published or accepted
expository / books / lecture notes
slides from talks
The links below marked "pdf" and "ps" allow you to download the same preprint versions that are also
available in the arXiv. Links to online published versions are also
given where available; in most cases these lead to pages with abstracts (publicly available) and downloadable PDF
files (available only to subscribers). If you're not viewing from a university or library that has subscriptions
to the relevant journals, you'll be offered the opportunity to pay an exhorbitant price for access to the
published articles; please do not do that. Instead, feel free to write me a quick e-mail
(my surname at math dot hu dash berlin dot de) and I'll be happy to send you PDF versions of any of the
published articles. I was tempted to post them all on this page, but I have trouble understanding precisely
whether that's legal, so I'm erring on the side of caution.
On symplectic fillings of spinal open book decompositions I: Geometric constructions (joint with Sam Lisi and Jeremy Van Horn-Morris)
Preprint arXiv:1810.12017 (October 2018)
This paper is the first in a two-part series introducing spinal open book
decompositions as a tool to study symplectic fillings of contact 3-manifolds.
Part 1 addresses most aspects of the subject that do not require
holomorphic curve theory, e.g. the existence/uniqueness of contact structures or
symplectic/Stein structures compatible with a given spinal open book or
Lefschetz fibration respectively, plus the construction of non-exact
symplectic cobordisms that realize a topological operation known as
spine removal surgery. Part 2 will then use holomorphic
curves to prove that contact manifolds supported by planar spinal open
books can have their fillings classified in terms of Lefschetz fibrations,
a result that implies the vast majority of known results on classification
of fillings, plus some new ones for contact circle bundles over surfaces.
- 1 page per side (68 pages): pdf (790 kB)
(Note: This paper completely subsumes the manuscript known as
sums, monodromy maps and symplectic fillings, which was occasionally
cited in the past as "in preparation" but is now officially abandoned.)
A pretty good synopsis of the ideas and results in this paper may be
found in this blog
post by Laura Starkston, summarizing a 3-part minicourse I gave at
a workshop in Minnesota
in Summer 2013. It was also advertised in this
talk by Jeremy Van Horn-Morris (video) at the 2012
Georgia Topology Conference.
Transversality and super-rigidity for multiply covered holomorphic curves
Preprint arXiv:1609.09867 (September 2016, last revised May 2019)
ANNOUNCEMENT: The gap that was previously found in this
paper has now been filled. For a full explanation, see
this blog post.
- 1 page per side (86 pages): pdf (978 kB)
This paper answers the long-standing open question about super-rigidity of holomorphic
curves in symplectic Calabi-Yau three-folds: namely, simple J-holomorphic curves in this setting are super-rigid
for generic compatible J, meaning there are only finitely many of them for each genus and the
Gromov-Witten invariants can be reduced entirely to a finite sum of obstruction bundle calculations.
In fact, super-rigidity holds generically for all simple closed index zero curves in dimension at least
six, and (by different arguments that are not so new) also in dimension four for curves of genus zero or one.
This paper also addresses the problem of regularity for multiply covered curves, proving in particular
that unbranched covers of closed curves are generically regular, and finding sharp criteria to
prove the same about branched covers.
The main technical theorem behind these results defines a smooth stratification of the space of branched covers
such that kernels and cokernels have constant dimension on each stratum. A prerequisite for this
is to understand the splitting of a Cauchy-Riemann operator for a multiple cover in terms of the irreducible
representations of its (generalized) automorphism group; this idea is adapted from
Taubes's work on the Gromov invariant, and
in that spirit, we also include a brief informal discussion of wall-crossing phenomena for
generic homotopies of almost complex structures.
(Note: readers interested in a less formal overview of the ideas and results in this paper might
enjoy the series of three blog posts I wrote about it, starting with
"Transversality for multiple covers, super-rigidty, and all that".)
Spine removal surgery and the geography of symplectic fillings (joint with Sam Lisi)
to appear in Michigan Math. J.
Preprint arXiv:1902.01326 (February 2019)
This note is a spin-off of the spinal open book project with Lisi and Van Horn-Morris:
we use spine removal cobordisms to prove that there is a bound on the
geography of minimal symplectic fillings for any contact 3-manifold
supported by a planar spinal open book. Contrary to what many readers
may expect, the argument does not involve Dehn twist factorizations or mapping
class groups. It does require a small amount of holomorphic curve technology,
but nothing fancier than what was current in 1996.
- 1 page per side (15 notes): pdf (264 kB)
Unknotted Reeb orbits and nicely embedded holomorphic curves (joint with Alexandru Cioba)
Symplectic Geom. 18 (2020), no. 1, 57-122
Preprint arXiv:1609.01660 (September 2016, last revised May 2019)
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, meaning it is
not only contractible but is also the embedded boundary of an embedded disk.
The same holds for all contact structures on reducible 3-manifolds.
The proof is a mixture of standard SFT-type techniques with
the intersection theory of punctured holomorphic curves, including
at least one new tool that we expect to have wider application: a
"local" adjunction formula for sequences of holomorphic annuli breaking along a Reeb orbit.
- 1 page per side (50 pages): pdf (647 kB)
Generic transversality for unbranched covers of closed pseudoholomorphic curves (joint with Chris Gerig)
Comm. Pure Appl. Math. 70 (2017), no. 3, 409-443
Preprint arXiv:1407.0678 (December 2014, last revised November 2016)
This is the first installment of a larger project to establish transversality
results for multiply covered holomorphic curves in all dimensions, without
abstract or domain-dependent perturbations. In this paper, we use an analytic
perturbation technique of Taubes to show that for generic tame almost
complex structures J, transversality can be achieved for all
unbranched covers of index 0 closed J-holomorphic curves.
As a consequence, the Gromov-Witten invariants (without descendants) in
dimension four can be computed as finite counts of
honest J-holomorphic curves,
including both simple curves and multiple covers (with rational weights).
- 1 page per side (31 pages): pdf (375 kB)
ps (541 kB)
- 2 pages per side (16 pages): pdf (464 kB)
ps (553 kB)
Subcritical contact surgeries and the topology of symplectic fillings
(joint with Paolo Ghiggini and Klaus Niederkrüger)
de l'École polytechnique - Mathématiques, 3 (2016), pp. 163-208 (open access)
(August 2014, last revised February 2016)
One of the main results of this paper is the fact that the contact prime
decomposition theorem does not extend to higher dimensions, i.e. in any
dimension greater than three, there exist tight contact structures on connected
sums that do not decompose as contact connected sums. This result arises
from a more general investigation of the possible higher-dimensional generalization of Eliashberg's
theorem stating that a symplectic filling of a contact connected sum in
dimension 3 is always obtained by attaching a 1-handle to another filling.
The higher-dimensional version applies to subcritical contact surgeries,
and we prove that at least up to dimension 5, a homotopy theoretic statement
along these lines is true: the belt sphere of a subcritical surgery must
be nullhomotopic in any symplectically aspherical filling.
- 1 page per side (42 pages): pdf (703 kB)
Contact hypersurfaces in uniruled symplectic manifolds always separate
This paper proves that nonseparating contact hypersurfaces can never exist in a closed
uniruled symplectic manifold, hence (by a result of G. Lu) the Weinstein conjecture is
known for all contact hypersurfaces in such settings. This is in some sense a
higher-dimensional followup to my earlier paper with Albers and Bramham, published in AGT.
J. London Math. Soc. 89 (2014), no. 3, 832-852
(February 2012, last revised December 2013)
- 1 page per side (24 pages): pdf
ps (457 kB)
- 2 pages per side (12 pages): pdf (385 kB)
ps (467 kB)
(The final version, which is three times the length of the original, implements the
Cieliebak-Mohnke framework for achieving transversality in Gromov-Witten theory, thus
it does not require unnatural assumptions such as semipositivity. It includes
an appendix showing that the forgetful map in this framework is a pseudocycle, and also
some discussion of a useful recent result of Mohsen involving
the intersection of a Donaldson hyperplane section with a pseudoconvex hypersurface.)
Non-exact symplectic cobordisms between contact 3-manifolds
We introduce a technique for symplectically attaching certain generalized
4-dimensional handles along transverse links and pre-Lagrangian tori in
contact 3-manifolds. This produces non-exact symplectic cobordisms in many
situations where exact cobordisms are known not to exist, e.g. we show that
all contact manifolds with planar torsion admit symplectic cobordisms to all
other contact manifolds, and we characterize a large class of contact
manifolds that admit symplectic caps containing symplectically embedded
J. Differential Geom. 95
(2013), no. 1, 121-182
(August 2010, revised February 2013)
- 1 page per side (50 pages): pdf
ps (1.1 MB)
- 2 pages per side (25 pages): pdf (677 kB)
ps (1.1 MB)
A hierarchy of local symplectic filling obstructions for contact
This paper introduces an infinite hierarchy of new symplectic filling obstructions known as "planar torsion", which
generalizes overtwistedness and Giroux torsion, and causes the vanishing of the
ECH contact invariant. The proof of this makes use of the existence
of a special stable Hamiltonian structure which admits non-generic holomorphic open books of arbitrary genus
(cf. the paper on open book decompositions below).
Math. J. 162 (2013), no. 12, 2197-2283
(January 2010, last revised February 2013)
- 1 page per side (65 pages): pdf
ps (1.5 MB)
- 2 pages per side (33 pages): pdf (884 kB)
ps (1.5 MB)
(Note: This paper supersedes the preprint formerly known as
Holomorphic curves in blown up open books. An old version by
that name remains available as
arXiv:1001.4109 since it has
been cited a few times in papers that are published.)
Weak and strong fillability of higher dimensional contact manifolds
(joint with Patrick Massot and Klaus Niederkrüger)
Math. 192 (2013), no. 2, 287-373
(November 2011, revised January and September 2012)
We generalize to higher dimensions several constructions and results that are
standard in 3-dimensional contact topology, including weak symplectic fillings,
Giroux torsion, and the Lutz twist. In particular, we find the first examples
in dimension 5 of contact manifolds that are weakly but not strongly fillable, as
well as examples in all dimensions that have various characteristics of tightness
(e.g. lack of contractible Reeb orbits, lack of flexibility) and yet are not
weakly fillable. As an ingredient in these constructions, we also generalize to
all even dimensions the existence of exact symplectic manifolds with disconnected
contact type boundary.
- 1 page per side (69 pages): pdf
Algebraic torsion in contact manifolds (joint with Janko Latschev; with an appendix by Michael Hutchings)
We extract a contact invariant from Symplectic Field Theory that defines a
higher order generalization of "algebraic overtwistedness", and thus
measures an infinite scale of "degrees of non-fillability" for contact
manifolds. We discuss examples in dimension three and use these to
derive some non-existence results for exact symplectic cobordisms, some of
which are complementary to the existence results in the paper on
non-exact cobordisms above. As far as we know, this is the first
known contact topological application of the "full" SFT algebra.
Geom. Funct. Anal. 21 (2011), no. 5, 1144-1195
Update February 2012: the published version of this paper contains a minor error in the appendix (see the
posted by Michael Hutchings on his blog). We have corrected this in the most recent update to the arXiv version.
(September 2010, last revised March 2012)
- 1 page per side (53 pages): pdf
ps (872 kB)
- 2 pages per side (27 pages): pdf (654 kB)
ps (892 MB)
Weak symplectic fillings and holomorphic curves (joint with Klaus Niederkrüger)
We introduce a large class of contact 3-manifolds that are
tight but not weakly fillable, or weakly but not strongly fillable,
including many that have no Giroux torsion. We also show that
weak fillings of planar contact manifolds are always deformable to
blow-ups of Stein fillings.
Ann. Sci. École Norm. Sup. (4)
44, fascicule 5 (2011), 801-853
(March 2010, revised May 2010)
- 1 page per side (42 pages): pdf (799 kB)
On non-separating contact hypersurfaces in symplectic 4-manifolds (joint with Peter Albers and Barney Bramham)
We prove obstructions to the existence of non-separating contact hypersurfaces
in symplectic 4-manifolds, e.g. they do not exist whenever the contact
manifold is planar or has Giroux torsion, or if the symplectic manifold is
a ruled surface.
This also introduces the concept of a partially planar contact manifold,
which will be important in some forthcoming papers.
Algebr. Geom. Topol. 10 (2010) 697-737
(January 2009, last revised July 2009)
- 1 page per side (30 pages): pdf (450 kB)
ps (682 kB)
- 2 pages per side (15 pages):
pdf (486 kB)
ps (694 kB)
Open book decompositions and stable Hamiltonian structures
A brief note proving that every planar open book decomposition of a contact
manifold can be made pseudoholomorphic. This proves an important special
case of a result by C. Abbas that was announced several years ago but
not available until recently. It also shows that every open book can
be lifted to a family of pseudoholomorphic curves for nongeneric data
(this will be used in some work in progress).
Expo. Math. 28 (2010), no. 2, 187-199
(August 2008, last revised June 2009)
- 1 page per side (13 pages): pdf (214 kB)
ps (364 kB)
- 2 pages per side (7 pages):
pdf (276 kB)
ps (370 kB)
- Strongly fillable contact manifolds and J-holomorphic foliations
This uses punctured holomorphic spheres in symplectic cobordisms to prove
several new results about symplectic fillings of contact manifolds,
fillable" and "Stein fillable" are equivalent notions when the contact
manifold is planar, and all strong fillings of the 3-torus are
equivalent up to symplectic deformation and blow-up.
Math. J. 151 (2010), no. 3, 337-384
(June 2008, last revised July 2009)
- 1 page per side (44 pages): pdf
ps (820 kB)
- 2 pages per side (22 pages):
pdf (577 kB)
ps (836 kB)
- Automatic transversality and orbifolds of punctured holomorphic
curves in dimension four
This paper generalizes several previously known transversality criteria
for holomorphic curves in dimension 4 to the context of non-simple and
non-immersed curves with cylindrical ends in a symplectic cobordism, and
then uses the setup to exhibit a natural class of moduli spaces that are
smooth for generic J despite containing (unbranched!) multiple
Math. Helv. 85 (2010), no. 2, 347-407
2008, last revised August 2009)
- 1 page per side (58 pages): pdf
ps (757 kB)
- 2 pages per side (29 pages):
ps (778 kB)
- Compactness for embedded pseudoholomorphic curves in
This is the first step in a large project to justify the statement that
"nice holomorphic curves degenerate nicely" (of which the smooth moduli
spaces studied in the transversality paper above are another example). It
possible degenerations of holomorphic curves in the symplectization of
a contact 3-manifold which also have embedded projections into the
3-manifold: in particular, multiple covers can never appear, thus the
compactified moduli spaces are smooth for generic J.
J. Eur. Math. Soc.
(JEMS) 12 (2010), no. 2, 313-342
(March 2007, last
revised March 2008)
- 1 page per side (32 pages): pdf
ps (601 kB)
- 2 pages per side (16 pages):
ps (613 kB)
- Finite energy foliations on overtwisted contact
A cleaner and more elegant presentation of the main result from my
thesis, that every overtwisted contact manifold admits a finite energy
Geom. Topol. 12 (2008) 531-616
(November 2006, last revised March 2008)
- 1 page per side (76 pages): pdf
ps (4.5 MB)
- 2 pages per side (38 pages): pdf
ps (4.5 MB)
- My Ph.D. thesis: Finite energy foliations and surgery on transverse links (January 2005)
Note: this is not the original version, but rather a revision from July 14, 2005.
The original had a gap in the main compactness proof, resulting from an erroneous
statement in Appendix B about degenerating Riemann surfaces with boundary.
This is why I generally try to use words like "obvious" and "clearly" as
little as possible. Anyway, the problem has been fixed in the revision.
- 1 page per side (279 pages): pdf (2.6 MB)
ps (7.6 MB)
- 2 pages per side (140 pages): pdf (2.6 MB)
ps (7.7 MB)
- Stein structures on Lefschetz fibrations and their contact
boundaries (joint with Sam Lisi)
In their article, Baykur and Van Horn-Morris find counterexamples to a
conjecture of Stipsicz and Ozbağci stating that all the Stein
fillings of a given contact 3-manifold should have "uniformly bounded
topology" in certain senses. Their counterexamples rely on a
special case of a basic
symplectic topological result proved in the paper in progress on
symplectic fillings of "spinal open books" by Van Horn-Morris, Lisi and
myself: every allowable Lefschetz fibration over a compact oriented
surface with boundary (not only the disk) admits a Stein structure that
fills a contact
structure uniquely determined by the spinal open book at the boundary.
Lisi and I provided our proof of this fact in this appendix since the
larger paper on spinal open books is not yet available. In the main body
of their paper, Baykur and Van Horn-Morris also provide their own
completely different proof of a very similar result, using symplectic
handle attachments and convex surface theory.
An appendix to the article Families of contact 3-manifolds with
arbitrarily large Stein fillings by R. İnanç Baykur and
Jeremy Van Horn-Morris
J. Differential Geom. 101
(2015), no. 3, 423-465
This is a book-in-progress created originally as lecture notes for the
graduate course Holomorphic Curves in Symplectic and
Contact Geometry at ETH Zürich and HU Berlin in 2010. The project is
currently slightly more than half finished: it includes a detailed development of the
technical background on closed holomorphic curves (though only part of the
compactness theory), leading to an exposition of Gromov's proof of the
nonqueezing theorem. I plan to add material on symplectic 4-manifolds,
punctured holomorphic curves and applications to contact geometry in
- Holomorphic Curves in Low Dimensions: From Symplectic Ruled Surfaces to Planar Contact Manifolds
Springer Lecture Notes in Mathematics, 2018.
This book developed out of some lecture notes
orginally written for a minicourse on holomorphic curves that I gave at IRMA Strasbourg
in October 2012. The stated goal of the notes was to review the appropriate
background material on analysis of holomorphic curves and topology of Lefschetz
pencils, and then explain (from a modern perspective)
complete proofs of the main results
paper on rational and ruled symplectic 4-manifolds. A second goal,
which ended up becoming significantly more prominent in the notes than in
the original minicourse, was to relate those results to subsequent progress
on certain questions in 3-dimensional contact topology, notably the Weinstein
conjecture and the classification of symplectic fillings. The final version
also contains an extra chapter explaining the main result of
McDuff's followup paper
on immersed symplectic spheres, its connection to Gromov-Witten invariants
and the characterization of uniruled symplectic 4-manifolds.
- Contact 3-manifolds, holomorphic curves and intersection theory
Cambridge Tracts in Mathematics, Cambridge University Press, 2020
Preprint arXiv:1706.05540 (September 2013, last revised August 2019)
This book is based on some
lecture notes written originally for a 5-part lecture series
I gave at the LMS Short Course "Topology in Low Dimensions"
at Durham University in August 2013. The notes were meant to serve as a well-motivated
and topologically oriented
introduction to the intersection theory of punctured pseudoholomorphic curves
(developed originally by Richard Siefring) and its applications in 3-dimensional
contact topology. They begin with a brief review of
the closed case and a sample application (McDuff's characterisation of
symplectic ruled surfaces), and then explain the essential ideas and results from
Siefring's intersection theory, concluding with an application to the
classification of symplectic fillings of planar contact manifolds.
The necessary technical background on holomorphic curves is provided (without proofs) but relegated
to the appendices, so that the main body of the text can focus on topological
issues instead of analysis.
The final revision of the appendices also includes self-contained proofs of
the similarity principle, positivity of intersections
and a weak version of the Micallef-White theorem (including a survey of the bits of
elliptic regularity theory that are needed for all this), and a
quick reference on Siefring's intersection theory aimed at researchers
who would like to use it.
- 1 page per side (170 pages): pdf
- Lectures on symplectic field theory
Preprint arXiv:1612.01009 (December 2016)
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.
This electronic version is missing the final three chapters, which will be
included in the printed version, to appear in the EMS Lectures in Mathematics
series. Updates on the publication of the book will be posted here periodically.
Comments and corrections from readers are welcome!
- 1 page per side (343 pages): pdf (3.1 MB)
- Lectures on Holomorphic Curves in Symplectic and
Contact Geometry (work in progress, last revised May 2015)
- 1 page per side (230 pages): pdf
ps (3.04 MB)
- 2 pages per side (115 pages): pdf
ps (3.12 MB)
Note: This is Version 3.3, which contains slightly more material than
the version (3.2) currently available on the arXiv
A beginner's overview of symplectic homology (May 2010)
Not even remotely intended for publication, this was the outcome of an
obsessive weekend in May 2010 after I gave the first talk in the
Symplectic Homology Learning Seminar, which was great fun.
Lecture notes on bundles and
connections (June 2007, last revised September 2008)
- 1 page per side (28 pages): pdf
For some of the talks below, a choice is offered between the
"full" and "basic" version. This usually means that the "full"
version is the actual file I used in the talk, including some
"moving images" of a very primitive type... these can be
illuminating, but if you'd rather do without them and just read
the content, that's what the "basic" version is for.
Back to Chris Wendl's home page
- Equivariant transversality, super-rigidity and all that,
Western Hemisphere Virtual Symplectic Geometry Seminar, April 10, 2020
full pdf version (1.1 MB)
- On symplectic manifolds with contact boundary, or "when is a Stein manifold merely symplectic?", mathematics colloquium at
the Université Libre de Bruxelles, February 9, 2018
full pdf version (2.7 MB)
- What can have a 3-sphere as its boundary, and why should you ask
This is a general audience talk for the UCL AdM Maths Society, given
March 3, 2014.
full pdf version (1.3 MB)
Here also are a couple of nice
videos (not by me) illustrating the 3-body problem.
- Some Tight Contact Manifolds are Tighter
Mathematical Forum on Algebraic Geometry and Symplectic Geometry,
Keio University, Tokyo, January 25, 2013.
(Note: I also gave a very similar talk at the
British Topology Meeting, University of Cambridge, September 8, 2012.)
full pdf version (953 kB)
basic pdf version (262 kB)
- On Contact Topology, Symplectic Field Theory and the PDE That
University pure maths colloquium, December 3, 2012
full pdf version (858 kB)
basic pdf version (227 kB)
- On Symplectic Cobordisms Between Contact Manifolds, from the
Geometry and Topology Workshop, June 2010.
pdf (361 kB)
ps (1.1 MB)
- Open Books and Fiber Sums, SFT and ECH: A Plethora of Obstructions
to Symplectic Filling, some slides that I used for illustration purposes in an otherwise mostly
blackboard talk at the MSRI workshop Symplectic
and Contact Topology and Dynamics: Puzzles and Horizons, March 2010.
pdf (149 kB)
ps (357 kB)
Here's a brief explanation of the slides:
For full details on most of this, see the paper A hierarchy of local symplectic filling obstructions for contact 3-manifolds listed above.
- Slide 1: a depiction of the holomorphic planes that arise from a Lutz twist and can be used to prove that
overtwisted contact manifolds have vanishing contact homology.
- Slide 2: the analogous picture of holomorphic cylinders arising from Giroux torsion, which imply vanishing of
ECH contact invariant, as well as algebraic 1-torsion in Symplectic Field Theory.
- The rest: these are the individual frames of a very low-tech "animation" summarizing the holomorphic curve argument by
which planar torsion in a strongly symplectically fillable contact 3-manifold leads to a contradiction.