RELATIVE FLOER THEORETIC INVARIANTS FOR
SEMI-POSITIVE SYMPLECTIC MANIFOLDS
THESIS PROPOSAL FOR UMUT VAROLGUNES
1. Introduction
This thesis is about a Floer theoretic construction which associates a unital
algebra to any open subset of a semi-positive symplectic manifold. I expect it to
have applications to displacability of open subsets, mirror symmetry, and finding
nontrivial elements in various symplectic mapping class groups. It is an ongoing
project, admittedly in its beginning stages, under the supervision of Paul Seidel. It
is expected to be completed towards the end of the Spring semester of 2018.
In what follows, I will briefly describe the construction of the invariant, and list
its expected properties, and potential applications.
2. Construction of the invariant
Recall that a semi-positive symplectic manifold is a manifold X 2n such that
for every A ∈ π2 (X), if ω(A) > 0 and c1 (A) ≥ 3 − n then c1 (A) ≥ 0. When
the symplectic manifold is open we also impose certain boundedness conditions at
infinity which we don’t explicitly specify here. Under these assumptions, one can
define Hamiltonian Floer homology relatively easily because sphere bubbles can be
excluded for generic choices of Hamiltonians and almost complex structures due
to dimension counting arguments [1]. It is possible that our invariants could be
defined in more general settings but we will not be concerned with that. Let us
also stick to the closed case in this section in order to highlight the new features of
the invariant rather than having to talk about the well-known analytic difficulties
that arise in the open case [12].
Since the symplectic action functional is not exact on the free loop space, one
has to work with Novikov coefficients here. For any module M over the Novikov
ring,
X
(1)
Λ≥0 := {
ai q αi | ai ∈ Z, αi ∈ R≥ , and αi → ∞},
i∈Z+
one can define Hamiltonian Floer cochain complex with coefficients in M . Namely,
if (Ht = Ht+1 , J) is an admissable pair, we define
M
(2)
CF ∗ (X, H, J, M ) =
M,
i∈P0 (X)
where P0 (X) is the set of contractible periodic orbits of H. For any two elements
γ, γ 0 of P0 (X), we obtain a map M → M , which is simply multiplication with the
1
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THESIS PROPOSAL FOR UMUT VAROLGUNES
following element of Λ≥0 :
(3)
X
#(M(γ, γ 0 , C))q E(C) ,
C
where the sum is over homotopy classes of maps
(4)
(S 1 × [0, 1], S 1 × {0}, S 1 × {1}) → (M, γ, γ 0 ),
M(γ, γ 0 , C) is the moduli space of the solutions of the Floer equation connecting γ
to γ 0 which represents C, and # is a signed count of orbits of the translation action
of R, which outputs zero in case the dimension of the moduli space is not 1.
The differential d : CF ∗ (X, H, J, M ) → CF ∗ (X, H, J, M ) is the matrix with
entries given by these maps. Its square is the zero map, making CF ∗ (X, H, J, M )
a chain complex.
Note that we do not have an ordinary Z-grading in general. There are more
involved notions of grading that do provide help in computations, which I plan to
exploit in the future [4]. I do not see anything that is special about signs in this
story. Let us ignore signs and gradings from now on.
It is important to notice that not every admissable homotopy (Ht,s , Js ) automatically defines a continuation map in this case. This is because the solutions
of the continuation map equation do not necessarily have non-negative topological
energy [2]. Hence the analogue of (3) in this case may not give an element of Λ≥0 .
One case in which this happens is when the homotopy is monotone, meaning that
H is increasing with the s coordinate. Clearly, the continuation maps in this case
are not quasi-isomorphisms except in trivial cases.
Let Λ be the Novikov field, i.e. αi ∈ R in (1). We also introduce the notation
Λ≥r ⊂ Λ for the elements with valuation ≥ r, and Mr for the quotient Λ≥0 -module
Λ/Λ≥r
Definition 1. SC ∗ (U ) := lim lim CF ∗ (M, H, J, Mr ), where
←− −→
• the direct limit is taken over admissible pairs (H, J) such that H is negative
in the closure of U with maps defined via monotone homotopies.
• the projective limit is taken over all real numbers r using the maps induced
by the canonical maps Ms → Mr for s ≥ r.
We have been a little sloppy when taking the direct limit because in principle the
continuation maps can be different for different choices of monotone homotopies.
Yet, in fact any two such continuation maps are chain homotopic to each other,
which implies that the chain homotopy type of the limit is independent of the
choices we make for the monotone homotopies.
In order to explain this definition, let us choose specific kinds of cofinal families.
Let us also make a simplifying assumption, which is to assume that cl(U ) is a
manifold with boundary with cl(U ) − U as its boundary. Almost all open subsets
can be Hamiltonian isotoped to such a subset so I don’t think of this as a real
restriction.
• Let H : X → R be a function which is first of all negative in cl(U ) and
sufficiently admissable. I want to choose it so that it has small first derivatives on cl(U ), increasing from inside to outside in a neighborhood N of
cl(U ) − U in X with cl(U ) − U as a regular level set, and finally it is strictly
bigger outside of U ∪N than the inside, but again has small first derivatives.
.
RELATIVE FLOER THEORETIC INVARIANTS FOR SEMI-POSITIVE SYMPLECTIC MANIFOLDS
3
• The cofinal family will be obtained by composing this function with functions hn : R → R in a way that pushes us by slow steps to the colimit in a
specific way.
• Then one has to perturb functions to get from the Morse-Bott situation
to a real Morse one, but I will assume that this will not cause any real
problems.
When this is done carefully it is not too hard to believe that for the orbits that
lie inside of U , once they are created, they will behave in the simplest way under the
continuation maps. On the other hand the ones that lie outside of U will at some
point get killed by the continuation maps by the choice of our coefficient modules.
Because of the inverse limit all Floer trajectories between “inner” orbits will be
taken into account in our definition.
In case cl(U ) − U is a contact hypersurface, Viterbo gives a definition of symplectic cohomology of cl(U ) [3], and our invariant is a q−deformation of that.
3. Expected Properties
I expect the following properties to hold without too much doubt:
• SH ∗ (U ) := H ∗ (SC ∗ (U )) is a unital algebra over the Novikov field.
• SH ∗ (M ) is isomorphic to the quantum cohomology algebra.
• (restriction maps) Whenever U ⊂ V we get algebra maps,
(1)
SH ∗ (V ) → SH ∗ (U ).
In particular SH ∗ (U ) is a module over quantum cohomology.
• (Hamiltonian isotopy invariance) Let U ⊂ V be open subsets, and φt is a
Hamiltonian isotopy of M such that φ0 = id and φt (U ) ⊂ V for all t, then
under the relabeling identifications SH ∗ (U ) = SH ∗ (φt (U )) the restriction
maps SH ∗ (V ) → SH ∗ (φt (U )) are the same.
• (BV operator) There should exist a BV operator ∆ : SH ∗ (U ) → SH ∗−1 (U )
satisfying the usual axioms.
• (locality) If U can be disjoined from itself by Hamiltonian isotopy, then
SH ∗ (U ) = 0.
Here are some properties that has some chance of holding. They are stated very
vaguely.
• (Mayer-Vietoris property) There should be some kind of (co)sheaf property
of these groups. I have yet to come up with an idea of exactly what to
expect but there are examples of this phenomenon in the literature [8], [6],
[7].
• (open string version) There is also an open string version of this invariant
which imitates the wrapped Floer theory construction [5], meaning that
we try to only get the Hamiltonian chords that lie in the given open subset. Namely, let L be a Lagrangian in M , then we define a vector space
HF ∗ (L, U ) over the Novikov field. Probably, there is a Lagrangian intersection version of this too, and hence a definition of a relative wrapped
Fukaya category.
• (open-closed maps) The standard open-closed and closed-open string maps
should descend to our invariants in some way. Yet, the existence of both
limits and colimits in our definition requires care in doing this.
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THESIS PROPOSAL FOR UMUT VAROLGUNES
• (interaction of relative HF ∗ with actual HF ∗ , module structure) If we have
any Lagrangian K in M , then we can define HF ∗ (K, (L, U )) by a similar
limit construction. This defines functors from F uk(M ) → Ch(K). We
can expect that for certain covers of X by open subsets, and ”generating”
Lagrangians inside, these functors would define enough test functors to
distinguish objects of F uk(M ).
• (fibrations) In case our manifold M is fibered over a base B, we can restrict
ourselves to open subsets which are preimages of open subsets in B, and
use Hamiltonians which are lifts of functions on B. This might lead to
some (co)sheaf like objects over B, and make it easier to generate the
Fukaya category of M in the sense of the previous item. This is particularly
interesting in the case of integrable systems.
4. Potential Applications
4.1. Displacability questions. As already mentioned an expected property is
that if U is an open subset such that SH ∗ (U ) 6= 0, then U can not be disjoined
from itself by a Hamiltonian isotopy. The proof of this would work by using the
displacing Hamiltonian isotopy to change the chain complex computing SH ∗ (U )
in a way that doesn’t change the homology, where the new generators are seen to
not exist by geometric reasons. This kind of argument is used in Rabinowitz Floer
homology for displacing hypersurfaces [10]. The role that is played by the leafwise
intersections should be played by the inner periodic orbits in our case.
Open subsets of the two-sphere provide interesting test cases for this property. Namely if one takes the open set as the complement of n disks with areas
A1 , A2 , . . . , An , an easy area argument shows that the displacibility is equivalent to
one of Ai ’s having area more than half the total area of the sphere. It seems like
SH ∗ (U ) 6= 0 also gives a necessary and sufficient condition.
I am not aware of many results for displacability of open subsets of symplectic manifolds in the literature. For displacability of other kinds of subsets, the
first that comes to mind is Lagrangians, for which Lagrangian Floer homology
has been used to prove numerous nondisplacability results. Secondly, as was also
mentioned before, Rabinowitz Floer homology can be used to find a necessary condition for the displacability of certain special types of hypersurfaces [10]. Finally,
Entov-Polterovich have deviced a method using quasi-states to find obstructions to
displacing what they call stems, which can be thought of generalizations of fibers of
the moment map for symplectic toric manifolds [9]. Most of these subsets have standard open neighborhoods, which we can use to turn these problems into problems
about open subsets.
4.2. Integrable systems and mirror symmetry. Mirror symmetry from my
perspective is a way of understanding the various Fukaya categories associated to a
symplectic manifold by constructing a (possibly non-commutative) algebraic space
for which the Fukaya category is some sort of category of sheaves. To present ideas
let me first restrict to the case of a complex algebraic hypersurface in a toric variety
(say (C∗ )n for now but we can deal with compactifications too).
Let X ∈ (C∗ )n be a hypersurface. Mikhalkin constructs a pair-of pants decomposition of X using a tropical degeneration [11]. More precisely, X forms a mildly
singular torus fibration over a tropical hypersurface Y ∈ Rn , where the vertices of
Y index the pair-of-pants in the decomposition, which are glued together according
RELATIVE FLOER THEORETIC INVARIANTS FOR SEMI-POSITIVE SYMPLECTIC MANIFOLDS
5
to the edges (1-cells) connecting these vertices in Y along the cylinders above those
edges. By Viro patchworking (i.e. carefully taking the real part), we can construct
a Lagrangian section of the torus fibration, consisting of compact and non-compact
connected pieces L1 , . . . , Lk . For n = 2, in the model pair-of-pants, this looks like
the places where the pants are stitched together.
While it is not true that these L0i s generate (or split generate) the wrapped
Fukaya category, it is true locally in some sense. Using the open string version of
the construction above for Li ’s, we can produce a diagram (based on the vertices and
edges of Y ) of A∞ -algebras. Now it is indeed enough to test Lagrangians against
this diagram of algebras, to distinguish objects of the Fukaya category. Strictly
speaking this is a conjecture, but one that is very likely to hold. In Heather Lee’s
paper for the n = 2 case a strong locality result was shown which makes it possible
for her to by-pass the relative Floer theoretic construction described here. It is
possible that this locality result generalizes to general type varieties, which would
make everything simpler. Regardless, we will think of this diagram, or if available
its more geometric realizations, as the mirror of X.
Once we have this mirror, which is unfortunately something like a non-commutative
non-archimedean rigid analytic space in the general case, one needs to construct a
mirror functor. Even though it is pretty clear what this should be geometrically,
to really get an analytic sheaf over the mirror one needs to show a bunch of convergence statements a la Family Floer homology [14]. Presumably, the required
analytic results already exist in the literature [15], [16].
It’s clear that these techniques are more generally applicable. For example, one
would expect to produce mirrors to any integrable system with mildly restricted singularities. I am tempted to believe that having local Lagrangian sections is enough.
Note that Mikhalkin’s construction doesn’t actually construct a Lagrangian fibration but I will pretend that it does here.
One advantage of constructing the mirror this way is that, under mild conditions,
symplectomorphisms of an integrable system, which in turn act on the Fukaya
category, will manifestly have actual space level mirrors as automorphisms of the
mirror space, rather than just automorphisms on the mirror category. I already did
a project analyzing a special case of this (paper currently being written) in which a
particular symplectomorphism was shown to be not symplectically isotopic to the
identity by analyzing the mirror automorphism.
A related problem is to investigate if there is a mirror statement to the following.
If a complex automorphism of a variety acts trivially in a formal neighborhood of
a point, then it acts trivially on the whole variety. I am not sure how to make the
analogous statement in the non-commutative, non-archimedean setting, but it is
sensible to expect that it exists. This would tell us nontrivial information about
“how many” of the torus fibres of an integrable system can a symplectomorphism
disjoin from themselves, given that only one of them is sufficiently fixed. More precisely, from an automorphism one gets a bimodule over the mirror which“contains”
a smalle piece of the diagonal bimodule. Then one expects that it contains almost
all of the diagonal bimodule.
Forgetting about the symplectomorphism for the moment, let us consider a single
Lagrangian, which gives a sheaf over the mirror diagram. It is sensible to think that
the mirror “lives over” something like a tropical base (i.e. the Clemens polytope)
which is the base of the integrable system as in SYZ mirror symmetry. One can
6
THESIS PROPOSAL FOR UMUT VAROLGUNES
then think about the notion support of this sheaf as a subset of the base of the
integrable system. General properties of support, which is designed to detect if the
Lagrangian has an essential intersection with the fiber above the point in the base
is an interesting question in itself. The symplectomorphism question can also be
thought of as the properties of the support of the corresponding bimodule inside
the product of the base with itself.
Let me finish off with one last question. Let us again consider open sets which
are preimages of open subsets in the base. An interesting question is whether
our invariant can distinguish open subsets in the base from its convex hull. The
invariant that was previously used in the place of our construction, the KontsevichSoibelman sheaf [13], is not able to do this. Unfortunately, I currently have no idea
how to approach this from our viewpoint.
References
[1] D. McDuff and D. Salamon, J-holomorphic Curves and Symplectic Topology, Second Edition,
2012.
[2] A. Floer and H. Hofer, Symplectic homology I, Mathematische Zeitschrift January 1994, Volume 215, Issue 1, pp 37-88
[3] C. Viterbo, Functors and computations in Floer homology with applications: Part II , Preprint
available online
[4] N. Sheridan, Homological Mirror Symmetry for Calabi-Yau hypersurfaces in projective space,
Inventiones Mathematicae 199 (2015), no. 1, 1-186
[5] M. Abouzaid, P. Seidel, An open string analogue of Viterbo functoriality, Geom. Topol.
14,627718 (2010)
[6] M. Abouzaid, D. Auroux, A. Efimov, L. Katzarkov, D. Orlov, Homological Mirror Symmetry
for punctured spheres, J. Amer. Math. Soc., 26 (2013), 4, 1051-1083
[7] H. Lee, Homological Mirror Symmetry for open Riemann surfaces from pair-of-pants decompositions, preprint available upon request
[8] A. Kapustin, L. Katzarkov, D. Orlov, M. Yotov, Homological Mirror Symmetry for manifolds
of general type, Central European Journal of Mathematics, December 2009, 7:571
[9] M. Entov, Quasi-morphisms and quasi-states in symplectic topology,. Proceedings of the International Congress of Mathematicians (Seoul, 2014)
[10] P. Albers, U. Frauenfelder, Rabinowitz Floer Homology: A Survey,. Global Differential Geometry, Volume 17 of the series Springer Proceedings in Mathematics, pp 437-461
[11] G. Mikhalkin, Decomposition into pairs-of-pants for complex algebraic hypersurfaces, Topology 43 (2004), 10351065.
[12] Y. Groman, Floer theory on open manifolds, preprint available online
[13] P. Seidel, Some speculations on Fukaya categories and pair-of-pants decompositions, Surveys
in Differential Geometry, vol. 17, 2012, pp. 411-426.
[14] M. Abouzaid, Family Floer cohomology and mirror symmetry, preprint available online
[15] K. Fukaya, Cyclic symmetry and adic convergence in Lagrangian Floer theory, Kyoto J.
Math. Volume 50, Number 3 (2010), 521-590.
[16] Y. Groman, J. Solomon, A reverse isoperimetric inequality for J-holomorphic curves, Geometric and Functional Analysis, October 2014, Volume 24, Issue 5, pp 1448-1515