashberg "Bishop disk" argument to study the special case of Giroux torsion via a Bishop family of holomorphic annuli with boundary on an "anchored overtwisted annulus". The second approach uses punctured holomorphic curves, and is based on the observation that every weak filling can be deformed in a collar neighborhood so as to induce a stable Hamiltonian structure on the boundary. This also makes it possible to apply the techniques of Symplectic Field Theory, which we demonstrate in a test case by showing that the distinction between weakly and strongly fillable translates into contact homology as the distinction between twisted and untwisted coefficients.

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arXiv2010

Holomorphic Curves in Blown Up Open Books

We use contact fiber sums of open book decompositions to define an infinite hierarchy of filling obstructions for contact 3-manifolds, called planar k-torsion for nonnegative integers k, all of which cause the contact invariant in Embedded Contact Homology to vanish. Planar 0-torsion is equivalent to overtwistedness, while every contact manifold with Giroux torsion also has planar 1-torsion, and we give examples of contact manifolds that have planar k-torsion for any $k \ge 2$ but no Giroux torsion, leading to many new examples of nonfillable contact manifolds. We show also that the complement of the binding of a supporting open book never has planar torsion. The technical basis of these results is an existence and uniqueness theorem for J-holomorphic curves with positive ends approaching the (possibly blown up) binding of an ensemble of open book decompositions.

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arXiv2009

Automatic Transversality and Orbifolds of Punctured Holomorphic Curves in Dimension Four

We derive a numerical criterion for J-holomorphic curves in 4-dimensional symplectic cobordisms to achieve transversality without any genericity assumption. This generalizes results of Hofer-Lizan-Sikorav and Ivashkovich-Shevchishin to allow punctured curves with boundary that generally need not be somewhere injective or immersed. As an application, we combine this with the intersection theory of punctured holomorphic curves to prove that certain geometrically natural moduli spaces are globally smooth orbifolds, consisting generically of embedded curves, plus unbranched multiple covers that form isolated orbifold singularities.

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