Symplectic Cohomology of Contractible Surfaces
by
David Sean Jackson-Hanen
A.B. in Mathematics, Harvard University, 2007
Submitted to the Department of Mathematics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Mathematics
ASSACHUSETTS INSi
OF TECHNOLOGY
JUN
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
JiBRARIES
June 2014
@ Massachusetts Institute of Technology 2014. All rights reserved.
Signature redacted
A u th or ..............................................................
Department of Mathematics
May 2, 2014
Signature redacted
C ertified by ..........................................................
Paul Seidel
Professor
Thesis Supervisor
Signature redacted
A ccepted by .........................................................
Alexei Borodin
Chairman, Department Committee on Graduate-Theses
17 2014
IE
Symplectic Cohomology of Contractible Surfaces
by
David Sean Jackson-Hanen
Submitted to the Department of Mathematics
on May 2, 2014, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Mathematics
Abstract
In 2004, Seidel and Smith proved that the Liouville manifold associated to Ramanujams
surface contains a Lagrangian torus which is not displaceable by Hamiltonian isotopy, and
hence that higher products of this manifold provide non-standard symplectic structures on
Euclidean space which are convex at infinity. I extend these techniques a wide class of
smooth contractible affine surfaces of log-general type to produce a similar torus. I then
show that the existence of this torus implies the non-vanishing of the symplectic cohomology
of the Liouville manifolds associated to these surfaces, thus confirming a portion of McLeans
conjecture that a smooth variety has vanishing symplectic cohomology if and only if it is
affine ruled.
Thesis Supervisor: Paul Seidel
Title: Professor
3
4
Acknowledgments
First and foremost I would like to thank my advisor, Paul Seidel. Over the past few years
he has provided a seemingly endless well of fantastic ideas and suggestions; his influence on
the following will be immediately apparent. Discussion with Sheel Ganatra, Mark McLean,
Ivan Smith, Mohammed Abouzaid, and Tomasz Mrowka were all very helpful. I would
also like to thank my fellow graduate students, in particular Tiankai Liu, Ailsa Keating,
and Nate Bottman, for support both mathematical and personal. It is no exageration to
say that I would not have been able to complete this process without them. The research
consitituting this thesis was partially suupported by NSF grant DMS-0943787
5
6
Chapter 1
Introduction
In [24], Seidel and Smith proved that the Ramanujam surface-the first known topologically contractible algebraic surface, constructed in [21]- considered with its natural exact
symplectic structure contains a Lagrangian torus which cannot be displaced from itself via
Hamiltonian isotopy. A itself is not homeomorphic to R 4 , but for n > 2 A" is diffeomorphic
to R 4n by the h-cobordism theorem. Thus, the A" provide examples of non-standard symplectic structures on Euclidean space. This was not the first example of an exotic symplectic
structure on Euclidean space-Gromov constructed the first in [10]-but it was the first such
structure which is convex at infinity, in the sense that outside a compact set it is isomorphic
to the positive half of the symplectization of a contact manifold. Convex symplectic manifolds are particularly appealing to study, since they admit good Floer theories that allow
one to define objects such as symplectic cohomology and, in many cases, Fukaya categories.
Over the next half-decade, work by McLean[18] and Seidel and Abouzaid[3] produced many
more exotic convex manifolds, eventually showing that all Liouville manifolds of dimension
at least 6 admit infinitely many pairwise non-isomorphic Liouville structures. Most of these
further exotica were produced via careful choices of Legendrian handle attachment, hence
leaving the algebraic category of the initial results (which is not, of course, to say that there
hasn't been further work on the symplectic geometry of affine varieties; e.g. McLean[19]
proves that for contractible surfaces log-Kodaira dimension is a symplectic invariant.) This
thesis extends the argument of [24] to more general contractible surfaces of log-general type.
Theorem 1.1. Let A be a smooth contractible surface of log-Kodairadimension 2, satisfying
Assumption 7.1 and Assumption 9.1. Then the symplectic cohomology SH*(A) is nonvanishing.
Proof. Overview Let X D A be a smooth projective surface containing A such that D
X \A is a curve with at worst normal crossing singularities, and equip X with a Kahler form
wx (These notions will be explained in more detail in section 3). Let J' be an w-compatible
almost complex structure on X with J' = Jtnrd near D. Let p E D be a crossing point,
and form Y as the blow-up of X x CP 1 = C U {oo} at the point (p, 0); Y is known as the
degeneration to the normal cone of X at P, with its projection map -r : Y -+ CP1. For
~ X.
t j 0 we have a natural identification Y = 7r-'(t)
7
Y has a symplectic form wy for which the natural extension of J' x JCpto Y is Wycompatible. Y contains a Lagrangian submanifold with boundary, K = UtE[_,,EKt, for
some small real E > 0, such that Kt = 7r- (t) n K is a Lagrangian crossing torus at p for Y
when t = 0, and a Clifford torus in the exceptional fibre for t = 0.
Suppose, for a moment, that the minimal energy of non-constant J' holomorphic discs
in (Y, Kt) goes to infinity as t -* 0+. By taking Kt for small t to K via symplectic parallel
transport, one then can produce the following situation: a fixed Lagrangian torus K C A
and symplectic form w, and sequence of almost complex structures J 1 , J 2 ,... such that the
minimal energy of J holomorphic discs with boundary on K is greater than n. We will
show in section 5 that this condition implies that SH*(A) # 0, where we will also define
symplectic cohomology.
In section 4, we show that if the minimal energy does not go to infinity, and if p is chosen
correctly (where the fact that such a correct choice exists is proved in section 9) then there
exists a J' holomorphic map f : CP1 -+ X with f -1(D) = {pt}; in fact the proof provides
energy estimates for f. If we somehow knew that f were an embedding for J' = Jtnd,
then f would provide an algebraic embedding A'
-+
A, which is impossible by [20 3.4.10;
unfortunately there is no way to guarantee that this is the case.
Instead, we prove in section 6 that for generic choices of J' which are regularfor holomorphic
curves of bounded energy intersecting D at one point, the homology class B = f,([CP'])
represented by any such curve must satisfy c1 (B) - B - D > 0. Section 7 then proves that
there no curves that look like Gromov-limits of curves satisfying this homological condition
with the standard almost complex structure, by examining intersection theoretic constraints
imposed by the relatively minimal model of (X, A, D). The brief section 8 then shows how
all of these pieces come together to prove that SH* (A) # 0
The above proof also serves as a rough outline of the thesis. Sections 2 and 3 provide
basic background on affine surfaces and the symplectic geometry thereof. We note here that
Assumption 7.1 and Assumption 9.1 hold in all examples known to the author, but they
play rather different roles in the argument; Assumption 9.1 is fundamental in the sense that
without it there are no incompressible tori at infinity, and thus no small perturbation of
the argument in this thesis could work; while Assumption 7.1 is more of a technicality that
could probably be removed.
8
Chapter 2
Background geometry
We begin by recalling the Liouville manifold structure on a smooth affine algebraic surface;
our presentation draws heavily on [251.
So let A be a smooth affine algebraic variety over C.
A " A'
-
Looking at the closure of A in
P", we see that there exists a projective variety X with A " X; since Hiron-
aka's resolution of singularities [13] proceeds by performing repeated blow-ups along parts
of the singular locus, we may assume X is smooth without changing A so that D = X \ A is
the support of an ample divisor in X; let H be an ample divisor with supp H = D. While
D will generally of course not be smooth, we assume for now that the singularities of D
are simple normal crossings. In other words, at every point p E D there exists an open
coordinate neighborhood p E U with p = (0, 0, ... 0) and
DfnU={(xl,... Xnlxilxi2 ...
ik = 0,il < i2 ... < ik}
Let L = O(H). Since L is ample, it admits a Hermitian metric whose curvature is a
strictly positive (1, 1) form. More precisely, let s E H 0 (X, L) be a holomorphic section with
s- 1 (0) = D. On A = X \ D, we consider the form w = -ddclog(Is[J), where the norm I -[c
is taken with respect to the Hermitian metric. Ampleness implies that for some choice of
metric this is a positive (1, 1) on A, equal (up to a constant) to the curvature of the metric.
Note that - log(IsIc) : A -* R is exhausting, i.e. proper and bounded from below. We also
have
Lemma 2.1. All criticalpoints of -
log(|s1)
are contained in a compact subset of A.
Proof. Let p E D. We show that a sufficiently small neighborhood V C X of p contains no
critical points of - log(jsIL) in V n A by showing that at particular directional derivative is
non-zero near p.
Let U be a small holomorphic coordinate ball around p. Since D is a simple normal crossing
divisor, we may assume that D n U = {(x1 , ... X) C UjxIx 2 ... Xk = 0} for some k < n
after reordering the variables, while p = (0,0 ... 0). Llu is trivializable, and we can pick a
trivialization such that s = x"' ... x', such that wi > 1, since s vanishes precisely along
D. While the metric I -[I need not agree with the standard Euclidean metric I -1, they are
positively proportional, i.e. there exists ) : U -+ R such that I - Jc = eP'I - I with respect to
9
this trivialization. Then
d(- log(IsI )) = d(- log(e|x " - - - xk$
=
-db - w 1 dlog
lxi
- - - --
)
wkdlog IxkI
Write x, in polar coordinates x, = r 1 ei01, and (9, the vector field pointing in the positive radial direction for x 1 when x1 0 0. 0 is smooth at 0, hence do( On) = a0/Or1 is
bounded near 0. On the other hand, widlogIx1l(ari) = wi/ri -+ cc as x 1 -+ 0, while
widlog IxiI(ari) = 0 for i 5 0. We conclude that d(-log(IsIc))(orJ)-+ -oo as z -+ 0, and
thus - log(Is c) has no critical points near p.
0
Note that proof relied essentially on the fact that p was at worst a normal crossing
singularity of D.
With this lemma, we see that q = -log IsI', gives A the structure of a finite-type Stein
manifold. In other words, 0 is an exhausting (proper and bounded from below) strictly
plurisubharmonic function, which is another way of saying that -dd'g is Kahler, and all
the critical points of # are contained in a compact #-1((-oo, C)) for some C >> 0. Shifting perspectives slightly, A is an exact symplectic manifold with primitive 9 = -dc.
The
Liouville vector field associated to 0, i.e. the vector field Z such that 9 = w(Z, .), is then
V0, the gradient taken with respect to the associated Kahler metric. In particular, for any
C >> 0, Z is outward pointing along the boundary of 0-((-oo, C]). This allows us to put
a Liouville manifold structure on affine varieties:
Definition 2.1. A Liouville domain is a compact exact symplectic manifold with boundary
(M, w, 0) with w = dO such that the Liouville vector field Z is outward pointing along
Y = OM. Note that a := 01y is then a contact form on Y. A complete finite-type Liouville
manifold is an open exact symplectic manifold without boundary (M, w, 9) with w = d9 which
can be written M = M Uy Y x [0, oo), where M is a Liouville domain with (4M, 0) = (Y, a)
and Y x [0, oo) is the positive half of the symplectization of Y, i.e. has symplectic form
d(era) for r the radial coordinate on [0, oo). We call Y x [0, oo) the conical end of M.
Note that on the conical end Z = Z, and hence the flow of Z is defined for all positive
times (and negative times, as the negative end is capped off by a compact manifold.) Thus
the exact symplectic structure on an affine variety is never a complete Liouville structure,
since the associated flow will always hit D in a finite time.
However, 0-'((-oo, C]) is a Liouville domain for C sufficiently large. We can then form
the associated complete finite-type Liouville manifold A4 by attaching the infinite cone
q-1(C) x [0, oo). A is diffeomorphic to A, and it is easy to see (using the flow of Z)
that different choices of C that are sufficiently large give rise to exactly symplectomorphic
Liouville manifolds, whence we suppress any mention of C in the notation A.
We had to make many choices to define the plurisubharmonic function # on A. The following lemma asserts that A0 does not depend on those choices up to the appropriate notion
of isomorphism.
Lemma 2.2. Let 0' be another strictly plurisubharmonicfunction on A arisingfrom a section of an ample line bundle with support on the complement of A in some smooth projective
10
completion, where the hypersurface at infinity has simply normal crossing singularities. 0'
induces an exact symplectic structure (w', 0') and hence another complete Liouville manifold
, such that p*0' = 0 + df, where f is
Ao. Then there exists a diffeomorphism p : A0 -+
compactly supported.
Proof. We will sketch the proof of this fact; for more details see (19] Section 8.
We first need a family version of Lemma 2.1. Suppose that we have two line bundles
Lo and L1 together with metrics I - lo and I - 11 and holomorphic sections so and si such
that so (0) = si (0) = D as sets; note that we do not assume that either Lo or L1 are
ample. Let 0 = - log solo and #1 = - log Isili which are both proper smooth functions on
A. Define #t = (1 - t)oo + to1 . Then all critical points of all (0t)tE[0,1 lie in some compact
subset of A; the proof is almost word for word the same as that of Lemma 2.1.
In particular, suppose that L1 and L 2 are both ample. Then #o and 11 are both strictly
plurisubharmonic, and hence so are all 4t. Pick C >> 0 larger than all the critical points
of all Ot. Then form Z, by attaching an infinite cone to Ot-((-oo, C]) for all t. Then
Ot = -dot is a finite-type deformation of Liouville structures on A. A Moser-style argument then proves the existence of a diffeomorphism as in the statement of the lemma. This
proves independence of the choice of ample line bundle and metric on that line bundle, for
a fixed nodal curve.
It remains to show that the structure is independent of the choice of curve D. Suppose
first that D' is a blow-up of D, i.e. there exists a blow-up morphism 7r : X' -+ X with
ir- 1 (D) = D' and the blow-up locus contained in D. D' also supports an ample divisor; let
Lo be an ample line bundle on X' with section so such that so (0) = D', metric I - lo, and
strictly plurisubharmonic function 0 = - log IsoI on X' \ D' :: A.
are the corresponding data on (X, D), let (Li, si, I-1, #1) = (7r*L, r*s, 7r*
1
1, 7r* X'\D,(0)). We have si (0) = D', and while Lj is not ample, #1 is still strictly plurisubharmonic, since 7r is a diffeomorphism over X' \ D'. Thus as in the previous paragraph the
complete Liouville manifold associated to (X' \ D', -do) is Liouville isomorphic the complete Liouville manifold associated to (X' \ D', -do,), and the later is isomorphic via 7r to
that of (X \ D, -d#). Finally, Hironaka's resolution of singularities implies that for any
two resolutions, there exists a third resolution which is a common blow-up of both of them.
Thus the Liouville structure on an affine variety is unique up to Liouville isomorphism. 5
Then if (L, s,
-|,
4)
We also need to specify the class of almost complex structures on complete finite-type
Liouville manifolds appropriate for constructing Floer theories. Given a complete finite
type Liouville manifold M with a fixed conical end, we say that an w-compatible almost
complex structure J is of contact type, or convex, if outside of a compact set -d(er)o J = 9,
with r the radial coordinate. This condition is equivalent to J(0) = R, where R is the
Reeb vector field on Y. On an affine variety the natural almost complex structure J is of
contact type, after possibly rescaling the plurisubharmonic function by a constant, and the
flow of Z = Z preserves J. Thus there is a natural extension of J to a convex almost structure j on A; this is in fact true for any J that is standard outside of a compact subset of A.
The preceeding discussion was entirely general; we now specialize to dimc(A) = 2.
11
In
this case, one can ensure that D is a singular Riemann surface with at worst simple normal
crossing singularities, i.e. nodal singularities. Thus the conditional results we have proved
in this section apply. We can describe such a divisor via its dual graph as follows: we draw
one vertex vi for each irreducible component Di C D, and we draw one edge connecting
vi and vj for each nodal singularity of D in Di n Dj. If i 4 j this number is obviously
IDi f D l, but nodal self-intersections are possible.
Lemma 2.3. : Suppose A is a smooth affine surface which is topologically contractible
(in the analytic topology), and (X, A, D) is a projective compactification of A with simple
normal crossings. Then the dual graph of D is a rationaltree, i.e. all irreducible components
of D are rational, and the dual graph is a tree.
Proof. This is a simple application of Alexander duality. From the long exact sequence in
singular homology we have
0
o H 3 (D, R) -+ H3 (X, R) -+ H
H1 (A, R) = 0
(X, D, R)
3
Hence by Poincare duality H1 (X, R) = 0, and so H, (X, R)
sequence we have
o = H 2 (A, R)
~ H 2 (X, D, R)
-*
H,(D, R)
But the only nodal Riemann surfaces with H, (D, R)
=
=
-
0. Again using the long exact
H(X,R) = 0
0 are rational trees.
Since Hi(X, D, Z) = H 4 -i(A, Z) = 0 for i = 2, 3, we have that H2 (D, Z) -+ H 2 (X, Z) is
an isomorphism. This allows us to determine the intersection form of X from D: it is a free
abelian group generated by the images of the irreducible components of D; the intersection
numbers of distinct components are either 0 or 1 depending on whether they intersect;
and the self-intersection numbers of the given components (for this reason, the dual graph
is typically decorated with integers at each vertex corresponding to the self-intersection
number.)
12
Chapter 3
Deformation to the Normal Cone
In this section we have a triple (X, A, D) where X is a smooth projective surface, D a curve
with normal crossing singularities supporting an ample divisor H and A = X \ D a smooth
affine surface.
Definition 3.1. Let p c sing D be nodal point. Let V - p be small coordinate neighborhood
of p with respect to which p = (0,0), D n V = {x, ylxy = 0}, and the Kahler form restricted
to V is standard, i.e. wIu = E dzi A dzi. This is possible by Lemma 3.1 below. Then a
crossing torus at p is a torus L C V\D which is isotopic inside V\D to La,b = {(x, y)I =
a, yj = b} for a, b c R. Note that La,b is a Lagrangian torus.
Given a crossing torus L = La,b, intuitively what we would like to do is shrink L towards
p, i.e. let a, b -+ 0, and examine the behavior of holomorphic discs with boundary on L.
We will realize this situation via the algebro-geometric notion of degeneration to the normal
cone at p, which has the advantage of being smooth even in the limit when the torus shrinks
"all the way" to p.
Definition 3.2. Forp E X, the Deformation to the Normal Cone of X at p is the variety
Y given by the blow-up of X x CP1 at (p, 0)
We have a fibration 7r : Y -+ CP1 . For the t : 0, the fibre Y is isomorphic to X,
while for t = 0, the fibre consists of two irreducible components, Y = Z U P, where P is
exceptional fibre (thinking of Y as a blow-up of X x CP', Z is isomorphic to the blow-up
of X at p. The intersection Zfn P is equal to a line in P = Cp2, and the exceptional divisor
in Z.
X x CPR is Kahler with the product metric, so by [9] p.185, Y also admits a Kahler metric.
It is worthwhile for our purposes to sketch the argument, since we will use details of the
construction later on.
Proof. that Y is Kahler. Consider L = O(H) an ample line bundle with metric I - kc,
0(1) -+ CP' equipped with the standard metric, and wxCPi the curvature form of the
product metric. Letting f : Y -+ X x CP' be the blow-up, the form f*WXxCp1 is then
semi-positive, and positive away from tangent vectors along P. Now consider 0(-P). In a
suitable neighborhood V of P O(-P) is the pullback of 0(1) under V -+ C3 x CP 2 -+ CP 2 ,
so on V there exists a metric on O(-P) whose curvature gives the pullback of the FubiniStudy metric. Away from V, 0(-p) is trivial, so one can take a constant metric with
respect to that trivialization.
13
Patching these metrics with cutoff functions then yields a form w-p which is semi-positive
outside of a compact subset disjoint from P, and positive along the exceptional fibre of Y.
Thus w-p is bounded from below, and for r > 0 sufficiently small wxCpi + rw-p will be
positive everywhere. Call this metric wy
0
Note that this proves that IC := n[L 0 0(1)] 0 O(-P) for n sufficiently large is ample.
Also, wy restricted to P is isomorphic to the standard Fubini-Study metric on P up rescaling by r.
Within P we have three distinguished lines, Co = P n Z (i.e. the singular locus of Yo),
and C1 and C2 the proper transforms of the two components of D x CP 1 passing through
(p, 0). Let Ko c P \ {Co U C1 U C2} be any Clifford torus disjoint from those lines, i.e.
viewing P as the quotient of C3, KO is the quotient of S 1 (ri) x Sl(r 2 ) x Sl(r 3 ) for some
choice of radii ri, r2 , r 3. Since wy restricted to P is a rescaled Fubini-Study metric, KO is a
Lagrangian subspace of P.
Away from Co, r : Y -* CP' is a submersion with symplectic vertical tangent spaces.
Hence wy defines a symplectic connection for 7r away from Co by taking the horizontal
tangent space to be the symplectic orthogonal complement to the vertical. Using this connection, we transport KO over [-e, e} to get K, a real 3-manifold with boundary. We will
need the following two facts about K.
(a) K C Y is Lagrangian. To see this, note that since Ko C Yo is Lagrangian and the
connection is symplectic, Kt C Y is Lagrangian. But for any x E Kt, TxK splits as a one
dimensional horizonal space direct sum TxKt, hence wy vanishes on K and K is Lagrangian.
(b) For each t $ 0 sufficiently small, Kt is isotopic to a crossing torus in Yt a X. To see
this, note that since the space of all connections for 7r is contractible, if K' is constructed
similarly to K using some other connection, then K' is isotopic to K via an isotopy that
covers 7r. If t is sufficiently small this isotopy will avoid the total transform of D x CP1,
since Ko = K01 n D x CP 1 = 0. But near t = 0 one can pick a local trivialization of the
situation, and the trivial connection with respect to that trivialization, for which it is clear
that Kt is a crossing torus. See [24] p. 15 for more details.
Consider the section of r*L vanishing along b=proper transform of D x CP', the section of
7*0(1) vanishing along Y,, and the already described meromorphic section of 0(-P)with
a pole along P. These together give a section sK of ICvanishing precisely along b U Y U P
with 4 = - log IstI a strictly plurisubharmonic function on Y \D U Y U P. For each t # 0,
t := 4y\st is a strictly plurisubharmonic function on Y \ Dt ~ X \ D, with associated
Kahler form wt = -ddq5.
Recall that the meromorphic section of O(-P) has norm 1
outside of a neighborhood of P; hence for some t > 0 #t is proportional to q5, the original
strictly plurisubharmonic function on A = X \ D. Thus wt provides a deformation of the
original Liouville structure on A.
3.1
Recall that we defined the Lagrangian K = UtE[EEIKt by symplectic parallel transport via
the symplectic connection induced by the fibration Y -+ CP1 . It will be useful for us to
14
ensure that this parallel transport preserves the divisor at infinity. More precisely, we have
D the proper transform of D x CP 1 , and for t E CP', t # 0, D := bn Y. For t, s #0 E CP ,
let py : Y -+ Y, be the symplectic parallel transport for any path from t to s avoiding 0.
Then we would like p.)(Dt) = D,.
The symplectic form on Y is given by wy = wxCP ±rw-p for some r > 0 sufficiently small.
Here wxxcp is the semi-positive form pulled back from X x CP' and w-p is supported in
a small neighborhood of P, the excepional divisor, and restricts to the Fubini-Study form
on P. wxCp, coming from a product metric, certainly preserves b, and hence so does wy
for fibres sufficiently far from 0. w-p, on the other hand, has a priori nothing to do with
D. Our aim in this section is to show that we can modify the symplectic structure on Y to
ensure that D is preserved under parallel transport. Namely
Proposition 3.1. There exists a choice of metric I -I on the ample line bundle L, giving
the symplectic form on X and hence defining the semi-positive form wX CP1 ; and a form
W' on Y which restricts to the Fubini-Study form on P; and a constant r > 0 such that
w/y = wxxCp1 + rw' is Kahler and if p' is parallel transport with respect to w' , then
p'(Dt) = D, for t, s 5 0 and y a path from t to s in CPF \ {0}.
Proof. Note that by considering the reverse path and noting that the composition is the
identity it suffices to show that p'(Dt) C D. The key to the proof is the fact that w-p is
supported in a small neighborhood of the exceptional fibre admitting a torus action, and
making w-p invariant under this torus action is sufficient to guarantee the conclusion of
Proposition 3.1.
At the crossing point p C X we have irrreducible components Dx and DY of D such that
{p} = Dx n Dy and an open coordinate chart V such that p = (0, 0), Dx n V = { (x, y) ly = 0}
and DY n V = {(x, y)x = 0}.
Lemma 3.1. After possibly shrinking V, we may assume the metric |
the associatedKahler form w is standard on V, in the sense that wIv =
on L is such that
E dzt A &-.
Proof. We first note that on any Kahler manifold M with form W and p E M with contractible neighborhood U and a different Kahler form wo on U, after shrinking U there
exists an exact perturbation of w which is equal to wo on U. More precisely, let h : M -+ R
be a smooth function with compact support such that on U,
h = Kahler potential of wo -
Kahler potential of wlu
Then w' = w + O5h is a global form which restricts to wo on U. Of course, for a general
choice of h, w' will not be a Kahler form, but if the functional form of h is chosen carefully
then w' will be Kahler. See [22] Lemma 7.3 for details on how to choose h.
Now apply this result to the open set V containing p and the standard Kahler form with
respect to the coordinates on V; let h : X -+ R be as in the previous paragraph. Modify
the metric I -I on L by I-I'= e-h/ I- . Then for the holomorphic section s with s-1(0) = D
we have - log IsI' = h/4 - log Isi, and so ddc(- log IsI') = 'Oah - ddc log Is1, which by
construction is standard on V.
E
15
I-I has
So we may assume that the metric
been chosen to be standard on V. Replacing
V by a polydisc centered at p contained in V, we have a an action of T 2 = R/Z x R/Z on
V given in coordinates by T(a,b) - (x, y) = (e 2 7iaX, e 2 7riby). The standard metric is obviously
invariant under this action, so wlv is T 2-invariant. This T 2 action induces two circle actions
given by the sub-circles T- = {0} x R/Z and Ty = R/Z x {0}. Note that the fixed set of Tx
is Dx n V and the fixed set of Ty is Dy n V.
This T 2 action extends to an action on V x CP1 trivially on the second factor. Let
V = ir- 1 (V x CPI') C Y under the blow-up map 7r. Then the T 2 action on V \ P extends smoothly over P; indeed, if [x : y : z] are homogenous coordinates on P induced by
the coordinates on V, then the extension is given by r(a,b) - [X : y : z] = [e2 7rax . e 2 7riby : 1
which is the restriction of the action of U(3) on Cp 2 to a torus.
The form w-p was constructed by taking the pull-back of the Fubini-Study form and multiplying by a cut-off function in a neighborhood of P. By shrinking the support of the cut-off
function we can ensure that there is an open set W c Y such that supp w-p C W C W C V.
Since we have a T 2 action on V, we can average w-p by this action. More precisely, for any
q E V, we define the form w' at q by
(w-p)q :=
J
i-ab)(W-P )(,b)qd
Where dp is Haar measure on T 2 . The Fubini-Study metric on Cp 2 is invariant under
the action of U(3), and since w-p restricts to the Fubini-Study metric on P, W'p also
restricts to the Fubini-Study metric on P.
Thus wT_, is a form which restricts to the Fubini-Study metric on P, is semi-positive in
some neighborhood of P, and is bounded from below. Hence exactly as in the proof that
wx~CP + rw-p is Kahler for r sufficiently small, there exists r' > 0 (potentially smaller
than the previous r) such that
Wy
pl + r'w-p
=
is Kahler.
The key fact to note about wr is that, since both terms defining it are invariant under
r, wy is also invariant under r.
Now we can prove that w' satisfies the conclusion of Proposition 3.1. Let t, s E CP' \ {0}
andy: [0, 1] -+ CP' \{O} any path from 0 to 1. Let q E Dt, and~
unique lift of y with y(0) = q. We wish to show that
(i) im
'c
W1.
In this case, w> restricted to im
(1) E D.
'
[0, 1]
-
Y be the
Consider two special cases:
is given simply by wXxCP1.
We al-
ready know that wXxCP1 preserves D, and so ~'(1) E D, as desired.
(ii) im 2y c V. In this case we must have
intersection of the proper transform of D, x
loss of generality, assume q E (Dx)t. Since w'
to it commutes with the action of T 2 via r;
q E (D,)t or q E (Dy)t, where (D.)t is the
CPI with Y, and similarly for (Dy)t; without
is i--invariant, parallel transport with respect
a fortiori it also commutes with the action of
16
T on V. Thus parallel transport must preserve fixed point sets of T,. q E (D2)t is fixed by
T , and so (1) E Y, is also fixed by T2. But the set of fixed points of the action of T' on
Yt is (D,)., n V, and so (1) E (D.), as desired.
Finally, note that any path can be decomposed into finitely many sub-paths that satisfy
either (i) or (ii), which completes the proof for any path.
El
17
18
Chapter 4
Gromov Limit Lemma
This section is devoted to proving Theorem 4.1 and related results. As before we assume
that A is a smooth surface, X a smooth projective compactification with D = X \ A a
curve with simple normal crossing singularities. Let U D D be a small open subset that
deformation retracts onto D.
Let p E D be any intersection point of two irreducible components of D, and form the
degeneration to the normal cone at p as in the previous section, so we have the total space
Y and the Lagrangian K C Y, K = Ut 4E[,,EKt. We say that p is a good intersection point if
for some crossing torus Kt at p with Kt C U, the map 7r,(Kt) -+ 7r(U \ (U nD) is injective,
in which case we say that K is locally incompressible. Since crossing tori are all isotopic,
local incompressibility of one implies local incompressibility of all.
Theorem 4.1. Suppose that p is a good intersection point, and for t > 0 let et be the
minimal energy of a non-constant holomorphic disc u : (D 2 , 0D 2 ) _- (At, Kt). Suppose that
lim inft-o+ et < oo, i.e. the minimal energy of holomorphic discs does not go to infinity as
t -+ 0+. Then there exists a holomorphic (and hence algebraic) map v : CP 1 -+ X, such
that v maps only a single point to the complement of A, i.e. v- 1 (D) = {pt}. The energy of
discs is computed with respect the the induced symplectic structure from At " Y.
Proof. Suppose that the minimal energy of holomorphic discs does not go to infinity as
t -* 0. Then there exists an E E R, a sequence {ti} C R with ti -+ 0 as i -+ oo and sequence
of holomorphic maps ui : (D 2 , OD 2 ) -+ (Ats, Kt2 ) such that E(ui) < E for all i. Note that
by precomposing with branched covers D2 + D 2 of the form z _+ Z if necessary, we may
assume E(ui) ;> -. Think of each ui now as a holomorphic map ui : (D 2 , aD2 ) _* (Y, K).
By construction, K C Y is a Lagrangian submanifold. Hence, by classical Gromov compactness for Riemann surfaces with Lagrangian boundary conditions ([10] for the original
argument, or [17] Chapter 4 for a textbook treatment) there exists a nodal Riemann surface
with boundary R and a stable holomorphic map u : (R, OR) -+ (Y, K). R in particular lies
in the Deligne-Mumford compactification of the space of discs, and hence is an arithmetic
genus 0 surface with boundaryt. More precisely, if we write R as the union of of irredudible
components R = R 1 U R 2 U - - - U Rn, then at least one of these components, say R 1 , is a
disc whose image has boundary on KO, while the others are either discs or copies of CP 1 ;
the intersections R, n Rj consist of at most a single nodal point; and the intersection graph
of R is a tree.
19
One property of the Gromov compactness is CO convergence of the images of the curves.
Since uk(D 2 ,0D2 ) C (Yk, Kt),we
conclude that u(R,4R) C (Yo, Ko).
Recall that Y = Z U P, where Z is the blow-up of x at p and P is the exceptional divisor of
Y, viewed as the blow-up of X x CP 1 at (p, 0). They intersect along the exceptional divisor
of Z, which is a component of D, the total transform of D under the blow-up Z -+ X.
Thus, we can partition the components of R into three sets: (I) Components whose image
under u lie entirely in b; (II) components whose image lies in P and intersect P \ D; (III)
components whose image lies in Z and intersect Z \ D. Note that 9R 1 c Ko C P, and so
(II) is non-empty. Less obviously, we have
Proposition 4.1. (III) is non-empty, i.e. there exists a component Ri of R such that
u(Ri) c Z and u(Rj) n Z \b =A 0.
Proof. Consider v = 7rou : R -* X, where
7r is the composite projection Y
-+
X x CPi -
X.
Looking only at the singular fibre, we can think of 7r as the contraction of P. Note that
Proposition 4.1 is false if and only if v(R) c D, since 7r contracts components of R mapping
to P to the point p, and is a diffeomorphism away from a finite set on the other components.
So we have to show that v(R) V D.
So suppose for contradiction that v(R) c D. Since ui -+ u as a Gromov limit, hence in
particular in C0, and 7r is continuous, we must have that, for every open set W c X x CP 1
with D c X x {O} c W, then im(ui) c V for all i sufficiently large. Recall that p is a good
intersection point, so that a crossing torus L at p is locally incompressible in X \ D, in the
sense that there exists an open set D c U such that L C U and the map 7ri(L) -+ 7ri(U \ D)
is injective.
Now consider the open set U x CP 1 . This clearly contains the copy of D in the zero
fibre, hence for i sufficiently large it must contain im(ui). Thinking of ui now as simply
a map to X, since it maps to the fibre of X x CP 1 -+ X, we have that im(ui) c U. By
construction ui was a map to A = X \ D, and so in fact im(ui) c U \ D.
In other words, we have ui : (D 2 , 6D 2) -+ (U \ D, Kti). But Kt is locally incompressible. Hence u(0D 2 ) is a contracible loop in Kti. Kti is Lagrangian, which is another way of
saying that Gt, is closed when restricted to Lti. Hence fs, u Oti = 0. By Stokes' Theorem,
E(ui) = fD2 Uwti = 0. But ui was non-constant by assumption, hence must have non-zero
energy, which is a contradiction.
0
So we have a holomorphic map v : CP' -+ X given by v restricted to some irreducible
component of R. We wish to show that v 1 (D) = {pt} is a single point. In any case since
v is algebraic we know that v-1 (D) is a finite set with b points; we have to show that b is
equal to one.
Let qo : A -+ R be the strictly plurisubmarmonic function induced from the section of
AC as in the previous section. By Lemma 2.1 and the fact that v is non-constant holomorphic, all singularities of f := #o V
VS:=CP1\v-1(D) are contained in the compact set
40 o v- 1 ((-oo, C) for C >> 0.
20
f
need not be a Morse function on S, but it is still exhausting. Thus, the fact that all
critical values of f are less than C implies that for any c > C, S retracts onto f-(-oo, c])
and c is obviously a regular value. Thus, f 1 (c) consists of a disjoint union of b circles.
We do not have a single plurisubharmonic function #0 : A -+ R, but a family #t
(-e, e) x A -+ R, coming from the fact that P U b U YOO is an ample divisor of Y. Set
R. By Sard's theorem, there exists c > C such that c is regular value
fi = 'ti o ui : D2
for all fi, so that f'~1(c) is a union of circles for all c. Note that liminfi fi(OD2 ) = 00,
and by Gromov compactness the domains fi-1((-oo, c]) converge to f-1((oo, c]). Hence
for k sufficiently large the manifold f,-'((-oo, c]) contains a connected component whose
boundary is a disjoint union of b circles; let t = tk.
Let Sk = f-j((oo,c]) and Tk = f-([c,oo)), so Sk U Tk = D 2 and Sk n Tk =
a union of at least b circles. Similarly to [191 Lemma 2.3 we have the following
Lemma 4.1. The inclusion Sk " D 2 induces
0
Sk is
an inclusion H1 (Sk, Z) " H 1 (D 2 , Z)
In particular, H 1 (Sk, Z) = 0. We state the lemma as we did because the obvious analogue applies, mutatis mutandis, to more general Riemann surfaces with boundary lying on
a Lagrangian.
Proof. We first claim that Tk must be connected. If not, write Tk = T 1 U T 2 , and suppose
that T 1 contains the boundary of the original D 2 = Tk U Sk. Let N =
t-1 ((-00, c]) c A,
so that (N, #t IN) is Liouville subdomain of the Liouville manifold (A, #t).
By construction, Uk : T 2 -+ A is holomorphic, with uk(T2) C A \ No and uk(oT2) c ON.
But then the integrated maximum principle ((2] Lemma 7.2) implies that uk(T2) C ON,
which contradicts the fact that Uk is transverse to ON = qt~1 (c). Hence we can conclude
that T 2 = 0 and T = T is connected.
Thus Tk is a connected Riemann surface with oTk = aSk U aD2 ; the aSk consists of the
circles mapping to ON, and aD2 the original boundary circle mapping to the Lagrangian
torus. Now we apply some elementary algebraic topology:
Lemma 4.2. : The map Hi1 (aSk, Z)
-+
H1 (Tk, Z) is injective
Proof. Suppressing Z coefficients, the homology long exact sequence gives
0 = H 2 (T )
-+
H 2 (T,OSk) _+ H1(OSk) -+ H1 (Tk)
But H 2 (Tk, aSk) e H 2 (Tk/OSk), and Tk/OSk is homeomorphic to a Riemann surface with
0
a single boundary component. Hence H2 (Tk, aSk) = 0.
21
Now we apply the Mayer-Vietoris sequence to D 2 = Tk UOSk Sk- We have
0 = H 2 (D 2 )
+
H 1, (Sk)
-+ H1(Tk ) ( H1 (Sk)
-*
H 1 (D 2 )
If (0, m) c H 1 (Tk) ( H1(Sk) is any element which maps to 0 (which is of course any
element, since H 1 (D 2 ) = 0, but the same argument would imply injectivity for more general Riemann surfaces), then it must come from n E H,(aSk). But by Lemma 4.2 n = 0, so
m = 0 and H1 (Sk) "+ H 1 (D 2 ) is injective, and so H1 (Sk) = 0. The only smooth Riemann
surfaces with vanishing H, are disjoint unions of discs and spheres, and A, being exact, has
no non-constant holomorphic spheres. Thus, all connected components of Sk are discs, and
since some component of Sk has b boundary components we conclude that b = 1.
L
We will also need a slightly more quantitative version of Theorem 4.1. We first need a
preliminary lemma about energy estimates on blow-ups. If (M,w) is compact symplectic
and J is w-compatible and integrable in a neighborhood of a point p, then the blow-up at
p, 7r : N -+ M has symplectic form, compatible with the natural almost complex structure
J', given by WN,r = rWE + WM, where WE is supported in a small neighborhood of the
exceptional divisor E and restricts to the Fubini-Study metric on E; wM = 7r*M, hence is
semi-positive; and r > 0 is sufficiently small.
Lemma 4.3. For any R > 0, there exists r > 0 such that for any homology class B
H 2 (N, Z) with a J'-holomorphicrepresentative and WN,r(B) < R, then wM(B) < 2R.
E
Proof. Choose r' sufficiently small that WN,r' is symplectic. Since only finitely many homology classes below a given energy level have holomorphic representatives ([17] Proposition
4.1.5), there is a finite set {B 1, ... Bk } of classes with WN,r'(Bi) < R. Since this list is finite,
there exists r < r' such that wM(Bi) > 2R ==* wN,r(Bi) > R, so by decreasing r' to r,
one eliminates all classes Bi with representatives of energy less than R.
Of course, new homology classes may be introduced with energy less than R. But for
any such class B', we have WN,r(B') <R WN,r (B'), which implies WE(B') > 0, and so
WM(B') = WN,r' - rWE(B) < R -WE(B') < R
SO WN,r
works.
Lemma 4.4. Fix R > 0, and choose r > 0 such that the previous lemma holds for wy =
rwp + wxxc1. Suppose that lim infin,0 ei < R. Then there exists holomorphic map v
CP" -+ X as in Theorem 2 with w(v.[CP') < 2R
Proof. Choose a sequence of non-constant holomorphic discs ui : (D 2 , OD 2 ) -+ (Atle,, Ltk)
such that E(ui) < R. Then the Gromov limit u satisfies rwp(u) + wxC1,i (u) = E(u) < B,
hence all components have energy less than R. By Lemma 4.3 we then have for each closed
component S of S that 7r*wxxcpi(u*[Sj]) < 2R. Since v is the image of one of these
components under blow-down, we have w(v[CP"]) = wxxCi (v* [CP']) < 2R.
L
22
We will need two more lemmas later on, both of which seem to fit most naturally in this
section. We recall that given an affine variety A with strictly plurisubharmonic function 0,
we let A = A be the associated complete Liouville manifold obtained by attaching a halfinfinite cone to 0-1((oo, C]) for C >> 0, with associated one-form 9 and symplectic form
cZ. By Lemma 2.2 AO is independent of k up to exact Liouville isomorphism, but q does
determine the conical end structure required to define convex almost complex structures.
We consider only C such that 4j~((oo, C]) D KE, and so there is an obvious embedding
KE c A.
Lemma 4.5. Fix R > 0, and let J be an w = wE-compatible almost complex structure on X
standard on U such that liminf et > R. Then there exists an w-compatible almost complex
structure JR on AO, convex at infinity with respect to 05, such that the minimal energy of
non-constant JR holomorphic discs in (AO, KE) is greater than R. Recall that e here is a
constant, fixed once and for all.
Proof. Let t E R> 0 be sufficiently small that et > 0. We have the symplectic parallel
transport map pt,, : Yt -+ T defined with respect to the symplectic fibration Y -+ CP,
and the inverse parallel transport map pE,t : Y -+ Y. pt,E : (X, wt) -+ (X, wE) is a symplectomorphism, pt,,(Kt) = K, by definition, and by Proposition 3.1 we may assume that
pE,t(DE) = Dt. Thus we have a symplectomorphism pt,E : (A, Kt, Wt)
-+
(A, KE, wE).
Let JR = p*t(j) and 0' = p*,(ot). Then JR is w-compatible and O'R is a proper exhausting function and -d(do's o JR) = w; in particular 0' is strictly plurisubharmonic with
respect to JR. The minimal w-energy of non-constant JR-holomorphic discs in (A, KE) is
then > R by construction. We wish to find a a-compatible almost complex structure JR
on A with the same property.
((-oo, cR)) D KE.
Since 0' is exhausting, there exists cR E R such that V := (0'
Pick C >> 0 sufficiently large that V C 0-1((-oo, C)). Form AO by attaching a cone to
4-1 ((-oo, C]). We then have the following result, an easy application of a version of the
maximum principle:
Let J be any w-compatible almost complex structure on A 4 such that J = JR on some
open neighborhood of V. Then if u : (D 2 , 6D 2 ) - (A 4 , K) is J-holomorphic, then im
U C V.
Indeed, suppose that this is false. Then there exists a J-holomorphic disc u : (D 2 , OD 2 ) (A 4, , KE) such that im u 0 V. Let fR = 0' o u defined on u~1(V), and let c' < c be such
that c' is a regular value of fR and c' is sufficiently close to c such that LE c (0'R)1((o, c')).
Let M = A 4 \ (4')~1((oo, c)) and S = u-1(M).
Then S is a non-empty Riemann surface with boundary, and M is an exact symplectic
manifold with concave type boundary, i.e. the Liouville vector field is inward pointing
along the boundary. Furthermore, u is a J holomorphic map, where J = JR near M and
hence is of contact type near the boundary, and u- 1 (OM) = OS. But then applying the
integrated maximum principle [2] Lemma 7.2 to this situation implies that u(S) c oM,
which is a contradiction.
23
Now let JR be any w-compatible almost complex structure on A such that JR = JR
on V, and JR is convex at infinity; such a JR always exists because the space of compatible
almost complex structures on a symplectic vector space is contractible. Then by the previous paragraph every JR-holomorphic disc with boundary on K, is contained in V. Since
c = w on V, any such non-constant disc satisfies cD(u) = w(u) > R. So this JR works.
The following lemma, a slight variant on the previous arguments in this section, will be
needed in Section 7
Lemma 4.6. Consider the triple (X, A, D), assuming only that A is smooth and D is a
curve with normal crossing singularitiessupporting an ample class. Let U D D a small open
neighborhood, and let J,J1, J 2 ,... be w-compatible almost complex structures on X which
are all equal to Jtndrd on U, and such that limi,
Ji = J. Let R > 0 and consider a
sequence ui : CP -+ X, where ui is Ji-holomorphic with E(ui) < R, uj1(U) # CP 1 , and
Iui'(D)l = 1.
After possibly taking a subsequence, we may assume that the ui Gromov-converge to a stable
J-holomorphic map i: 3 -+ X. Then there exists a component S1 C S with corresponding
map vj such that vi 1(U) $ S1. Furthermore,for any such component Sk of S with image
not contained in U, we have |vj 1(D)I = 1
Proof. The fact that there exists a component of S whose image is not contained in U
follows from the fact that no ui has image in U and the Co convergence of images of
Gromov-convergent sequences.
So let Sk be any component whose image is not contained in U, and suppose v- 1 (D)
contains b points. Let q : A -+ R be a strictly plurisubharmonic function with respect to
J determined by a section of the ample bundle vanishing along D. Think of Vk now as a
proper map on a b-puntured Riemann sphere to A and the ui as proper maps from C to A.
Since strict plurisubharmoniticity is an open condition with respect to the almost complex
structure, # is also strictly plurisubharmonic for Ji, i sufficiently large. Let r C R be such
that (i) r is a regular value of # o Vk and 0 o ui for all i; and (ii) r is sufficiently large that
0 k has no critical values greater than r. Then q o vk 1 ((-oo, r]) is a compact smooth
genus-0 Riemann surface with b boundary components.
Since the ui Gromov converge to U, for i sufficiently large o fi-1 ((oo, r]) must contain a connected component S[ which is a genus-0 Riemann surface with b boundary components. By
an analogue of Lemma 4.1-the proof is almost word for word identical, with the a puncture
replacing the boundary component lying on a torus-the map H1 (SF, Z) -+ H1 (C, Z) = 0 is
injective. Since the rank of H1 (Si, Z) is b - 1, we conclude that b = 1.
24
Chapter 5
Criteria for Non-vanishing of
Symplectic Cohomology
We begin by providing a brief overview of the symplectic cohomology of complete finite-type
Liouville manifold M, as introduced by Cieliebak-Hofer-Floer in [4] and Viterbo in [26]. In
addition to the original sources there are many other treatments of this material, such as
[25] and [1], so we will choose to highlight certain features important for our purposes.
Let M be an exact symplectic manifold with dO = w. Let H = Ht, t E R be a family
of Hamiltonian functions with Ht+1 = Ht. If x : S1 = R/Z -+ M is any loop, we define the
action of x to be
AH(X)=-
x*
JS1
+
jJS1
H(x(t))dt
A loop is a Hamiltonian orbit for H if it is an integral curve of XH, the Hamiltonian
vector field of H. Now let J = Jt be a 1-periodic family of w-compatible almost complex
1
structures on M. This defines a family of Riemannian metrics It. Let u : R x S -+ M be
a smooth map. We define the energy of u to be
ds A dt
E(u) = JrxS a
/
Is
2
The Floer equation for u is then
-9 + J at
as
XH) =0
We call a solution to Floer's equation with E(u) < oo a Floer trajectory. Floer's equation
can be thought of as the negative gradient flow equation for the action functional on the
infinite dimensional loop space; more prosaically, suppose u is a Floer trajectory. Then,
assuming that the orbits of H are non-degenerate (which can be arranged with a small
perturbation of H) one can show that lim8 +±o u(s, -) = x±, where x+, x- are Hamiltonian
orbits of H. We say that u is Floer trajectory from x+ to x-. Stokes' Theorem then implies
E(u) = AH(x-) - AH(x+)
Let K be a field (which we will immediately suppress), and let CF*(M, H) = CF*(M, H; K)
25
be the free K-vector space generated by time-1 Hamiltonian orbits of H. There exists a
grading on the generators of CF*(M, H) called the Conley-Zehnder index cz(x); we will
not define this index as we will not make use of any of its features save existence. There
are several competing conventions for defining Lcz, including whether it determines a homological or cohomological grading; we implicity follow those of [25]. The Conley-Zehnder
index is in general only defined mod 2, hence in general provides a Z/2Z grading, although
if c1(M) = 0 it determines an honest Z grading on CF*(M; H).
For simplicity we now only consider H with finitely many Hamiltonian orbits. Standard
transverality arguments (see e.g. [[23] Theorem 1.24; the reader can see Section 6 for an
unrelated argument of similar general flavor) imply that for generic choices of t dependent
Hamiltonians and almost complex structures (H, J), the space M(x, y, H,J) of Floer trajectories from x to y for any two orbits is a disjoint union of finite dimensional manifolds of
dimension congruent to tcz (y) - Lcz(x). Note that there is no s-dependence on H or J, so
there exists a smooth R action on M(x, y, H, J); Let M(x, y, H, J) = M(x, y, H, J)/R, so
M (x, y, H, J) is a union of smooth manifolds of dimension congruent to tcz (y) - tcz (x) -1
mod 2 (or simply equal if c1 = 0.)
Thus for any x we can define the Floer differential Jx = JH,jx to be such that the coefficient of y in Jx is the count of isolated (=0 dimensional) non-constant Floer trajectories
in M(x, y, H, J). If char K : 2 this obviously must be a signed count, and hence one needs
to orient M(x, y, H, J). If 62 = 0, we then define
Definition 5.1. The Floer Cohomology HF*(M, H) is the homology of the complex (CF*(M,H), 6).
Of course, we have swept several issues under the rug, such as (i) Compactness, which
is required for any "count"; (ii) the fact that 62 = 0; and (iii) Dependence on the choice
of H and J. Issues (i) and (ii) are genuine problems on arbitrary exact symplectic manifolds.
So now assume that M is a complete finite-type Liouville manifold and J is a family of
almost complex structure convex at infinity. Letting p = er o u on the conical end of M,
then for certain choices of Hamiltonians (such as the ones that are linear at infinity; see below for the definition) a maximum principle applies to p, which guarantees that any Floer
trajectory stays inside a compact subset. Gromov compactness and the a priori energy
bounds by action difference then implies that any sequence of Floer trajectories from x to y
converges to a broken trajectory from x to y with sphere bubbling. This resolves issue (i),
and since M has no J holomomorphic spheres for any t examining the ends of 1-dimensional
moduli spaces together with a gluing theorem solves (ii).
The standard approach for tackling (iii) is via continuation maps. Namely given (H+,J+)
and (H-,J-), consider a smooth generic family of (s, t) dependent pairs (H',J) with
(H 8 , J) = (H-,J-) for s << 0 and (H',J) = (H+, J+) for s >> 0. Then one counts
isolated solutions of the continuation map equation
-
-9
- XH-
('-
=
This, when defined, is a chain map, and different choices of paths (H', J) give rise to
chain homotopic maps. On a compact symplectic manifold, these maps are always defined,
26
and so (modulo resolving issue (ii) in this case) one can show that HF*(M, H) is independent of all choices; this is essentially Floer's original proof of the Arnol'd conjecture for
monotone symplectic manifolds. For non-compact symplectic manifolds, these continuation
maps do not exist between arbitrary Hamiltonians, and HF*(M, H) depends on the choice
of H. Thus to define an invariant of M we must restrict our class of Hamiltonians.
Let Ht be a Hamiltonian such that, outside a compact subset, Ht(r,y) = h(er) with respect to the conical end of M, where h is a time-indepedent real-valued function with
h'(er) # 0 for r sufficiently large. Then outside a compact subset the Hamiltonian vector
field XH = h'(er)R, where R is the Reeb vector field on Y, so closed time-1 orbits of H are
in bijective correspondence with time-h'(er) periodic Reeb orbits on (Y, a).
Thus if h(er) = er + c for some constant c, Ht(r,y) = h(er) outside a compact subset,
and r is not the period of any Reeb orbit on Y, then -rHwill have no time-1 orbits outside
of a compact subset. Any Hamiltonian equal to a linear function of er outside a compact
subset is called a linearHamiltonian. Then we can define
Definition 5.2. SH*(M)<r := HF*(M; -H) for any linear Hamiltonian of slope 1 such
that Hamiltonian-Floertheory is well-defined.
The notation implicitly assumes that this is indepedent of the choice of H. Indeed,
assume r+ < r-, and we have slope-1 linear Hamiltonians H+ and H- with almost
complex structures J+, J-. Then let (HS, J) be a smooth family interpolating between
(r+H+,J+) and (r-H-, J-) such that H, is a linear Hamiltonian of slope r,, and assume
8
0. This negativity of derivative ensures that the continuation map using (HI, J )
as <
has energy bounds given by the difference in actions of the endpoints, and furthermore
the maximum principle applies to such trajectories, and hence gives a well-defined map
HF*(M,r+H+,J+) - HF*(M,r-H-,J-)
In particular taking r = r = r+, we have maps going both ways that compose to the
identity, so HF*(M,rM, J) = SH*(M)<r is independent of the choice of H and J. Furthermore when r < r' and -r,r' are not the periods of Reeb orbits on Y, we have natural
maps SH*(M)<r -+ SH*(M)<T'. Since the the set of periods on a contact manifold is
discrete, we can finally define
Definition 5.3. The symplectic cohomology SH*(M) is the direct limit of SH*(M)<r for
-r not a period of a Reeb orbit of (Y, 0|y)
Note that we defined symplectic cohomology with respect to a fixed conical end on M,
which need not be preserved under Liouville isomorphisms. However, if 0 : M - M' is a
Liouville isomorphism, then again by looking at the maximum principle for continuation
maps we have an induced map 0. : SH*(M)<T -+ SH*(M')<T' when -r<< r' (there needs
to an exponential decay-type conidition on r, for the maximum principle to apply), and
hence an isomorphism o. : SH*(M) -+ SH*(M').
Symplectic cohomology is in general hard to compute; in the case of cotangent bundles
Viterbo [27] proved that symplectic cohomology is isomorphic, up to grading shifts, to the
homology of free loop space, but the later is often hard to compute as well. Rather than
a full computation, we will rely on two features of the theory to prove non-vanishing, the
first of which is the following:
27
Proposition 5.1. There exists an element 1
only if 1 = 0.
E SH-I(M) such that SH*(M)
= 0 if and
Proof. Sketch Let H be a linear Hamiltonian of slope 1 that is time-independent and C2
small inside the conical end. Then for r > 0 sufficiently small, in particular smaller then the
period of any Reeb orbit on Y, CF*(M; H) is generated by stationary orbits corresponding
to the critical points of H and Floer trajectories correspond to Morse flow lines, and so
HF*(M; -rH)~ H*+n(M) (the shift is a result of the implicit convention we are using for
LCZ)Thus as part of of the direct limit defining SH*(M) we have a map H*+n(M) -+ SH*(M).
SH* (M) has a product structure given by counts of solutions to Floer's equation on thrice
punctured spheres (so-called "pairs of pants"), under which it is an algebra over H*+n(M).
Thus looking at the image of 1 E HO(M) gives a multiplicative identity for the product on
SH*(M), from which the result follows.
[
The abstract existence of 1 E SH*(M) is not enough for our purposes; since we will be
working at the chain level shortly, we shall need a chain level representative e E CF*(M)
representing 1.
Given regular families (H, J) with Ht a linear Hamiltonian of appropriate slope, let J'
be a fixed time-independent convex almost complex structure. Consider a solution to
-+Js
Os
-
at
-
XH
=
(*)
Where (H', J') = (H, J) for s << 0 and (Hs, J-) = (0, J') for s >> 0. Then u is
holomorphic for s >> 0, and finite energy, so by the removable singularities theorem it u
extends smoothly over the puncture to give a map u : C - M which is holomorphic near 0
and converging to a time-1 orbit of H.
For a generic choice of J' and path, we then define Ej E CF*(M, H) such that the coefficient of x in e is the count of isolated solutions to (*). Then eJ is cocycle mapping to
1 E SH*(M). Note that at the chain level ej may depend on the choices we have made.
The other ingredient is a chain-level map to the Novikov ring induced by a Lagrangian
submanifold. Let L C M be an exact Lagrangian submanifold of a Liouville manifold, (i.e.
with the infinite cone attached) or more generally a Lagrangian bounding no non-constant
holomorphic discs with respect to some almost complex structure. Then there exists a natural map SH*(M) -+ H*+n(L), arising from counts of half-infinite Floer trajectories with
boundary on L (a more precise definition is below.) One can then show that the composition
H*+n (M) -+ SH* -+ H*+n(L) is the ordinary restriction map in singular cohomology. In
particular, it maps the unit to the unit, and hence the middle group cannot be zero. Thus
we obtain Viterbo's result that the existence of an exact Lagrangian submanifold implies
the non-vanishing of symplecitc cohomology.
For our purposes, this is not quite strong enough. The problem is that the Lagrangian
we have constructred could bound non-constant discs; we only know that, in the limit as
we vary the Kahler metric in a prescribed manner, the minimal energy of a non-constant
28
disc diverges to infinity. Thus we need a strengthening of the non-vanishing result for this
limiting situation.
Theorem 5.1. Let M be a Liouville domain and L c M a compact Lagrangian with
the following property: there exists a sequence of w-compatible almost complex structures
J1, J2 ... which are convex at infinity with respect to a fixed conical structure on M, and
there exists a sequence Ci where Ci is less than the energy of any non-constant (w, Ji)
holomorphic disc with boundary on L, and lim infi Ci = oo. Then SH*(M) # 0
Our argument follows [25] Section 5b, with some care paid to the actions involved. First
a bit of perturbation housework. Let H 1 , H 2 ,... be a cofinal sequence of linear Hamiltonians for the limit defining SH*(M). Let J1 , J2.... be a sequence of w-compatible timedependent perturbations of J1, J2, such that Ji = Jj outside a compact subset (in particular
is time-independent), and such that HF*(M; Hi, Jj) is well-defined for all i, j. If J"- J'
is sufficiently small, then by Gromov compactness for any t the minimal energy of nonconstant (J.)t-discs with boundary on L is greater than C., so we will assume that we have
chosen our perturbation to satisfy this condition. Finally, let J' be a time-independent convex almost complex structure coming with paths (H; , J')from (0, J') to (Hm, J.) defining
identity cochains Em,n.
1
We consider Floer half-cylinders with boundary on L, i.e. maps u : S x [0, oo) -+ M
satisfying Floer's equation, u(t, 0) E L and lim 8 so0 u(-, s) = x(-) a time-1 orbit of H. When
L is exact evaluation at (0,0) induces a chain map from time-1 orbits to pseudo-cycles
on L, and hence the map SH* -* H*+n(L). In our case we will consider only those cyllnders mapping to the fundamental class of L, i.e. Poincare dual to the one point pseudo-cycle.
Recall that K is the base field for symplectic cohomology; as we have been rather cavalier with orientations perhaps it is best to assume char K = 2. We then denote by A the
Novikov ring over K, i.e. the field of all formal sums Ej aiTbi with ai E K, bi C R and
limi bi = oo, and by A> 0 C A is the subring where all powers of T are strictly positive.
Recall that 0 is a primitive for w and for any Floer half-infinite cylinder let a E HI(L) be
the homology class of the lower boundary loop, taken with opposite orientation. Let y E L
be any point.
For any m we define the homomorphism q5: CF*(Hm) -+ A by taking a weighted sum
of isolated half-infinite cylinders with boundary on L satisfying (*) as well as u(0, 0) = y,
where the weight assigned to any cylinder is tfc '.
To ensure this makes sense, note that
since we are taking the opposite orientation for a we have
E(u) = -AHm(a) - AHm(x) =
0 ja
<
Hm(u(-t,O))dt - AH(x)
Js1
j9-AHm(X)+C
For some constant C, since L is compact. Thus for a fixed generator x E CH*(Hm) and
power of t, we have an energy bound, and so by Gromov compactness the zero dimensional moduli space is finite. We will write Om,n for the homomorphism determined by
(Hm, Jn). We will assume that the Hm have been arranged such that there exist inclusions
CF*(M; Hm) -+ CH*(M; Hm,) for m' > m which are compatible with bm,n and mr',n29
Lemma 5.1. Let H be a linear Hamiltonian, and J a convex almost complex structure.
Let Cj be less than the infimum of the energies of non-constant J-holomorphic discs with
boundary on L. Then if x E CF*(H) is any generatorwe have #i(6x) E A>AH(X)-C+Cj for
constant C > 0. Here 6 is the Floer differential.
Proof. As is typical with this kind of statement, the proof proceeds by examining the ends
of one-dimensional moduli spaces. Fix x with AH(X) > a. q(Jx) counts broken Floer
trajectories as in the left side of Figure 5-1. Any such trajectory can be smoothed into a
one parameter family of genuine Floer trajectories. Note that if u is any such smoothed
trajectory, we have
E(u)
j
-AH(x)±C
If there were no bubbling, then #(6x) would then be represented as a sum over boundary
components of a compact one-manifold, and hence #(Jx) = 0 and # would be a chain map.
However, disc bubbling along the boundary is possible, as in the right side of Figure 5-1.
Let u' be the main component of such a configuration and v be the disc bubble component.
We then have
E(u') + E(v) = E(u) j 9- AH(x) + C
But v is a non-constant J holomorphic disc, and so by definition E(v) > Cj. Thus
0 < E(u')
0
- AH(x) + C - CJ
Or
AH(x)
-
+ CJ
j
1
Since the power of T in any term given by a trajectory is fa 9, this proves the result.
l
Corollary 5.1. Fix m and thus the linear Hamiltonian H = Hm. Then for n E N sufficiently large #,n(,nx) E A> 0 , where Jmn is the Floer differential determined by Jn.
Proof. Since the slope of H is admissable all Hamiltonian orbits of H are contained in a compact subset of M, and since they are isolated there are only finitely many. Hence the action
AH(x) is bounded from below for Hamiltonian orbits. By assumption lim infij Cj1 = 00,
and so for n sufficiently large 0 < AH(x)-C+CJn and the result follows from Lemma 5.1 E
Lemma 5.2. Fix m. Then for n >> 0 sufficiently large, qm,n(em.n) E 1 + A> 0
Proof. The proof is very similar to that of Lemma 5.1. By the definition of Em,n, km,n(em,n)
consists of weighted counts of configurations of discs as in the left side of Figure5-2. This
includes the constant disc with value y, which obviously contributes 1 to the count. Now
consider non-constant configurations mapping with boundary loop -a. These can be glued
into one parameter families of smooth discs, which by Stokes' Theorem satisfy E(u) <
fc, 9+ C. The other possible end of the configuration space is broken trajectories as in the
30
X
elX
-
Disc Bubbling
q5(6x)
Figure 5-1: On the left, configurations contributing to
of smoothings thereof.
4(Jx); on
the right, disc bubblings
right side of Figure 5-2. The main component u' has non-negative energy, and the disc
bubble is non-constant, and so we have
0 < E(u') < E(u) - E(Disc bubble) <
0 + C - C
and so
CN - C
For n sufficiently large Cn > C, and so
f,
I
0 > 0. For such n we therefore have Om,n(Em,n) E
1 + A>O
Proof. of Theorem 5.1 SH*(M) = 0 if and only if the identity element 1 is equal to zero,
i.e. any identity cochain is a coboundary. SH*(M) is the direct limit lim,-r
SH*(M)',
So if
large.
for
sufficiently
r
for
in
SH*(M)r
1
vanishes
iff
in
SH*(M)
vanishes
so
1
and
and
with
Hamiltonian
linear
a
=
Hm
H
SH*(M) = 0 there exists m sufficiently large and
z 1 E CF*(H) with 5m,iZi = El.
Let n E N be sufficiently large that Corollary 5.1 and Lemma 5.2 apply to (H, Jn). Let
T : CF*(H) -+ CF*(H) be the continuation map from (H, Ji) to (H, Jn), i.e. the map
obtained by counting trajectories
au
ou
-- + J9(- - X H) = 0
With J, = J1 for s >> 0 and J, = Jn for s << 0. T is a chain map, and is action increasing,
in the sense that if x E CF*(H) is a generator, and x' is any loop with non-zero coefficient
in T(x), then AH(X') >: AH(X)Now let E'n,n = T(Em,i). We would like to appeal to Lemma 5.2 to conclude that Om,n(E') E
31
HS = 0
4
HS =
0
H' = H
H- = H
Disc Bubbling
Figure 5-2: On the left, configurations contributing to O(E); on the right, disc bubblings of
smoothings thereof.
1 + A>O, but while em,n and e' , agree in cohomology, they are distinct cochains, since
they are defined by different paths (Hms, Jn) and ((H' )", (J')s) from (J',0) to (Hm, Jn).
However, examining the proof of Lemma 5.2, it is clear that the only facts about (Hm,* Js)
we used were that H," vanishes for s large and Jn = Jn for s very negative. Since those
are both true of ((H,)s, (J')S) the same argument as in the proof of Lemma 5.2 implies
1 + A> 0 .
qOm,n(E'M,n)
T(zi), all the loops of zn are Hamiltonian orbits of H, and so by Lemma 5.1
6
qm,n(m,nzn) c A>O. But m,nzn = Em,n since T is a chain map and 4m,n(c'n) E 1 + A>O.
This is a contradiction.
Thus if zn
=
32
Chapter 6
Transversality with tangency
conditions
As usual we have the triple (X, A, D) with an open set U -D D and a symplectic form W
on X arising from a section of the ample line bundle supported on D. We will use the
same symbol D for both the nodal curve and the associated class in H 2 (X, Z). We have an
auxilliary nodal rational curve E C X, whose role in the story will become apparent in the
next section. Our goal in the section is the following lemma:
Lemma 6.1. Let R > 0, and B e H 2 (X,Z) a homology class with w(B) < R. Let J =
J(U, E) be the space of w-compatible almost complex structures J' such that J' = Jstndrd
along U U E. Then there exists a Baire-dense subset ICR C j such that if J' E ICR and
there exists a simple J' holomorphic map u : CP1 -+ X such that (i) im u 0 U U E; (ii)
u,[CP2 ] = B; and (iii) u- 1 (D) = {pt}, then B must satisfy
c(B) - B - D > 0
We recall that a simple holomorphic map on CP1 is one that cannot be factored as
u = v o g where g : CP1 a (CP is a non-trivial holomorphic branched cover.
The idea of the proof of Lemma 6.1 is that the above quantity represents the virtual dimension of a certain moduli space of holomorphic maps, and ICR is then the set of regular
almost complex structures for this problem. We begin with the general version of transversality for pseudo-holomorphic curves with prescribed tangencies along a union of complex
submanifolds that we shall need. The result and proof are a slight generalization of [5]
Section 6; related results are in [15].
Let (X, w) be a a compact symplectic manifold, and Jo an w-compatible almost complex
structure. Let D = D 1 U ... Dk a union of Jo holomorphic submanifolds. We require
transversality of the irreducible components of D in the sense that for I c {1,... , k} and
i V I Di is transverse to Di := njEiDj. We also consider an auxiliary union of holomorphic
submanifolds E = E 1 U ... U El. Let V C X be an open subset with V n D = 0. In our
application D will be the boundary divisor of our affine surface and E will be the divisor
blown down to obtain a relatively minimal model of (X, A, D).
Definition 6.1. Let J = J(V,E, w) be the space of smooth w-compatible almost complex
structures such that J = Jo outside of V and J = Jo restricted to E.
33
We equip 3 with Floer's Banach metric. Let 1i,... lk e Z>o, and m, p such that mmax({li}) (here n = dimcX). Let Bm'p = B m 'P(CP',X) be the space of L?7 maps from
SCP 1 to X with k marked points z = {zi,... zk}. Importantly, we do not require that the
marked points are distinct.
Proposition 6.1. Let B E H 2 (X, Z) and M(l1 ,... lk, B) := M4'P(CP, X, B, w, 3, D, {li,. . . })
{(f, J) C Bzm'' x Jj|f = 0, f(zi) E Di, with a tangency to Di of order li at zi, f.([CP1 ]) =
B, f is simple, and im f 0 E U VC}. Then M(l1,... lk, B) is a Banach manifold cut out
transversally by the 0 and appropriate normal maps. Note that m > max{li} in order
p
to make sense of order li tangencies.
There exists a Baire-dense set of almost complex structuresJ$eg E 3, such that for J' E Jreg,
the subspace M(l1,... lk, B, J') = {(f, J') E M(1 1 , ... lk, B)} is a transversally cut out manifold of dimension
2n + 2c,(f.[CP]) + 2k -
2(li + 1) dimc Zi
Note that M(1 1 ,...l, B) seems to omit certain curves, namely those with image in E,
but we will not be doing any curve counting, and in any case will later restrict to a subspace
on which such curves cannot appear as Gromov limits. To prove this result we need to be a
bit more precise in our understanding of tangency for holomorphic curves; we again follow
[5], where one can find the proofs of the subsequent statements, most of which are simple
if somewhat tedious calculations.
Consider C" with an almost complex structure J such that C m = C x {0} C C" is Jholmorphic. Let U C C be an open subset of 0, and f : U -* C" a smooth map with
f(0) = 0. We say that f is tangent to C' of order at least k if for all i, j with i + j 5 k,
a-RL (0) E Ck where s +it are coordinates on U; we abbreviate this condition on derivatives
as dkf C Ck. If f is J holomorphic, then one can trade a t derivative for an s derivative at
the cost of introducing lower order terms. Thus, a holomorphic map is tangent to order k
iff 9 (0) E C m for i < k. Note when m = n - 1 that a tangency of order k contribute k +1
to the local intersection multiplicity, e.g. a tangecy of order 0 is a transverse intersection.
Definition 6.2. If f : U
k +1'st normal vector jg
Thus if
f
-*
C" is J holmorphic and tangent to Cm of order k, then the
f(0)
is defined as
[
i{]
E Cn/Ck.
is tangent to order k it is also tangent to order k + 1 ifC
jf(o) =
0.
Proposition 6.2. Suppose that J' is another almost complex structure on Cn such that
C" is J' holomorphic and g : (C, Ck, J) -* (C", Ck, J') is a local biholomorphism with
g(O) = 0. Then for any i, j&mg o f(0) = g*jimf (0), for g. : C"/C" _+ Cn/Ck the induced
map on normal spaces at 0.
Proof. This is a relatively simple computation, see [5] Corollary 6.3.
Thus, we can extend this definition of tangency to any almost complex manifold X with
an almost complex submanifold Z C X: if
f
:U
-+
X satisfies f(0) = p E Z, pick a coordi-
nate chart near p such that Z C X looks like C m x {0} C C", and and compute j
34
m
f(0).
=
By proposition 2 this is independent of the choice of local coordinates, as are j2mf(0) and
all higher normal maps if the lower ones are zero; we call these maps J f. Thus we can now
be more precise about the "appropriate normal maps" in Proposition 1; they are simply
the j'f maps at the intersection points. We will, as a slight abuse of notation, sometimes
write d'f(z) E Z for f being tangent to Z at z of order 1.
T
p = T Bzm'p, the space of (m,p)
Given f E B."~', we have the tangent space at f, B3
1
l}, for any
c B7Pldio(zi)
,
= 0 for j
=
sections of f*TX over CP , and define B72,
1 < m - , which ensures that B7'p C C1.
Similarly, we can define E7'' the space of (m, p) sections of AO1f*TX; near each point zi we
trivialize this bundle to make sense of the j'th jet space with respect to a trivialization, and
for any / E m'Cthe condition dio(zi) = 0 does not depend on the choice of trivialization.
l}
Thus as in the previous paragraph we can define Ef' = { E 9f'PId3b(zi) = 0 for j
The following lemma is the key technical tool for establishing transversality, essentially
[5] Lemma 6.6.
Lemma 6.2. For any 1 E Z>O, m, p with m > 1+ 1, and (f,J) E Wm,p(S
simple and J holomorphic, and im f 9 E1 U ... U Em U V', the map
x T
F: BmP
f,1+1
2
, X) x 3, f
EM-1,P
fl
Given by
F((4, Y)) = Dfq$+
Y(f) o df o j
is surjective, where D 1 is the linearizationof the a operator at f.
Proof. This is a generalization of standard surjectivity arguments without derivative conditions, see e.g. [17] Proposition 3.2.1. The added complication comes from the fact that
we cannot here take m = 1, a choice which in the standard case means that one only ever
has to consider the dual of LP spaces, rather than the spaces of distributions dual to higher
Sobolev spaces.
m and p are such that
[im - 2] = 1 + 1. F is an elliptic differential operator,
and imF C m-1,P c gm-,P. If im F were a proper
in
gm-1,P,
hence has a closed image
subset of Em", then by the Hahn-Banach theorem (E6'-1,P)J would be a proper subset
Assume
of (im F)' inside the dual space (g7 -1,p)*. We will prove that (imF)' c (C7,
will thus imply the lemma.
P)±, which
Let E(im F)'. Recall that (EM~1'P)* = WI--m'(CPI, f*A 0 'ITX) with . +1 = 1, i.e.
the space of (1 - m, q) Sobolev distributions on the bundle W := f*AOlTX. Now let
0 E Wm'P(CP', f*TX) be any section with supp q ; CP 1 \ {z 1 , . . . zk}, so that 0 E B
Then since E (im F)', in particular 0 = (Df #) = (D*)(4). But D* is elliptic, and so by
elliptic regularity there exists 0 E C *(CP 1 \ {zi,. .. zk}, W) such that iCPl\{z1...zk}=
We will show that 8 = 0.
35
Let S* C CP' \ {zi,. . . ZJ} the set of injective points of f, i.e. the points z such that
f '(f(z)) = {z} and dfz 5 0, and let V* = s* n f -(V n (E U ...Em)C) (recall that V was
the set on which we are allowed to vary our almost complex structure). Since f is a simple
curve S* is dense inside CP', and by assumption V* is a nonempty open set.
Suppose that #(z) $ 0 for some z E V*. Then by an elementary linear algebra argument (see e.g. [17] Lemma 3.2.2) there exists Y E End(TXf(Z)) with YzJf(z) + Jf(z)Yz = 0
and (#(z), Y o df o j) > 0. Extend Y to a section Y of TJj such that Y vanishes outside
of a small neighbouhood of f(z). Then (#(z'), Y(z') o df o j) = 0 for z' outside of this
neighborhood, and is greater than zero inside, so (/, Y o df o j) > 0, and since # represents
on S*, 6(F(0, Y)) > 0, contradicting 6 Eim F1-.
Thus 3 vanishes on the open set V*. But # satisfies the Cauchy-Riemann type equation D*# = 0, so by unique continuation # is identically 0. 3 represents 77 restricted to
S 2 \ {z,... zk}, so supp 6 C {zI,... zk}. Any distribution with finite support is a sum of
derivatives of Dirac delta distributions ([14] Theorem 2.3.4), i.e.
k
E) E ak,a 09)
i=1 laI<N
where a = (a,, a2) is a multi-index and N is an integer. Now recall that 6 E W1-m,q(S 2, f*AOlTX);
since the Sobolev norm is 1 - m, one can take at most m - 1 derivates of any Dirac delta
appearing in 6, and so N < m - 1. In fact, since the Sobolev embedding theorem is
sharp for any a > [M - i - ], there exists a smooth section k with jq|#m.p = 1 and a
of arbitrarily large norm. Since # is a bounded operator we must thus have N < [m - 1- ].
p
We have l= [m-1-2], soN <l. IfHOE
((#)
=
m-1P, we then have
('0)
(zi) = 0
Since d1-1#(zi) = 0. Thus 6 E (Em7 lP), which is what we wanted to prove.
m and p with l+ 1 < m - ; let m' be such that l + 1 = [m' -].
Let
,0 E Sm-,lp. Since m' < m, 4 C 97 -P. Thus there exists (#, Y) E
x TJj such that
F((#, Y)) = 4. F is an elliptic operator, so by elliptic bootstrapping # E Bmj and so F
Now take arbitrary
is surjective.
We can now proceed to the main step in the argument for Proposition 6.1. The proof of
the following Lemma is similar to that of [5] Lemma 6.5, with the additional complication
given by the fact that our points zi are not necessarily distinct. We will use the normal
crossing condition on the Z's to resolve this issue.
Lemma 6.3. Suppose that M(l1, ... lk, B) is a smooth manifold of the correct dimension.
Let j be an index such that lj > max{l1,. . . lk} - 1. Then for any (f, J) C M(l1,... 1),
36
the map
(TB'P x {O})
n Tf M(l1,...
k)
Tf(,j)X/Tf(Z3 )Zi
-+
Given by
(b'i,0)
-*
j
V(zj)
is surjective
Proof. Near each point zj, a choice of local coordinates picks out a non-vanishing (0, 1) form
dz, and hence provides a local identification of (0, 1) forms with functions. With respect to
this identification, the linearized 9 operator has the form
Dfb =
as
+ J(f(z))
+ A(z) b
Where A(z) is a z-dependant linear operator. After a linear change of coordinates, we
1
may assume that z- = 0 E C, f(zj) = 0 c C", Z c X = C"- x {0} C C", and J(0) = i,
the standard almost complex structure. We will say these local coordinates are adapted to
j; we say adapted to j rather than zj since we could (and in our situation generally will)
have z, = zj for i $ j.
Indeed, let j := {i E [1, k] zi = zj}. As part of the normal crossing condition we have
that the manifold Z_ = niE Zi is transverse to to Z3 .
So let [v] e Tf(z,)X/T(zj)Zj. We may then choose a lift of [v] to v E Tf(zj) = C" such that
v E Tf(zj)Zz 3 . Near zj, with respect to a choice of coordinates adapted to j, define the map
4 to X via
(z) = z'i+ 1 v/(lj + 1)!
= 0.
is holomorphic with respect to the standard complex structure on C", i.e. 3+i
1
4'(zj) = [v]. But J(f(zj)) = J(0) = i, and so
It is also clear that d'i4(zj) = 0 and j
letting B(z) = J(f(z)) - i, a z-dependant linear operator on C with B(O) = 0, we have
:=DfeO=
+ A(z)o =
+ J(f(z))
=
+i
+ B(z)-5F + A(z)4'
+ A(z)4
B(z)
Now using the fact that d'i4(0) = 0, if one takes at most 1j partial derivatives of the
right hand side at 0 the only potentially non-zero terms involve taking all the derivative of
the -, i.e. abusing notation slightly
d'a (0) = B(0)4a
B( 8d-t
37
But B(0) = 0, and so dli i = 0.
Extend 4 to all of CP1 via a bump function supported in a small neighborhood of z3
disjoint from all zi with i V Jj, and extend i = Dfi. We then have d'ii(zi) for all i, i.e.
S
'.Then by the previous lemma, there exists (N',Y) E BmIP x TjJ such that
Df V5'+
1 (Y 0 f 0 i)v'
-
Let
=
+'.V) We have Dfpi + .(Y o f o i)4V =. For every i we have di +14'(zi)
1
d 4 = 0, and so d'i4(zi) = 0. Furthermore, for i V Ij
0 and
ii+1O(zi) =4+14'(zi)=O
While for i E E
ji+
1
4(zi) = j+ 1 4'(zi)
=
[v]
which is 0 for i 54 j. But recall that lj ;> max(li,... lk) - 1, so in fact ji O(zi) = 0 for
all i. Thus (0, Y) E Tf,jM (li,... lk), and the image of (4, Y) under the lj + 1'st normal at
at zj is [v]. Thus this map is surjective.
0
Proof. of Proposition 6.1: The case li = -1 for all i, i.e. that M(-1,... - 1, B) is a
Banach manifold when no intersection conditions are imposed at all, is a standard result ([17] Proposition 3.2.1). Then by the previous lemma, for any (l',...l') and j with
l> : max(l',...l') - 1, then M(l',...li + 1,... l') c M(l',... ,>...l')is a submanifold of real codimension 2 dimc Zj. Thus increasing l's always maintaining the condition
S max (li, . .. l1) - 1 proves that M(li,... ik, B) C M(o,... 0, B) is a submanifold of codimension Ei 2(l4 + 1) dimc Zi.
By the Sard-Smale theorem, the set of common regular values of all projections 7r : M(l,..
J, (f,J) '-+ J, for l' ; li, is Baire-dense in J. For any such generic J, M(l,... l; J)
7r 1(J) n M(li,. . . lk) is a Ei 2(li + 1) dimc Zi submanifold of M(O,... 0; J). We have
. l')
=
dimM(0,... 0; J) = 2n + 2k + 2c,(f,[S 2 1)
and so
dim M(1 1 ,... lk; J) = 2n + 2c,(f,[S 2 ]) + 2k - 2
2(l4 + 1) dimc Zi
We have thus far been considering moduli spaces with fixed marked points; we will
actually need the analagous result for varying marked points.
Definition 6.3. Fix a partitionP of {1,... k}, i.e. P is a collection of disjoint subsets of
{1,...k} whose union is {1,.. .k}. Let M(l1,...lk; P; B; J) be the moduli space space of
simple J holomorphic maps f : CP1 -+ X representing B C H 2 (X) with varying marked
38
points satisfying zi = zj iff i =p j, with tangencies of order li at zi to Zi C D and im
f g E U Vc. Let M(l1,... lk; P;A; J) = M(11 ,... lk; P; A; J)/Aut(S2).
Lemma 6.4. For a generic choice of almost complex structure, M(l1,... lk; P; A; J) is a
smooth manifold of real dimension
dim M(li,.
=
.
.,;
2n + 2c 1 (A) + 21P1 - 6 -
;A; J)
2(li + 1) dimc Zi
Proof. The case where li = -1 for all i is again standard. The passage to higher
proceeds by similar arguments to the proof of Proposition 6.1.
l then
Proof. of Lemma 6.1
Fix a homology class B E H 2 (X, Z) with w(B) < R and an w-compatible almost complex
structure J' on X such that J' = J on U and J'IE = JIE. Define M,(B; J') be the set
of simple J' holomorphic maps f : S 2 -+ X with im f n V : 0, im f 0 E, f,([S 2 ]) = B
and f- 1 (D) = {pt}. Note that every simple f E M 1(B, J') lies in some M(li,... li; B).
Indeed, if f(pt) is a smooth point of D, then i = 1 and li + 1 = B - D = B - Di (e.g. a
tangency of order zero corresponds to a multiplicity 1 intersection point, etc.), where D 1
is the irreducible component containing f(pt); while if f(pt) E sing D then since D has
normal crossing singularities i = 2, z1 = z2 and (11 + 1) + (12 + 1) = B -D. In the first case,
the real virtual dimension of the moduli space with fixed J' and varying f is
2n + 2k' + 2ci(B) - 6 - 2(l + 1)codimcDi = 4 + 2 + 2c 1 (B) - 6 - 2B - D
= 2ci(B) - 2B - D
and in the second case the dimension is
2n + 2k' + 2c 1 (B) - 2(l + 1)codimcDi - 2(12 + 1)codimcD
= 4 + 2 + 2c 1(B) - 6 - 2B
-
2 =
2B - D2 = 2c 1 (B) - 2B - D
(Recall that k' is the number of distinct marked points, so while k = 2 in the second
case k' = 1) It is geometrically reasonable that these moduli spaces must be of the same
dimension, since in the second case one could repeatedly blow up X at the intersection
point until the point eventually lies on a smooth point of the divisor, and the moduli space
dimension should not be able to "see" this process.
In any case, if J' is regular for all the relevent M (li,.... li; B)-and since there are only
finitely many such spaces, the set of such almost complex structures is Baire-dense-the
virtual dimension is equal to the actual dimension of the smooth manifold of maps. Hence
l
for any such J' we must have ci(B) - B - D > 0 as claimed.
39
40
Chapter 7
Relatively Minimal Models
The goal of this section is to use the notion of a relatively minimal model from the theory
of algebraic surfaces to prove the following lemma. Fix R > 0. We recall the set KR C J
from Lemma 6.1, a subset of the space of w-compatible almost complex structures that by
Lemma 6.1 contains the standard almost complex structure J in in its closure. We introduce
the following
Assumption 7.1. Let E C X be the (non-irreducible) curve blown down by the map to
the relatively minimal model X -+ X# (see Theorem 7.2 for the definition of a relatively
minimal model.) Then we assume that E and D have no common irreducible components.
This holds, in particular, if (X, A, D) is already relatively minimal (again, see below for
the definition), which is true for all the examples of contractible surfaces of log-general type
I know of.
Lemma 7.1. Consider a triple (X, A, D), where A is smooth of log-general type, D is a
rational tree, and (X, A, D) satisfies Assumption 7.1.
Fix a norm | - | on F(End(TX)), and J the standard integrable almost complex structure on X. Then there exists e > 0 such if IJ - J'J < e and J' E KR, then there are no
J'-holomorphic maps u : CP 1 -+ X such that (i) im u 0 U U E; (ii) u-'(D) = {pt}; and
(iii) w(u) < R.
We will need certain basic elements of surface theory. Proofs can be found in [16] and
[20]. All statements, as usual, are for surfaces over C.
Theorem 7.1. Let X be a smooth surface, and D C C an effective Q divisor (i.e. a
divisor represented by a sum of curves with positive rationalcoefficients.) Then there exists
a unique decomposition D = D+ + D- such that
(i) D+ and D- are effective Q-divisors.
(ii) The intersectionform on the Q-vector space generated by the irreducible components
of D- is negative definite (where we take any form on the zero vector space to be negative
definite.)
(iii) D+ is NEF, i.e. D+ - C > 0 for any curve C.
(iv) D+ - F = 0 for any irreducible component F of DD+ is called the NEF component of D, and D- the negative component.
41
Proof. [20] Theorem 1.19.1
This is known as the Zariski-Fujita decomposition. By uniqueness D = D+ iff D is
NEF, in particular if D is ample.
We have a triple (X, A, D) with A = X \ D, with D having simple normal crossing singularities and A smooth. The point of the next theorem is that there exists a simple geometric
realization of the NEF component of K + D. We first recall the following weak form of the
adjunction inequality.
Proposition 7.1. Let C C X be an irreducible curve on a smooth surface X, then
2ga(C) - 2 < K - C+C
2
with equality iff C is smooth.
Proof.
The following is essentially [16] Theorem 1.1
Theorem 7.2. Assume (X, A, D) are a triple with 9(A)
0. Then there exists a triple
(X#, A#, D#), with X# smooth, and D# an effective rational divisor with coefficients between 0 and 1 with supp D# = X# \ A#, such that
(i) There exists a surjective birationalmorphism 7r : X -+ X# with ,r-(A#) = A.
(ii) K# + D# is NEF on X#, where K# is the canonical class on X#.
(iii) ir*(K# + D#) = (K + D)+.
Furthermore,-9(A) = 2 iff (K# + D#)
model of (X, A, D).
2
> 0. (X#, A#, D#) is called the relatively minimal
Proof. Sketch First we recall the algebraic procedure for constructing the NEF component of an effective divisor. Starting with a rational divisor D = EZ ayDj and an index
i, set a = a3 if D - D > 0 or j 5 i, and a = a2 - Di - D/D? 2 0 otherwise. Then
D' = EZ a D satisfies Dj -D' > 0. Carry out this process for i going from 1 to n, then repeat; the non-trivial part of the theorem is showing that this converges to a rational divisor.
Assume inductively we have a surface triple (X', A', D'), D' rational and effective, and
there exists a curve E with E - (K' + D') < 0. Log-Kodaira dimension is a birational invariant, so R(A')
0, so K' + D' is equivalent to an effective divisor and thus E 2 < 0. There
are two possibilities:
(i) E Vsupp D'. In that case E -D' > 0, and so K'. E < 0. By the adjunction inequality
E is a rational curve with E 2 = K' - E = -1, so we may blow down E to get a new triple
(X", A", D") with a blowdown map 7r : X -+ X". We have
r*(K"/) = K' - E = K' - ((K - E)/E 2 )E
and
r*D"1 = D' + (D' - E)E = D' - ((D' - E)/E 2 )E
and so
42
7r*(K"I + D") = K' + D' - ((K' + D' - E)/E 2 )E
Which is precisely the numerical procedure described above. Note that this step can
happen only finitely many times.
(ii) E c suppD'. In that case write D' = rE + Do, where 0 < r < 1. Since E 2 < 0, we
have
E -D'= rE2 + E - Do
_
E2
and so
2
0 > (K' + D') -E > K' - E + E
which implies by adjunction that E has arithmetic genus 0. If K - E < 0, then E is
an exceptional curve and we can reduce the coefficient r to 0 and reduce to case (i). If
K - E > 0, then
((K' + D') -E)/E
2
= r + (E - Do + E -K)/E
2
<
r
Thus replacing D' with D" := D' - ((K' + D') - E)/E 2 E yields an effective divisor D"
with (K'+ D") -E > 0, which is simply the numerical procedure for constructing the NEF
component. For the statement about log-Kodaira dimension 2 see [16] 2.19.
L
Now assume the triple (X, A, D) has A contractible of log-general type. We set E =
of negaEZ Ej, the total divisor blown down under X -+ X#; E consists of rational curves
1
tive self-intersection. Recall that A has no algebraically embedded copies of A , and so for
each i either E C D or IEi n DI = 2; since we are enforcing Assumption 7.1 only the later
case can happen. Let D C U be a small neighborhood.
Proof. of Lemma 7.1 Suppose the statement is false. Then there exists a class B E H 2 (X, Z)
with w(B) < R, a sequence J1 , J 2 ,... of almost complex structures with Ji E KR and
limi Ji = J, and a sequence of U1 , u 2 ... of maps from CP1 such that ui is Ji holomorphic,
U U E, and (uj).[CP1] = B. By Gromov compactness for varying almost complex
im ui
structure, after perhaps passing to a subsequence, we may assume that ui Gromov-converge
to a stable J holomorphic map U: 3 -+ X defined on a stable genus 0 Riemann surface S.
ul is not necessarily simple, but it will factor as ui = i o g, where ii1 is simple and
g : CP1 -+ CP 1 is an m-fold cover. Thus (fii),([CP']) = B/m, and w(B/m) < R since
m > 1, and since J 1 C KR we must then have
c 1 (B/m) - (B/m) -D > 0
and so
c 1 (B) - B - D
43
(*)
Write S = So U Si u ... S1 as a union of irreducible components. Since im u 0 U for all
i, im U 0 U. So 3 has an irreducible component, say So, with u(So) V U. By Lemma 4.6,
uI-'(D) = {pt}.
Let Ci = U(Sj) the image curve. Then, conflating homology classes and curves representing
them, we have B = U,(S) = E miCi, where mi the multiplicity of the map Ulsi : Si -+ Ci.
Recall that D and E share no common components by assumption, and assume U has
been chosen sufficiently small that it contains no components of E. Then none of the Ci are
components of E; if Ci c U this is true by assumption, while otherwise we have ICinD = 1
by Lemma 4.6, whereas 1Ejn Dj > 2 for all j. Let 7r: X -+ X# be the relatively minimal
blow-down.
Since Ci
E for any i, Ci - (K + D)-
0, and so
,r(C) - (K# +D#) = Ci .- r*(K# +D#) = Ci - (K+ D)+ < Ci (K + D)
But K# + D# is NEF, and so
0
<
7r(Co) - (K# + D#) 5
mir(Ci)- (K# + D#) 5
B - (K + D) = B -D - c1(B)
m Ci- (K + D)
0
Where the last step follows from (*). Hence 7r(Co) - (K# + D#)
iff (K# + D#) 2 > 0, and so by the Hodge Index Theorem 7r(Co)
=
2
0. Recall that
< 0.
K(A) = 2
On the other hand, Co is not smooth, since JCo n DI = 1 and A has no algebraically
embedded copies of A' [20] 3.4.10. Since blow-downs cannot resolve singularities, 7r(Co)
is also singular. Also 7r(Co) intersects the support of D#, and so 7r(Co) - D# > 0, which
implies 7r(Co) - K# < -1. The adjunction inequality for singular curves says that
-1
and so
7r(Co)
2
>
<
7r(Co) -K#
+ 7r(Co)
2
0, which is a contradiction to the previous paragraph.
44
Chapter 8
Non-vanishing of Symplectic
Cohomology
Recall again that A is a smooth contractible surface of log-general type, X a smooth projective compactification of A and D = X \ A a curve with simple normal crossing singularities.
Proof. of Theorem 1.1 By Lemma 7.1, there exists a sequence of almost complex structures Ji, J2 ... on X which equal the standard almost complex near D, such that (i)
1
limi,, Ji = J; and (ii) The minimal energy of (w, Jk)- holomorphic maps f : CP -+ X
1
with f- (D) = {pt} is greater than 2k.
By Theorem 9.1, one can pick a good crossing point p C D, and perform degeneration
to the normal cone at p for each Jk. By Lemma 4.4, for some choice of blow-up metric
on the degeneration for (X, Jk) the minimal energy of Jk holomorphic discs with boundary
on Kt is greater than k as t -+ 0, since otherwise one could produce a rational curve of
energy less than 2k. By Lemma 4.5, there thus exists a convex almost complex structure Jk
on AO such that the minimal energy of non-constant (w, Jk) holomorphic discs in AO with
boundary on K, is greater than k. Theorem 5.1 then implies that SH*(A) : 0.
45
46
Chapter 9
3-manifold Topology
In this section, we will construct a locally incompressible torus near infinity. We first
introduce the following
Assumption 9.1. Given a rational tree D C X inside a smooth surface, assume that
dual graph of D is minimal in the sense that all components with self-intersection -1 have
valency either 1 or 2, and that the dual graph has at least 2 vertices with valency at least 3.
We will prove the following
Theorem 9.1. Let X be a projective variety, with A C X affine and topologically contractible and D = X \ A a rational tree satisfying Assumption 9.1. Suppose that the dual
graph of D has at least two vertices of valency at least 3. Then there exists an open set
U with D C U, and a torus T C U \ D, Lagrangian with respect to a choice of Liouville
manifold structure on A, such that the map
7ri(T) -+ 7r,(U
\ D)
is injective.
Recall that the dual graph r of a SNC divisor D is a decorated graph, with one vertex
for each irreducible component of D, with the vertices decorated with the self-intersection
numbers of the corresponding components, and for any two vertices v and w, the number
of edges between v and w is equal to the number of intersection points between the two
components (including self-intersections if v = w. ) Recall that all components of D are
rational and F is a tree.
Thinking of A as Liouville manifold, there exists a compact Liouville domain A! with
6-A:= Y a contact 3-manifold with
A=A
U
Yx[1,oo)
6A=Yx{1}
Let U be a sufficiently small open neighborhood of D in X-for example, on can take
U = -1 ((c, oo) U D, for c sufficiently large, where 4 : A -+ R is the strictly plurisubharmonic function # = - log IsI for s a holomorphic section of an ample line bundle vanishing
along D. Then U \ D deformation retracts onto Y (more precisely, onto a contact 3-manifold
47
diffeomorphic to Y; but since we consider only the smooth topology of Y in this section
the conflation is harmless.) Moreover, without loss of generality we can assume that the
crossing torus T C Y; depending on our choices this may require a non-Hamiltonian isotopy, and hence the image of T may not be Lagrangian; but since the isotopy is contained
in U \ D and the theorem refers only the induced maps on homotopy groups this will still
imply the theorem.
The key fact for proving Theorem 9.1 is that r provides a plumbing diagram for Y;
Definition 9.1. Given a connected graph r with integer labels on each vertex and smooth
Riemann curves Di associated to each vertex vi, we say that a r provides a plumbing diagram for a three-manifold Y if Y is diffeomorphic to a manifold constructed as follows:
For each vertex vi C F, take a copy Y of the total space of the S' fibration ,ri : Y -+ Di
with Euler class equal to D?, where Di is the irreducible componenent of D corresponding
to vi. For each edge Eij C F connecting vi with vj, one chooses small open discs Dij C Di
and Djj c Dj with Di n Dik = 0 if j =, k. Set Yj = Y \ir-1 (Dij); Yi is a closed manifold with boundary, with OYij diffeomorphic to S x S1; choose the identification such that
S1 x {1} is a section for the circle bundle &Yij -+ oDij. Finally, to construct Y, delete
all the 7r- 1 (Di j's and then glue OYi to 49Yji via map that, with respect to the trivializations above, flips the S1 factors, so that the fibre direction of one is identified with the base
direction of the other.
That the r dual to D in (X, A, D) we consider provides a plumbing diagram for Y
follows from the fact that over the smooth locus of D, Y is simply the boundary of a topological tubular neighborhood of D and the the singular locus of D consists normal crossing
singularities.
Note that 3-manifolds constructed via plumbings have embedded tori Tij for each edge
of the crossing diagram, corresponding to the glued boundary components. From the perspective of complex geometry, these tori can be viewed as crossing tori of normal crossing
divisors. In other words, for a normal crossing point p, choose local complex coordinates
(x, y) such that p = (0, 0), and the branches of the crossing correspond to IxI = 0 and Iy = 0.
Then the plumbing torus will be isotopic within U \ D to the tori {(x, y)I|xI = a, Iy = b}
for any a, b E R sufficiently small. The advantage of this viewpoint from our perspective
is that we will be able to choose the Kahler form such that these crossing tori are all Lagrangian, which is obviously the means by which the topological results here connect with
the symplectic consequences in the main body of the paper.
We now recall some basic topological definitions and facts about 3-manifolds. In the following assume that Y is an irreducible 3-manifold, i.e. every embedded sphere in Y bounds
a ball. Many of the following results can be found in [12]
Definition 9.2. A closed surface T C Y is incompressible if for every embedding
(D 2 , S 1 ) -+ (Y, T), there exists an embedding g : D 2 -+ T with gIsi = f Is'-
f
An algebraic analogue of incompressibility is injectivity of the map 7r 1 (T) -+ 7ri(Y).
Algebraic incompressibility always implies geometric incompressibility, while geometric implies algebraic for orientable manifolds, although the proof of the later is non-trivial, relying
48
on the Loop Theorem for 3-manifolds [12] Section 3)
There is an analogous notion of boundary incompressibility for surfaces with boundary
(S, aS) c (Y, aY). As we will not use the notion in detail we will not spell it out further,
although it has a natural algebraic analogue, viz. the fact that 7r,(S, 4S) -+ iri(Y, OY) is
injective.
If N C Y is a codimension-1 submanifold with trivial normal bundle, then we define the
(topological) cut of Y along N, YIN, to be the manifold obtained by removing a small open
tubular neighborhood N x (-1,1) of N. Cutting creates two new boundary components
diffeomorphic to N.
We will need the following simple
Claim 9.1. Let TT' c Y be embedded tori, with either T = T' or TnT' = 0. If TnT' = 0
then assume further that T is incompressible. Then T' C Y is incompressible if and only if
T' c YIT is incompressible, where if T' = T then T' C YIT is either of the copies of T in
the cut.
Proof. The "only if" direction is clear: any disc in YIT can also be viewed as a disc in Y,
hence can be replaced with a disc in T with the same boundary.
For the "if' direction, let f : (D 2 , SI) _+ (Y, T') be any disc. The goal is to replace f with
2
another map f', agreeing with f on S1, mapping the interior of D to Y \ T. Then f' can
be viewed as a map to YIT with boundary on T', so can be replaced by a map to that torus.
We may assume without loss of generality that f is transverse to T. Thus f -(T) is a
2
finite disjoint union of embedded circles. If f-1 (T) = aD or 0, then by the previous paragraph we are done. If there are interior circles, choose an innermost circle K C f- 1 (T),
i.e. a circle whose interior disc does not intersect the pre-image of T. Letting C be the
disc bounded by K, we see that f restricted to (C, K) is a disc with boundary on T and
otherwise disjoint from it, hence by assumption can be replaced with a disc mapping to T.
Hence f can be replaced with a map f mapping all of C into T. A small homotopy of f
then removes this intersection. Hence we have reduced the number of interior circles by
one, and the result follows by induction.
Our strategy of proof for Theorem 9.1 is now as follows: we cut Y along a certain
collection of crossing tori, and prove that the resulting boundary components are all incompressible. Then Claim 9.1 implies that the original tori were incompressible in Y
Our main tool for proving incompressibility will be the notion of a Seifert Fibering
2
Definition 9.3. The standardSeifert fibering of the solid disc D x S1 of type (p, q) (p and
q relatively prime, q > 1) is the foliation of the solid torus by circles obtained as follows:
start with the trivial foliation by {x} x S' for x G D 2 , cut the torus along a meridian disc,
then glue the discs back together via a 27rp/q rotation.
49
A Seifert Fibration of a 3-manifold Y is a foliation by circles such that all leaves have
neighborhoods isomorphic to a neighborhood of a leaf in a standardSeifert fibering.
We will sometimes refer to leaves of the foliations as fibres, for reasons to be explained
shortly. Fibres that do not have neighborhoods isomorphic to (p, 1) standard neighborhoods
for some p e Z are called exceptional. Note that exceptional fibres are interior and isolated.
Identifying the fibres of a Seifert fibration gives a quotient map Y -+ B, where B is an
orbifold whose underlying space is a surface; we will use B to also denote this surface. Note
that the orbifold points of B correspond to the exceptional fibres of Y. In particular, 9B is
a union of circles, and since all exceptional fibres are interior &Y is a union of circle bundles
over the circle, hence if Y is orientable they are tori.
Definition 9.4. Let Y be Seifert fibered, and S C Y a smooth surface. We say that S is a
Vertical surface, if S can be written as a union of circle fibres; S is a Horizontal surface, if
S is transverse to all circle fibres; and S is -parallelif it is isotopic fixing OS C OY to a
surface contained in 4Y.
Note that vertical surfaces admit circle foliations, hence in particular no discs can be
vertical. As for horizontal surfaces, transversality implies that the map S -+ B is a branched
covering, branched over the points of B corresponding to exceptional fibres, with branching
degree q at an exceptional fibre of type (p, q) (note that a meridian of a standard Seifer
fibered solid torus intersects each non-exceptional fibre q times.). Hence the RiemannHurwitz formula says
X(B) = x(S)/n + Ej(1 - 1/qj)
Where i ranges over the exceptional fibres of Y. The key theorem about Seifert fibered
manifolds is the following
Theorem 9.2. Let S C Y be an incompressible, boundary incompressible, non a-parallel
surface in a Seifert fibered 3 manifold. Then S is isotopic, fixing aS, to either a vertical or
a horiztontal surface.
Proof. Sketch Giving the proof in full would take us a little far afield into the land of 3manifolds, so we will be content to outline the argument. For details of the argument given
here see [12] p.15-18]. Let us call any surface which is incompressible and a-incompressible
essential.
The first step is to prove that any essential surface in Y = S x D 2 is isotopic to a meridian
disc. Isotope S such that
consists of either meridians circles or is transverse to all meridian circles, and let D be some meridian circle, which after an isotopy of S we may assume
is transverse to S. Circles in Sn D may be eliminated, innermost first, by incompressibility
of S and irreducibility of Y via an argument analogous to the proof of Claim 9.1, while
arcs in S n D actually cannot occur due to a-incompressibility of S and the fact that aS is
transverse to all meridian circles. Thus S n D = 0, and aS consists of meridian circles. aincompressibilty then implies that
is a single circle, and then S is isotopic to a meridian
disc.
aS
aS
50
For the general case, consider the surface B 0 c B with small open neighborhoods of the
orbifold points and a single smooth point if there are no orbifold points removed, and let
Yo = 7r~1(Bo). S can be isotoped such that all boundary components are either horizontal
or vertical in Y, and such that all vertical fibres in S are contained in Y and S is transverse to the fibres removed in constructing B 0 , and thus So = S n Yo also has all boundary
components either vertical or horizontal.
Choose a collection of arcs that cut Bo into a disc, and let A be the pre-image of these
arcs, so that A is a union of annuli and Y = YoIA is a solid torus. By similar arguments
to the above using incompressibility we can eliminate circles in So n A bounding discs in A
and arcs in So n A with boundary on a single boundary component of A, so that A n So
consists of either vertical circles or horizontal arcs. Thus Si = SoIA also has all boundary
components either vertical or horizontal.
One can arrange that Si is incompressible via a homotopy of S. One can also show that
the only incompressible surfaces which are not 0-incompressible are a-parallel annuli, so
Si must be a disjoint union of essential surfaces and a-parallel annuli. a-incompressibility
of S allows us to isotope S to eliminate intersections with exceptional fibres if Si contains
a-parallel annuli with horizontal boundary. So we may assume that Si contains components
that are either essential or 0-parallel annuli with vertical boundary.
Yi is a solid torus, and so horizontal and vertical circles in 4Y always intersect, thus
Si must contain either essential surfaces or 0-parallel annuli with vertical boundary, but
not both. In the former case Si is isotopic to a meridian disc, and hence S is isotopic to a
horizontal surface, while in the later case Si can be isotoped to a vertical surface rel OS 1 ,
which provides an isotopy of S to a vertical surface.
Armed with this, we can now prove
Lemma 9.1. Let Y be Seifert fibered, and T c 0Y be a boundary torus which is not
incompressible. Then Y is a solid torus with a standard Seifert fibering.
Proof. Let D c Y be a disc with 9D C T not bounding a disc in T. Then D is not 0-parallel,
and discs are always incompressible and boundary incompressible. Hence by Theorem 9.2,
D is isotopic to a horizontal surface, since there are no vertical discs. Recalling that B is
the base of the Seifert fibration, the branched covering D -+ B gives
1
1
X(B) = - + Ei(1 - -)
n
qi
>0
The only surface with boundary with positive Euler characteristic is the disc. Furthermore, each factor of 1 - 1/q > 1/2, so there can be at most one exceptional fibre, since
otherwise X(B) > 1, contradicting the fact that B is a disc. But a Seifert fibering over the
disc with a unique exceptional fibre is clearly isomorphic to a standard Seifert fibering of
the solid torus.
51
Now we return to the plumbed 3-manifold of a normal crossing divisor. We start with
some terminology about graphs.
Definition 9.5. Let F be a connected graph. We call a vertex a branching node if it
has valency at least 3. A leaf is a univalent vertex. A chain is a linear subgraph all of
whose interior vertices are bivalent. A maximal chain is a chain whose end vertices are not
bivalent. A maximal twig is a maximal chain with a leaf at one end, while a connecting
chain is a chain with branching nodes on both ends.
Note that since F is a tree, the number of branching nodes is one more than the number
of connecting chains. We are assuming there are at least two branching nodes.
For each edge E E F, pick once and for all a crossing torus T C Y corresponding to
that edge, and for each connecting chain C1... C8 , pick an edge Ej C Ci and corresponding
torus T. Then Y I{T1,... T,} has s + 1 connected components in one-to-one correspondence
with the branching nodes of F. We will label the component of YI{T, ... T} corresponding
to a branching node v, Y, and F, the component of F containing v after deleting the chosen
edges in the connecting chains.
We first need a simple lemma about the self-intersections of components of D; the proof
uses concepts from Section 7, so the reader who has skipped ahead can take the statement
on faith. The setup is a triple (X, A, D) with A smooth and 9(A) > 0, D is a rational tree,
F the dual graph of D, and F is minimal in the sense that all -1 vertices are branching nodes.
Lemma 9.2. Let vi E F be a vertex which is not a branching node lying on a maximal twig.
Then if Di is the curve correspondingto vi we have D? < -2.
Proof. We proceed by induction on the distance from vi to the leaf of the twig containing
= -2 - D?. Now
assume vi is a leaf. Then Di intersects one other component of D at a single point, and so
D-Dj=1+D?. Thus (K+D).Di=-1.
vi. Note that since Di is a rational curve by adjunction we have K - Di
Since R(A)
0 K + D is linearly equivalent to an effective divisor and we can K + D =
(K + D)+ + (K + D)-, where (K + D)+ is nef, and the intersection form of the graph of
the support of (K + D)- is negative definite. Thus since (K + D) -Di = -1 we must have
Di C supp(K + D)-. Thus D? < 0. But no leaf can be a -1 curve, and so D? < -2.
Now suppose that vi is a bivalent vertex. It is connected to two other vertices vi_ 1 and
vi+ 1 , and assume inductively that D+ 1 C supp(K + D)-. Then D - Di = 2 + D?, and so
(K + D) - Di = 0. This implies that (K + D)- - Di < 0. But we know that Di positive
intersects the component Dj+1 of (K + D)~, and so we must have Di C supp(K + D) and
thus D? < -2 as in the previous paragraph.
Lemma 9.3. The irreducible components of Y|{T1 ... T} are Seifert fibered. For each
branching node v, the boundary components of Y correspond to connecting chains out of v,
while the exceptional fibres of the Seifert fibration correspond to branches out of v.
52
Proof. Pick a branching node v. Cut Y along all the remaining edges in F, then think of
Y as being built from the resulting pieces. The topological picture of Y, is then as follows.
There is a central component, M, which is diffeomorphic to B x S1, where B is a disc with
at least two punctures. For each boundary component of M, one has either a (possibly
empty) sequence of annuli attached (these correspond to connecting chains), or a (possibly
empty) sequence of annuli capped off with a solid torus (the leaves at v).
1
Mv has a Seifert fibering from an isomorphism with B x S . The plan is then to extend this Seifert fibering over the rest of Y component by component.
First, suppose we have an annulus one of whose boundary components is foliated by circles.
Thinking of the annulus as T2 x I, this foliating extends linearly over the whole annulus,
so there is no obstruction to extending the Seifert fibration over these components.
Now consider a solid torus with a foliated boundary T. In H 1(T) a fibre of this foliation is equal to pm + ql, with p and q relatively prime, 1 the class of a longitude and m
the class of a meridian. Then as long as q $ 0 this foliation extends to a standard Seifert
fibration of the solid torus of type (p, q). For our purposes this is not quite enough, since we
need the central fibre of this standard fibration to be exceptional. So we need to prove that
the foliation of the boundary torus of a leaf component one gets by extending the foliation
of the central component is of type (p, q) with q j 0, 1
In order to extract such numerical information, a slight change of perspective is helpful.
Recall that (p, q) Dehn surgery on a knot in a three manifold involves removing an open
solid torus tubular neighborhood of that knot, and then gluing in another solid torus via a
map which takes a meridian to pm + ql (and hence a longtitude to t(-qm + pl)). Then
plumbing in a disc bundle over a disc with Euler number n corresponds, on the boundary
3-manifold, to Dehn surgery along a fibre with surgery coefficient n ([11] Section 5.3).
The advantage of the surgery perspective is that, unlike plumbing, one can surger with
arbitrary rational coefficients. This allows one to collapse chains of integral surgeries into a
single rational surgery, via a technique known as the slam dunk (see [11] p. 163). Namely,
consider a surgery on a simple two component link, with integral coefficient n on the first
and r E Q U {oo} on the second. Perform the first surgery, then isotope the loop of the
second surgery into the surgered torus. Since the first surgery was integral the resulting
loop will go around once, hence be isotopic to the central fibre of the solid torus. Thus
surgery on the second loop just cuts out this torus and glues it back in, so the result is
a single Dehn surgery on the first loop. Keeping careful track of the identifications shows
that this single Dehn surgery has coefficient n - 1/r
So if we have a branch of r, with surgery/plumbing coefficients a,, a2,... an (an corresponding to the leaf), we can instead view it as a Dehn surgery on a regular fibre with
coefficient the continued fraction
1
al which we will write [a, :
a2, --
1
a.] (this notation is slightly non-standard, since in standard
53
notation one adds rather than subtracts). The loop at the center the surgery is a nonexceptional loop in a Seifert fibration, so fibres of the original foliation are longitudes of
the pre-surgered torus. After a general p/q surgery, longitudes are take to -q/p loops, so
the boundary of this surgered torus is foliated by [0 : a, : ...: an] loops. Each ak 5 -2,
and so by uniqueness of continued fraction expansions [0 : a,, -- -an] is a non-integral finite
rational number. Thus the Seifert fibration extends over the leaves, with each leaf providing
an exceptional fibre.
0
Proof. of Theorem 9.1 Since U \ D deformation retracts onto Y it suffices to consider Y.
Consider the cut of Y along connecting chains Y{T1 ,... T 8 }; since there are at least two
branching nodes in D this is a non-trivial operation. Each component Y, is Seifert fibered
by Lemma 9.3, with boundary a crossing torus. If any of these boundary components were
compressible in Y, then by Lemma 9.1 Y, would be a solid torus with a standard Seifert
fibering. Solid tori with standard fibering have precisely one boundary component and one
exceptional fibre. However,
#
boundary components of Y +
#
exceptional fibres of Y, = valency of Y, > 3
So all the boundary components of the Y, are incompressible.
together they remain incompressible.
54
After gluing them back
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