considerations show that $C$ must intersect $P^{-1}(\star)$ a signed total of
$k$ times, and the monotonicity lemma applies again to get a contradiction. ∎

######  Remark 5.12.

For genus one, there are splittings of $L$ as $S^{1}\times\Sigma_{g}$ other
than the ‘obvious’ $\gamma_{0}\times\Xi_{g}$. Assume that there is a compactly
supported symplectomorphism $f$ of $Y_{r,s}$ taking $\gamma_{0}\times\Xi_{g}$
to $\gamma_{0}\times\Xi_{g}^{\prime}$ and which does not respect the
splitting. Then the above proof, using $f(P^{-1}(\star))$, etc, would show
that the count of points in
$\bar{\mathcal{M}}_{J}(\bar{Y}_{r,s};f(A),f(\alpha))$ is well-defined. On the
other hand, as noted in the proof, we know that it is non-zero for a unique
choice of $A\in H_{2}(\bar{Y}_{r,s},L\sqcup L^{\prime};\mathbb{Z})$ and
$\alpha\in H_{1}(L;\mathbb{Z})$ Maslov zero primitive such that $\partial
A=\alpha\sqcup\alpha^{\prime}$. This gives a contradiction: $f$ must in fact
have respected the splitting.

####  5.3.3. Conclusion

Start with a monotone Lagrangian $\gamma_{2n}\times\Xi_{g}\subset Y_{r,s}$,
where recall that

$\Xi_{g}=\Xi_{g}(\\{k_{0},l_{0},m_{0}\\},\ldots,\\{k_{g},l_{g},m_{g}\\}).$

Suppose you’re given a compactly supported symplectomorphisms of $Y_{r,s}$.
Choosing a compactification of $Y_{r,s}$ to $\bar{Y}_{r,s}$ which contains the
whole domain, the previous section shows that the count of holomorphic annuli
$\bar{\mathcal{M}}_{J}(\bar{Y}_{r,s},\gamma_{2n}\times\Xi_{g};A,\alpha)$, for
any regular $J$ and the one class $(A,\alpha)$ for which it’s non-zero, is an
invariant of $\gamma_{2n}\times\Xi_{g}$ which is unchanged under compactly
supported symplectomorphisms. In particular, if we’re also given
$\Xi^{\prime}_{g}=\Xi_{g}(\\{k^{\prime}_{0},l^{\prime}_{0},m^{\prime}_{0}\\},\ldots,\\{k^{\prime}_{g},l^{\prime}_{g},m^{\prime}_{g}\\})$
with $k_{1}\neq k_{1}^{\prime}$, we see that the Lagrangians
$\gamma_{2n}\times\Xi_{g}$ and $\gamma_{2n}\times\Xi^{\prime}_{g}$ must be
distinct up to compactly supported symplectomophisms. Putting everything
together, we get the following theorem.

######  Theorem 5.13.

Fix $g$. For any sufficiently large $r$ and $s$, we can construct an infinite
family of homologous monotone Lagrangian $S^{1}\times\Sigma_{g}$ in
$Y_{r,s}=\\{x^{3}+y^{3}+z^{r}+w^{s}=1\\}$, with fixed arbitrary even Maslov
class and monotonicity constant, distinct up to compactly supported
symplectomorphisms of $Y_{r,s}$. (This includes the exact case.)

######  Remark 5.14.

The conclusion of Theorem 5.13 also hold for well-behaved non-compactly
supported symplectomorphisms, namely all of those which can be extended to the
compactifications.

###  5.4. Holomorphic annuli counts in 2 dimensions

As annuli have Euler characteristic zero, by [Liu02, Theorem 1.2] the expected
dimension of a moduli space of annuli only depends on their Maslov class, and
not on the dimension of the ambient space. This means that we can hope for
well-defined counts Maslov zero annuli in complex dimension two. (In complex
dimension four or higher, one immediately runs into trouble because of the
possible existence of discs with negative Maslov indices.)

In particular, we’re able to get a two-dimensional version of Theorem 5.13 in
the case of tori.

######  Theorem 5.15.

For any sufficiently large $r$, we can construct an infinite family of
homologous monotone Lagrangian tori in $X_{r}=\\{x^{3}+y^{3}+z^{r}=1\\}$, with
fixed arbitrary even Maslov class and monotonicity constant, distinct up to
compactly supported symplectomorphisms of $X_{r}$. (This includes the exact
case.)

###### Proof.

We will just use the family of Lagrangian tori given by $T_{k,l,m}$ (see
Figure 11), although completely analogous arguments can be carried out for
other families, with changes to the annuli count formulae. Our proof will
follow from inspecting the arguments of Section 5.3 and noticing that they can
be replicated, indeed with some simplifications. We briefly note what changes
are to be made.

Let $T=T_{k,l,m}$. As before, let us first consider the case where $T$ is
monotone but not exact. Define moduli spaces
$\mathcal{M}_{J}(\bar{X}_{r};A,\alpha)$ and
$\bar{\mathcal{M}}_{J}(\bar{X}_{r};A,\alpha)$ following Definition 5.6. For
our favoured almost-complex structure $\tilde{J}$ from Section 5.2, one gets
an analogue to Proposition 5.7, with the one difference being that for the
sole choice of $(A,\alpha)$ such that
$\mathcal{M}_{\tilde{J}}(\bar{X}_{r};A,\alpha)$ is non-empty (in which case it
is as before one-dimensional), the signed count of points in
$\bar{\mathcal{M}}_{\tilde{J}}(\bar{X}_{r};A,\alpha)$ is $k_{1}$. To see that
this count is a well-defined invariant as $J$ varies, let us re-visit the
proof of Proposition 5.8. The boundary of the abstract moduli space of
holomorphic annuli gets avoided as before; and the arguments to exclude disc
bubbling carry over, in fact with small simplifications: in this dimension, if
a moduli space of index $2n$ discs is regular, then after quotienting out by
holomophic automorphisms it has expected dimension $2n-1$, so $n>0$.

Let us now turn to the case where $T$ is exact. We let $T^{\prime}$ be a small
_non-exact_ deformation of $T$. We construct this analogously to Lemma 5.10,
but using $\Pi_{r}$ instead of $P_{r,s}$: we use a close parallel copy of the
immersed $S^{1}$ in the base of $\Pi_{r}$ (which is now playing the role of
$\gamma_{0}$), and don’t touch the ‘meridional’ $S^{1}$ (which is in a fibre).
We’re then able to follow the proof of Proposition 5.11; as in the previous
case, the expected dimensions of spaces of holomorphic discs get adjusted in
our favour. ∎

Note that the assertions of Remark 3.9 about the ‘linking’ of the tori
$R_{k,l,m}$ and $S_{n,p,q}$ now readily follow by considering similar such
holomorphic annuli counts.

###  5.5. Extensions and limitations

Because of their reliance on partial compactifications, the proof of Theorems
5.13 and 5.15 will not survive under a general exact symplectic embedding of
Liouville domains: instead, if one wants the same conclusion for e.g. the
Milnor fibre of a ‘larger’ singularity, one would need it to have a
compactification in which holomorphic annuli still appear.

In the two-dimensional case, by Remark 3.3, the conclusion of Theorem 5.15
would also hold for $\\{x^{2}+y^{4}+z^{r}=1\\}$ for sufficiently large $r$.

More generally, we see that the constructions and arguments of Sections 5.1
through 5.4 (and in particular Section 5.2) can also be made whenever
$x^{3}+y^{3}$, i.e. the singularity $D_{4}$, is replaced by $x^{3}+xy^{u}$,
i.e. the singularity $D_{u+1}$ (while not Brieskorn–Pham, this is still
weighted homogeneous, so Remark 2.10 applies).

On the other hand, suppose we try to replace $D_{u}$ by an even larger
singularity in two variables. Suppose we’re given two disjoint vanishing
cycles on the associated Milnor fibre. One can check the following: if
removing those vanishing cycles from the Milnor fibre disconnects it, then
both of the resulting components (there can’t be more than two) have positive
genus. This means that the analogue of counting pseudo-holomorphic annuli
would now be to count higher genus pseudo-holomorphic curves (with Maslov
index zero, and two boundary components). While in dimension 3 these have
expected dimension zero, at least naively one can’t hope for a well-defined
count, as we can’t rule all of the possible configurations in the boundary of
the abstract moduli space of such holomorphic curves: there is no a priori
reason to be able to rule out an internal connecting node on the curve (as in
[Liu02, Figure 8]).

##  6\. Floer-theoretic properties

###  6.1. Distinguishing Lagrangians using Floer theory

We’ll show that the families of Lagrangians which we construct in $X_{r}$ and
in $Y_{r,s}$ (in Section 5.1) can be distinguished using their Floer homology
group with a fixed Lagrangian sphere. From one perspective, this is a weaker
invariant than the pseudo-holomorphic annuli counts of Section 5, as it shows
that the Lagrangians in the family are different up to Hamiltonian isotopy,
rather than arbitrary compactly supported symplectomorphisms; on the other
hand, this invariant survives under e.g. exact symplectic embeddings of
Liouville domains. We’ll also see that it allows us to distinguish some
Lagrangians for which the holomorphic annuli counts were all the same, or for
which we didn’t define the count (for instance, genus $g$ Lagrangians in
$X_{r}$).

For the standard almost-complex structure $J$, there are no non-constant
holomorphic discs in $X_{r}$ with boundary on $T_{k,l,m}$, as an immediate
consequence of the open mapping theorem. In particular, the count of Maslov
index two discs with boundary on $T_{k,l,m}$ vanishes (of course this is
automatic if $T_{k,l,m}$ is exact or has minimal Maslov number four or more,
so we would in fact only have to worry about the case where a ‘longitude’ of
$T_{k,l,m}$ has Maslov index two). This means that the Floer cohomology group
of $T_{k,l,m}$ with e.g. a Lagrangian sphere $L$ is well defined. We will take
our Floer cohomology groups to have coefficients in $\mathbb{C}$, and decorate
the Lagrangians with a choice of rank one local system, following the set-up
in [Sei08, Chapter 2]. In the case where $T_{k,l,m}$ is exact,
$HF(T_{k,l,m},L)$ can be equipped with a $\mathbb{Z}$–grading; in general, it
can be equipped with a $\mathbb{Z}/2|k-l|$–grading.

######  Proposition 6.1.

Then we can find Lagrangian spheres in $\\{x^{2}+y^{4}+z^{18}=1\\}$, say
$S_{1},S_{2}$ and $S_{3}$, which are vanishing cycles for the singularity
$x^{2}+y^{4}+z^{18}$ (and thus given by matching cycles in the base of a
Lefschetz fibration given by $\epsilon(x,y)+z$ for a small generic linear
$\epsilon$), such that for any choice of rank one local system on $T_{k,l,m}$,

$\text{rk }HF(T_{k,l,m},S_{1})=m+2k+1;\qquad\text{rk
}HF(T_{k,l,m},S_{2})=2k+1;\qquad\text{rk }HF(T_{k,l,m},S_{3})=2l.$

In particular, the Lagrangians $T_{k,l,m}$ and
$T_{k^{\prime},l^{\prime},m^{\prime}}$, constructed in
$\\{x^{2}+y^{4}+z^{18}=1\\}$ using the same basic configurations (i.e.
configurations of critical points of type $BB$ and $BC$) cannot be Hamiltonian
isotopic whenever $(k,l,m)\neq(k^{\prime},l^{\prime},m^{\prime})$.

###### Proof.

Let $S_{1}$ be the matching cycle $S$ of Figure 11. (Following Remark 3.3, we
drop the $d$ curve and work in $X=\\{x^{2}+y^{4}+z^{18}=1\\}$.) While $S$
isn’t one of the distinguished collection vanishing cycles for
$x^{2}+y^{4}+x^{r}$ which are given by matching cycles in Figure 4, it is
given by applying to one of them (a ‘type $a$’ matching cycle) negative Dehn
twists in a sequence of vanishing cycles which appear further down the list in
the distinguished collection (these also correspond to ‘type $a$’ matching
cycles). In particular, $S$ itself in a vanishing cycle for
$x^{2}+y^{4}+z^{r}$.

The sphere $S=S_{1}$ intersects $T_{k,l,m}$ in $m+2k+1$ points, labelled
$p_{1},\ldots,p_{m},q_{m+1},\ldots,q_{m+2k+1}$ on the figure. We claim that
the standard almost-complex structure is regular, and that for that choice
there are no holomorphic discs between intersection points.

First, we have that

$CF(S,T_{k,l,m})=\mathbb{C}\langle p_{1},\ldots,q_{m+2k+1}\rangle.$

Moreover, as a holomorphic disc in $X$ would project to one in $\mathbb{C}$
with the same boundary conditions, single variable complex analysis heavily
constrains the possibilities for holomorphic discs: by inspection, the only
possible differentials are from $p_{i}$ to $p_{j}$ with $j>i$, $p_{i}$ to
$q_{j}$, or $q_{i}$ to $q_{j}$ with $j>i$; see Figure 27 for a possible disc.

Figure 27. Image in $\mathbb{C}$ of topological discs between $p_{1}$ and
$p_{2}$. We’ve omitted some labels for legibility; these can be found in
Figure 11.

On the other hand, the Maslov index calculations of Lemma 3.6 imply that these
topological discs have either Maslov index zero, in the case of $p_{i}$ and
$p_{j}$, or negative Maslov index, in the case of $p_{i}$ (or $q_{i}$) and
$q_{j}$. Thus they must have zero or negative symplectic area, and cannot have
holomorphic representatives (even the area zero ones are not constant).
(Alternatively, one could note that by also by an application of the open
mapping theorem, the holomorphic discs contributing to the differential would
be the same for $T_{k,l,m}$ and $T_{k,k,m}$ – this just uses the fact that the
image of such a disc can’t cross the inflection point between the two ‘lobes’
of $T_{k,l,m}$ – and then make use of the $\mathbb{Z}$–grading on
$CF(S,T_{k,k,m})$.) This proves the claim for $S_{1}=S$.

Going back to Figure 11, we can write down matching cycles $S_{2}$ and
$S_{3}$, constructed similarly to $S_{1}$ but starting at a type $a$ critical
point in two different $BB$, such that $S_{2}$ intersects $T_{k,l,m}$ in
$2k+1$ points, and $S_{3}$ in $2l$ points; a completely analogous argument
shows that with the standard almost-complex structure there are no holomorphic
discs contributing to the differential on $CF(T_{k,l,m},S_{i})$, $i=1,2$,
which completes the proof. ∎

######  Remark 6.2.

If we just cared about determining the triple $(k,l,m)$, the Floer cohomology
with $S_{1}$, together with the Maslov class and the homology class, would
suffice – the characterisation using only Floer groups will be useful later.

There are clearly similar statements for $R_{n,p,q}$ and $S_{u,v,w}$ the
triples $(n,p,q)$ and $(u,v,w)$ are determined by the rank of the Floer
cohomology groups of $R_{n,p,q}$ and $S_{u,v,w}$ with some vanishing cycles
for $x^{3}+y^{3}+z^{r}$. Similarly for the parameters definining the Klein
bottles of Section 3.6 (and plenty of other multi-parameter families of tori
or Klein bottles one might construct analogously).

We also get the following corollary.

######  Corollary 6.3.

Fix any $r$ large enough such that the family
$\Lambda_{g}(\\{k_{1},l_{1},m_{1}\\},\ldots,\\{k_{g},l_{g},m_{g}\\})$ can be
realised in $X_{2,4,r}$. Then all of the parameters
$(k_{1},l_{1},m_{1}),\ldots,(k_{g},l_{g},m_{g})$ are determined by the ranks
of Floer cohomology groups of $\Lambda_{g}$ with vanishing cycles for
$x^{2}+y^{4}+z^{r}$.

Similarly for other genus $g$ Lagrangians built using the same techniques,
such as $\zeta_{g}$ of Section 4.2.1 and $\Xi_{g}$ of Section 5.1 (with
vanishing cycles for $x^{3}+y^{3}+z^{r}$), as well as further variations by
taking a connected sum of a mixture of tori and Klein bottles.

###### Proof.

The Floer cohomology calculations of Proposition 6.1 are unaffected by the
Polterovich surgeries: open mapping theorem type considerations still apply to
show that there can be no Floer differentials for the standard almost-complex
structure. ∎

Note that the calculations will remain valid under an exact symplectic
embedding of Liouville domains, e.g. by the integrated maximum principle
[Sei08, Lemma 7.5]. This completes the proof of Theorem 1.1.

Upgrading to complex dimension three, we get the following.

######  Proposition 6.4.

Fix $n\in\mathbb{Z}_{\geq 0}$, $g\geq 1$, and $r,s$ large enough such that the
family of Lagrangians $\gamma_{n}\times\Xi_{g}$ of Section 5.1 can be
constructed in $Y_{r-1,s-1}$.

Then there are Lagrangian spheres in $Y_{r,s}$, which moreover are vanishing
cycles for $x^{3}+y^{3}+z^{r}+w^{s}$, whose Floer cohomology groups with
$\gamma_{n}\times\Xi_{g}$ recover the coefficients $\\{k_{0},l_{0},m_{0}\\},$
…, $\\{k_{g},l_{g},m_{g}\\}$ determining $\Xi_{g}$.

###### Proof.

Let $S$ be one of the test Lagrangian 2-spheres used for determining the
coefficients of $\Xi_{g}$ in $X_{r}$ (this uses the fact that $\Xi_{g}$ itself
can be constructed in $X_{r-1}$, to have a ‘extra’ critical value for the
second end of the matching path for the sphere). As it’s a vanishing cycle for
$x^{3}+y^{3}+z^{r}$, we can find a matching path in the base of $P_{r,s}$
which intersects $\gamma_{n}$ in precisely one point, and such that in the
fibre above that point, the corresponding matching cycle, say $\mathbb{S}$,
restricts to $S$, and $\gamma_{n}\times\Xi_{g}$ restricts to $\Xi_{g}$. It is
given back in Figure 22 for the case of a general $n$ ($n=0,2$ are similar),
and in turn uses the assumption on $s$ to get the ‘spare’ critical value
needed to close off that matching cycle.

As the images of $\mathbb{S}$ and $S^{1}\times\Xi_{g}$ in the base of
$P_{r,s}$ intersect transversally in a single point, using the standard almost
complex structure, it’s immediate that the Floer cohomology of $\mathbb{S}$
and $S^{1}\times\Xi_{g}$ is precisely given by the Floer cohomology of $S$ and
$\Xi_{g}$, and the result then follows from Corollary 6.3. ∎

######  Remark 6.5.

The conclusion of Proposition 6.4 applies much more broadly:

  * •

to families of Lagrangians of the form $S^{1}\times\Sigma$, where the surface
$\Sigma$ is a connected sum of an arbitrary number of tori and Klein bottles
which could be constructed using any of the variations of Remark 5.3. (In
particular, the bounds on $r$ and $s$ needed in order to get infinite families
of $S^{1}\times\Sigma_{g}$ can be significantly improved from the naive ones
one would get using the numbers in Section 5.1.)

  * •

to families of Lagrangians of the form $L=\\#_{i=1}^{l}S^{1}\times\Sigma_{i}$,
obtained by taking Polterovich connected sums of the $S^{1}\times\Sigma_{i}$
with matching spheres. (As before, the surfaces $\Sigma_{i}$ are themselves
the connected sum of an arbitrary number of tori and Klein bottles.) This
connect sum procedure is the three-dimensional analogue of the constructions
$\Lambda_{g}$, $\Xi_{g}$ etc, and the same argument applies regarding ranks of
Floer cohomology. More precisely, just as in the two-dimensional case,
consider a matching cycle $\mathbb{S}$ use to detect the parameters used to
define $\Sigma_{i}$, and whose image in the base of $P_{r,s}$ only intersects
the image of $S^{1}\times\Sigma_{i}$ for one $i$. The open mapping theorem
applies as before to show that discs contributing to the differential on
$CF(L,\mathbb{S})$ are exactly the same as the ones contributing to
$CF(S^{1}\times\Sigma_{i},\mathbb{S})$.

The discussion of Floer cohomology groups in the 3-dimensional case has been
conducted in $Y_{r,s}$, i.e. $\\{x^{3}+y^{3}+z^{r}+w^{s}=1\\}$. However, if we
only care about the conclusions of Theorem 1.2, we claim that it’s enough to
take $\\{x^{2}+y^{4}+z^{r}+w^{s}=1\\}$, for sufficiently large $r$ and $s$, as
ambient space: we only needed $Y_{r,s}$ in Section 5.1 in order to construct
$\Xi_{g}$ (Figure 19), with the carefully arranged overlay of
$S_{k_{0},l_{0},m_{0}}$ and $R_{k_{1},l_{1},m_{1}}$ above some self-
intersection points of $\gamma_{2n}$, in order to get a non-trivial count of
holomorphic annuli. Working simply with $T_{k,l,m}$, we can write down similar
but simpler constructions in $\\{x^{2}+y^{4}+z^{r}+w^{s}=1\\}$, with the chief
difference being that we get, above any self-intersection point of the base
$S^{1}$, $T_{k,l,m}\sqcup\rho(T_{k,l,m})\subset X_{2,4,r}$ for some $\rho$
such that $\Pi(T_{k,l,m})\cap\Pi(\rho(T_{k,l,m}))=\emptyset$, where
$\Pi:X_{2,4,r}\to\mathbb{C}$ is the same Lefschetz fibration as before.

This completes the proof of Theorem 1.2

####  6.1.1. Further diffeomorphism types

We briefly note how the ideas Sections 3.7 and 4.2.1 might be further
developed when the ambient manifold is (for instance) $X_{2,4,r,s}$ for
sufficiently large $r$ and $s$. For concretness, let
$\zeta_{g}(\\{k_{1},l_{1},m_{1}\\},\\{k_{2},l_{2},m_{2}\\})$ be as in Section
3.7 ($g$ odd as before), and $\rho$ the product of positive Dehn twists in
vanishing cycles for $X_{2,4,r}$ such that $\rho$ fixes $\zeta_{g}$ setwise
but acts on it by an order $(g-1)/2$ rotation pointwise. Now note that at the
cost of enlarging $r$, we can use Propositions 2.5 and 2.7 to find another
product of positive Dehn twists in vanishing cycles for $X_{2,4,r}$, say
$\sigma$, such that $\sigma(\zeta_{g})\cap\zeta_{g}=\emptyset$; and also
$\tilde{\sigma}$, again a product of _positive_ Dehn twists in vanishing
cycles, such that $\tilde{\sigma}\sigma(\zeta_{g})=\zeta_{g}$, where the
latter identity holds pointwise (after deforming by a suitable Hamiltonian
isotopy).

Taking $s$ to be very large, we can now use $\rho,\sigma$ and $\tilde{\sigma}$
to mimick the constructions of Section 5.1 to get monotone Lagrangians in
$X_{2,4,r,s}$ which are $\Sigma_{g}$ bundles over $S^{1}$ (as always immersed
in the base of $P_{r,s}$), with monodromy given by $\rho$. (Recall that
$X_{\mathbf{a}}$ is defined in general at the start of Section 2). The base
$S^{1}$ can be arranged to have arbitrary even Maslov index; and Floer
cohomology with matching cycles in $Y_{2,4,r,s}$ allows us to distinguish such
Lagrangians up to isotopy. We can also use matching cycles to take connected
sums of several copies of such Lagrangians, and / or with Lagrangians of the
form $\Sigma_{g}\times S^{1}$; and one can come up with variations replacing
$\Sigma_{g}$ with certain connected sums of tori and Klein bottles. (Also,
dropping monotonicity would readily allow for e.g. the case where $g$ is
even.)

For concretness, we record the one exact case, using the Maslov index
calculation of Section 3.7.

######  Proposition 6.6.

Let $E$ be the $\Sigma_{3}$ bundle over $S^{1}$ with monodromy of order two
described above, and let $L$ be an arbitrary connected sum of copies of $E$
and $S^{1}\times\Sigma_{g}$, where $g$ can vary. Then for all sufficiently
large $r$ and $s$, there exists an infinite family of exact monotone
Lagrangians in $Y_{2,4,r,s}$ which are diffeomorphic to $L$, homologous, and
distinct up to Hamiltonian isotopy. This is preserved under exact symplectic
embeddings of $Y_{2,4,r,s}$.

######  Remark 6.7.

We can’t hope to get non-zero holomorphic annuli counts for non-trivial
$\Sigma$ bundles over $S^{1}$ without significantly refining our
constructions: the set-ups of Figures 18 and 19 can’t readily be amalgamated
while preserving all the features one would need.

###  6.2. Floer cohomology for families of 2-dimensional Lagrangians and
comparison with cluster mutations

Aside from acting by symplectomorphisms, in the two-dimensional case a well
understood technique for getting new Lagrangians from old ones is to use disc
surgeries, also known as geometric mutations. More precisely, given a
Lagrangian surface $L$ in a symplectic four-manifold together with a
Lagrangian disc with boundary on it, one can construct a new Lagrangian
surface $L^{\prime}$ via so-called ‘disc surgery’ on the original one [Yau].
If $L$ is monotone, then under suitable conditions $L^{\prime}$ is too;
similarly with exactness. This construction has been shown to have rich
connections with the theory of cluster mutations, explored e.g. in [STW,
PT20], where the disc surgeries are refered to as (geometric) mutations.

The torus case is particularly well understood; we follow the discussion in
[PT20]. Given a Lagrangian seed, i.e. a monotone Lagrangian torus $T$ together
with a collection of Lagrangian discs with boundary on $T$ [PT20, Definition
4.7], the authors explain how to iteratively perform mutations on the torus
[PT20, Definition 4.9]. Moreover, the local model for a single mutation is
given by passing from the Clifford to the Chekanov torus in
$\mathbb{C}^{2}\backslash\\{xy=1\\}$ (see [PT20, Sections 4.5–4.7]). Further,
the Floer cohomology for this local model is known, see e.g. the calculation
in [Sei, Proposition 11.8], which closely follows [Aur07]. In this local
model, depending on the choice of rank one local systems, the two tori are
either isomorphic or the Floer cohomology between them vanishes. (The matching
of local systems to get non-zero Floer cohomology is what gives the wall-
crossing formula.) In the exact case, the same remains true of $T$ and its
geometric mutation $T^{\prime}$.

Yau also introduces disc surgeries on higher genus Lagrangian surfaces [Yau];
they are also included in the discussion in [STW, Section 2] (though the
reader may wish to recall the caveats of e.g. [STW, Sections 1.2.4 and 2.2]
regarding iterations of mutations). Note that there isn’t a single model for
disc surgery in this case, as it depends on the isotopy class of the boundary
of the disc on the Lagrangian. However, given the result on local systems and
microlocal sheaves of [STW, Equation 1], one might nonetheless expect that
depending on the choice of rank one local system, an exact Lagrangian surface
$L$ and its geometric mutation $L^{\prime}$ are either isomorphic or have
Floer cohomology zero.

In contrast, our constructions yield the following.

######  Theorem 6.8.

Let $L$ be a copy of $T_{k,l,m}$ with monotonicity constant $\kappa$, and, for
$\lambda>0$, let $L^{\prime}$ be a copy of $T_{k+\lambda,l+\lambda,m}$ with
monotonicity constant $\kappa^{\prime}$, constructed using the same basic
configurations in $X_{u,v,r}$, for some $u,v$ such that $1/u+1/v\leq 2/3$ and
$r\geq 18$.

Let $(\alpha,\beta)\in(\mathbb{C}^{\ast})^{2}$ parametrise choices of rank one
local systems on $L$, with respect to the same basis for $H_{1}(L,\mathbb{Z})$
as in Lemma 3.6; similarly, we’ll use
$(\alpha^{\prime},\beta^{\prime})\in(\mathbb{C}^{\ast})^{2})$ for
$L^{\prime}$. Then we have that

$\text{rk
}HF((L,(\alpha,\beta)),(L^{\prime},(\alpha^{\prime},\beta^{\prime}))=\begin{cases}2\lambda&\text{
if }\beta=\beta^{\prime}\\\ 2\lambda&\text{ if }\beta=-\beta^{\prime}\\\
0&\text{ otherwise }\end{cases}$

This property is preserved under exact symplectic embeddings of Liouville
domains. Moreover, in the case where $u=v=3$, we can combine this with Theorem
5.15 to get families of Lagrangian tori not related by compactly supported
symplectomorphisms, in particular Dehn twists.

######  Remark 6.9.

If instead $r\geq 18g$ (or greater), one can similarly go about calculating
the Floer cohomology groups of e.g. the genus $g$ Lagrangians
$\Lambda_{g}(T_{k_{1},l_{1},m_{1}},\ldots,T_{k_{g},l_{g},m_{g}})$ for varying
values of $k_{i},l_{i}$ and $m_{i}$. While we don’t do this in detail here, it
will be clear from the proof of Theorem 6.8 that one can obtain infinite
families of homologous monotone genus $g$ Lagrangians (all with the same
Maslov class and monotonicity constant) such that the Floer cohomology between
them, for suitable rank one local systems, takes arbitrary large rank finite
rank.

#### Discussion

There is a general expectation that the smoothing of the singularity
$x^{u}+y^{v}+z^{r}$ (i.e. its Milnor fibre) should be mirror to a resolution
of this singularity: Brieskorn–Pham singularities fall into the framework of
Berglund–Hubsch [BH93], and are their own Berglund–Hubsch transposes; some
versions of mirror symmetry have been proved in [FU11, FK19], or [CU13] in the
case of $A_{n}$. In particular, one might expect the (wrapped) Fukaya category
of $X_{u,v,r}$ to be mirror to a category of coherent sheaves associated with
the resolution of the singularity $x^{u}+y^{v}+z^{r}$. The mirror-symmetric
counterpart to the statement about mutations is that given their Floer-
theoretic properties, the family of tori we construct can’t correspond to
cluster charts on some (fixed) mirror variety, or more generally to structure
sheaves of points of $(\mathbb{C}^{\ast})^{2}$ affine charts inside such a
mirror variety. This contrast with what we understand of a number of families
of tori in two-dimensional examples, e.g. in the case of $A_{n}$ [Aur07, CU13,
LM14], for $\mathbb{C}P^{2}$ [Aur07, Via16] or del Pezzo surfaces [AKO06,
Via17], and arguably most remarkably for log CY surfaces [GHK15, GHKK18,
Pas19]; cluster structures associated to Grasmannians have also recenty been
used to construct families of exotic Lagrangian tori in them [Cas].

###### Proof.

We’ll again use fibredness over $S^{1}$ to apply tools from standard complex
analysis. First, we note that irrespective of $\kappa$ and $\kappa^{\prime}$,
we can always pick Hamiltonian deformations of $L$ and $L^{\prime}$ which are
still fibred, and such that the intersection between them is naively minimal,
as given in Figure 28 in the case of $T_{2,2,0}$ and $T_{1,1,0}$: namely, we
first arrange for there to be $2\lambda$’s worth of fibrewise $S^{1}$
intersections, and then deform each of these $S^{1}$s in the fibre direction
to get two intersection points (the local model in the fibre is the zero-
section in $T^{\ast}S^{1}$ and a small Hamiltonian perturbation of it). Given
a fibred $T_{k+\lambda,l+\lambda,m}$, one can draw a fibred $T_{k,l,m}$ by
using its projection to $\mathbb{C}$ as a guide, running closely parallel to
it and then ‘skipping’ some twists in both the left and right-hand sides
lobes. (This was done in Figure 28.) To see that one needn’t worry about
possible variations in $\kappa$ and $\kappa^{\prime}$, note that one can first
take a Hamiltonian deformation of $L^{\prime}$ (in particular, preserving
$\kappa^{\prime}$) such that the relative sizes of the area contribuations
from the parts of the left and right lobes that get ‘skipped’ are arbitrary.

Figure 28. Calculating $HF(T_{2,2,0},T_{1,1,0})$. There are two intersection
points above each of the marked dots, in each case coming from perturbing a
fibrewise $S^{1}$. The colour of the dot encodes the orientation of the fibre
type $b$ cycle for $T_{2,2,0}$, using the conventions of Figure 11. The shaded
region is a potential projection of a holomorphic disc.

With respect to these choices,
$CF((L,(\alpha,\beta)),(L^{\prime},(\alpha^{\prime},\beta^{\prime}))$ has
$4\lambda$ generators. Let’s calculate the differential. Work with the
standard $J$. Suppose there’s a holomorphic disc $u:D\to X_{u,v,r}$ with
boundary on $L$ and $L^{\prime}$. We use the same trick as before: $\Pi\circ
u:D\to\mathbb{C}$ must be a holomorphic map with boundary on
$\Pi(L),\Pi(L^{\prime})$. We claim that such holomorphic maps to $\mathbb{C}$
must be constant.

Let us first consider potential holomorphic discs between two intersection
points both on the left lobe. Start with the example of Figure 28. Let
$m_{1},M_{1}$ be the intersection points above the left green dot, and
$m_{2},M_{2}$ be the ones above the left blue dot, ordered so that
$\text{deg}(M_{i})=\text{deg}(m_{i})+1$. Using holomorphicity of $\Pi\circ u$,
we see that the only possible discs with non-constant projections have
projections similar to the one in Figure 27, which, given our ordering of $L$
and $L^{\prime}$, would in our case give a differential from $m_{1}$ or
$M_{1}$ to $m_{2}$ or $M_{2}$. (To rule out other discs, it’s perhaps easiest
to note that the boundary of the projection of any other potential holomorphic
disc would differ from the one already described by an integer multiple of at
least one of the two immersed $S^{1}$s in the base.)

On the other hand, the Maslov index calculations in the proof of 3.6 show that
our potential disc would have index $-1$ if it were from $M_{1}$ to $M_{2}$,
or from $m_{1}$ to $m_{2}$; and index zero from $M_{1}$ to $m_{2}$. If we
generalise to work instead with $\lambda>1$, we now see that the indices of
potential holomorphic discs (still between intersection points both on the
left lobe) would become more and more negative.

The situation for holomorphic discs between intersection points both on the
right lobe is completely analogous. Finally, let us consider holomorphic discs
going between the two lobes. Holomorphicity of the projection map now implies
that the point in right-hand lobe must project to the outermost green point;
the shaded region in Figure 28 gives the potential holomorphic disc
projection; now notice that such configurations have already been studied when
thinking about the case with both intersection points in the left-hand lobe.

This mean for index reasons, we only need to worry about holomorphic discs
whose image is contained entirely in a fibre of $\Pi$. There are $4\lambda$
such discs, clearly regular: two in each of the fibres where $L$ and
$L^{\prime}$ intersect. Tracking local systems, there are two possible
configurations: the meridian in $T^{\ast}S^{1}$ and a push-off with the same
orientation (green dot), in which case the contributions from the two discs
cancel themselves out precisely when $\beta=\beta^{\prime}$; and the meridian
in $T^{\ast}S^{1}$ and a push-off with the opposite orientation (blue dot), in
which case the contributions from the two discs cancel themselves out
precisely when $\beta=-\beta^{\prime}$. ∎

######  Remark 6.10.

One could simplify the argument above, at least in the exact case (where the
Lagrangians carry an absolute grading) at the cost of increasing $r$ and
modifying the construction of $T_{k,l,m}$, for instance by adding two $BB$
type elementary configurations in each lobe to ensure that the indices of the
intersection points were all further apart.

######  Remark 6.11.

By constructing Lagrangians whose projections to $\mathbb{C}$ are disjoint, we
can of course also construct (finite) families of Lagrangians all of which are
Floer-theoretically disjoint.

###  6.3. Floer cohomology within families of 3-dimensional Lagrangians

With suitable care, one would expect to be able to leverage the bifibration on
$Y_{r,s}$, together with the fibred nature of the Lagrangians we construct, to
calculate the Floer cohomology groups between at members of various families
of Lagrangians from Section 5.1. We record the following observation for the
exact case.

######  Proposition 6.12.

Fix $g$. For suitably large $r,s$, there exists an infinite family of
homologous, monotone exact Lagrangians of the form $S^{1}\times\Sigma_{g}$
inside $Y_{r,s}$ such that for suitable choices of rank one local systems, the
Floer cohomology between members of that family can take arbitrarily large
rank.

As before, we only need $x^{2}+y^{4}+z^{r}+w^{s}$, and the conclusion persists
if we take connected sums of such Lagrangians, or under inclusion of Liouville
domains.

###### Proof.

This is similar to before. Let’s look at the case of
$\gamma_{0}\times\Xi_{g}$, for two different choices of
$\Xi_{g}=\Xi_{g}(\\{k_{0},l_{0},m_{0}\\},\ldots,\\{k_{g},l_{g},m_{g}\\})$.
Considering a Hamiltonian perturbation of one of them whose projection to the
base of $P_{r,s}$ is given by Figure 29, we see that there are two collections
of intersection points between the two Lagrangians (above each of the dots in
the base), in natural one-to-one correspondence. Moreover, for each pair of
such points, there are two holomorphic discs between them, whose contributions
cancel for suitable choices of rank one local systems (this is merely a
slightly unusual presentation of a standard configuration in $T^{\ast}S^{1}$;
a neighbourhood of the zero section therein is immersed in the same of the
fibration). Using the same circle of index and open mapping type
considerations as before, the Floer cohomology calculation then boils down to
finding the holomorphic discs in the fibres above each of the dots, which we
already know from the proof of Theorem 6.8.

Figure 29. Calculating the Floer cohomology between Lagrangians of the form
$\gamma_{0}\times\Sigma_{g}$. (Some labels are suppressed for legibility –
we’re using the same configuration as Figure 20. The Hamiltonian perturbation
of a Lagrangian $\gamma_{0}\times\Sigma_{g}$ is fibred over the curve
$\gamma_{0}^{\prime}$.

∎

##  7\. Generalisations to higher dimensions

Starting with our three-dimensional examples, we can iteratively use the
monodromy tricks provided by Propositions 2.5 and 2.7, as well as Polterovich
surgery, to construct interesting families of monotone Lagrangians in higher-
dimensional Brieskorn-Pham hypersurfaces.

More precisely, starting with the Lagrangians of Theorem 1.2 and iterating one
dimension at a time, we can do the following:

  1. (1)

Given a monotone Lagrangian $L$ in
$X_{1}=\\{z_{0}^{a_{0}}+\ldots+z_{m-1}^{a_{m-1}}=1\\}$, get a monotone
Lagrangian $S^{1}\times L$ in
$X_{2}=\\{z_{0}^{a_{0}}+\ldots+z_{m-1}^{a_{m-1}}+z_{m}=1\\}\cong\mathbb{C}^{m}$
by taking a product with an embedded $S^{1}\subset D\subset\mathbb{C}$ such
that the Lefschetz fibration
$z_{m}+\epsilon(z_{0},\ldots,z_{m-1}):X_{2}\to\mathbb{C}$ is essentially
trivial above $D$. (The function $\epsilon(z_{0},\ldots,z_{m-1})$ is a generic
linear perturbation.) The $S^{1}$ factor has Maslov index two.

  2. (2)

Given a monotone Lagrangian $L$ in
$X_{1}=\\{z_{0}^{a_{0}}+\ldots+z_{m-1}^{a_{m-1}}=1\\}$, for sufficently large
$a_{m}$, get a monotone Lagrangian $S^{1}\times L$ in
$X_{2}^{\prime}=\\{z_{0}^{a_{0}}+\ldots+z_{m-1}^{a_{m-1}}+z_{m}^{a_{m}}=1\\}$
fibred over an immersed $S^{1}$ in the base of the Lefschetz fibration
$z_{m}+\epsilon(z_{0},\ldots,z_{m-1}):X_{2}\to\mathbb{C}$, analogously to the
constructions of Section 5.1. The total monodromy about the immersed $S^{1}$
gives a symplectomorphism of the fibre $X_{1}$ which is, say, the $l$th power
of the total monodromy $\varrho$ of $X_{1}$ viewed as the Milnor fibre of a
singularity: using an auxiliary Lefschetz fibration of the form
$z_{m-1}+\epsilon^{\prime}(z_{0},\ldots,z_{m-2}):X_{1}\to\mathbb{C}$ and the
notation of Section 2.3, one would have $\varrho=\Phi^{k}$, meaning that in a
large compact set, $\varrho$ is induced by a $2\pi$ rotation of the base of
this fibration. (In particular, $\varrho^{l}$ acts as the identity on $L$.)

The Maslov index for this $S^{1}$ factor is calculated as in Proposition 5.1:
twice the total winding number of the immersed $S^{1}$ in the base, adjusted
down by $2l$. In particular, the $S^{1}$ factor can take arbitrary even Maslov
index.

  3. (3)

Using Polterovich surgery, for sufficiently large $a_{m}$, we can get any
connected sums of the Lagrangians in (2), with the induced Maslov indices.

The examples in (1) are clearly null-homologous. In (2) or (3) one can check
that they are primitive in homology (with $\mathbb{Z}$ coefficients in the
orientable case and $\mathbb{Z}/2$ otherwise) – for instance, for any such
Lagrangian we can find a matching cycle which intersects it in exactly one
point. Let $(\dagger)$ the collection of possible pairs of diffeomorphism
types and Maslov classes described by the iterative process above. (Note that
for a given diffeomorphism type $L$, we can get as Maslov class any element of
$H^{1}(L,\mathbb{Z})$ which is not excluded by orientability-type
considerations.) It is also clear that in situations (2) and (3) – whenever
$a_{m}$ is sufficiently large – we can typically tell two examples apart by
calculating their Lagrangians Floer cohomology with a fixed matching cycle. In
particular,

######  Proposition 7.1.

Fix an $m$-dimensional manifold $L$ and $\mu\in H^{1}(L;\mathbb{Z})$ such that
$(L,\mu)\in(\dagger)$, and any constant $\kappa>0$. Then for sufficiently
large $a_{i}$, there exist infinitely homologous monotone Lagrangian
$L\subset\\{z_{0}^{2}+z_{1}^{4}+z_{3}^{a_{3}}+\ldots+z_{m}^{a_{m}}=1\\}=X$,
distinct up to Hamiltonian isotopy, with Maslov index $\mu$ and monotonicity
constant $\kappa$. The bounds on the $a_{i}$ only depend on the diffeomorphism
type of $L$. The class $[L]\in H_{m}(X)$ is primitive, where coefficients
should be taken in $\mathbb{Z}$ or $\mathbb{Z}/2$ depending on orientability
of $L$.

This property is preserved under exact symplectic embeddings.

In particular, we get families of examples of monotone Lagrangians, including
Lagrangian tori, which are primitive in homology but have arbitrarily high
minimal Maslov number.

###  7.1. Further diffeomorphism types

We briefly remark that with a little care, we can mimick the constructions of
Section 3.7 and 6.1.1 in higher dimensions. Given a pair
$(L,\mu)\in(\dagger)$, fix a monotone Lagrangian
$L\subset\\{z_{0}^{a_{0}}+\ldots+z_{m-1}^{a_{m-1}}=1\\}=X$. Use this to
construct a monotone Lagrangian $\tilde{L}$ in
$\\{z_{0}^{a_{0}}+\ldots+z_{m-1}^{2ka_{m-1}}=1\\}=\tilde{X}$ as follows: first
take $2k$ copies of $L$, arranged cyclically using the automorphism
$z_{m-1}\mapsto\zeta z_{m-1}$ of $\tilde{X}$, where $\zeta$ is a $(2k)$th root
of unity; now perform Polterovich surgery on the copies of $L$ together with
$2k$ (_not_ $2k-1)$ matching cycles, say $k$ of them corresponding to a fixed
vanishing cycle, and $k$ of them corresponding to a second vanishing cycle,
which is disjoint from the first one. By modifying the area enclosed by the
chain of matching cycles we can arrange for the resulting Lagrangian to be
monotone, and for there to exist products of positive Dehn twists on
$\tilde{X}$, say $\rho,\sigma$ and $\tilde{\sigma}$, such that $\rho$ fixes
$\tilde{X}$ setwise and acts as an order $k$ rotation pointwise;
$\sigma(\tilde{L})\cap\tilde{L}=\emptyset$; and
$\tilde{\sigma}\sigma(\tilde{L})=\tilde{L}$, where this identity holds
pointwise up to Hamiltonian isotopy. For sufficiently large $a_{m}$, this then
allows us to construct monotone Lagrangians in
$\\{z_{0}^{a_{0}}+\ldots+z_{m-1}^{a_{2km-1}}+z_{m}^{a_{m}}=1\\}$ which are
non-trivial $\tilde{L}$ bundles over $S^{1}$, with monodromy of order $k$.
Further, as in Proposition 6.6, for fixed ‘soft’ invariants these can often be
distinguished up to Hamiltonian isotopy by the Floer cohomology with a fixed
matching cycle.

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