ON THE DIRECTED FUKAYA CATEGORY OF CERTAIN CUSP
SINGULARITIES
UMUT VAROLGUNES
Abstract. We compute the directed Fukaya category of T (2, q, r) singularities
using the methods of A’Campo. We then show that this implies the homological mirror symmetry conjecture in this case, by constructing an explicit
sequence of mutations.
1. Introduction
Let T (p, q, r) be the singularity at the origin of the complex function xp + y q +
z r + λxyz, for 2 ≤ p < q ≤ r, and λ sufficiently large1. It is conjectured in [5], [6]
that there exists a triangulated equivance,
(1)
Db Fuk→ T (p, q, r)−→D
˜ b CohP1p,q,r .
Here, P1p,q,r represents the weighted projective line with three orbifold points of
isotropies p, q, and r.
In an unpublished paper [16], Takahashi sketched an argument which would
prove (1) in the case p = 2, but he did not provide any details. We provide in this
paper a detailed account of this, going along the same lines with Takahashi. The
following is our main theorem.
Theorem 1. Let A be the path algebra (over C) of the ordered quiver (the unnumbered vertices can be numbered in any order respecting the arrows) in Figure 1, with
relations cX = bY = aX + aY = 0.
Then, Db Fuk→ T (2, q, r) is isomorphic to the bounded derived category Db (mod-A)
of right modules over A in the triangulated sense.
It is known that if one replaces Db Fuk→ T (2, q, r) with Db CohP12,q,r in Theorem
1, the statement holds. See [18], for a comprehensive survey article, which also
contains this result. Hence, we get (1) as a trivial corollary.
1we do not lose any generality by taking λ to be large.
3
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Figure 1
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This work was partially supported by MIT through a UROP grant. I would like
to thank Paul Seidel for useful conversations.
2. background
This section consists of two parts. First, we give a short introdution to the
directed Fukaya category of singularities as in [13], and [14]. This is nowhere near
a complete discussion, and the technical details are almost always omitted. One
should consult the papers cited above for more details. In the second part, we
summarize the computational techniques that will be used in the proof. This is
really just a list of statements, but we give references to the relevant papers.
2.1. Overview of the directed Fukaya category of a singularity. Let f :
Cn → C have an isolated singularity at the origin. By abuse of notation, we
denote this singularity also by f . The directed Fukaya category, Fuk→ (f ), of f is a
categorification of the notion of the intersection form with respect to a distinguished
basis of vanishing cycles. We refer the reader to [4], for the definition and general
properties of these objects from the classical singularity theory.
Assume that we are given a Morsification f of f and a choice of distinguished
basis of vanishing cycles {L1 , . . . , Lµ } on a generic (Milnor) fibre X. In addition to
the classical story, f : Cn → C fits into a symplectic framework (after restricting
the range and domain appropriately), called an exact Morse fibration by Seidel. In
this setting X is a symplectic manifold with vanishing cycles as Lagrangian submanifolds, and the geometric monodromy operators as symplectic automorphisms
of X.
We would like to construct a category of which objects are {L1 , . . . , Lµ } with the
morphism spaces corresponding to cochain complexes of Lagrangian Floer theory,
and the compositions as the pair-of-pants product. This is what Fuk→ (f ) is except
some technical modifications. We list some of these below.
• We have to work with graded Lagrangians instead. In Seidel’s papers this is
done by choosing a relative Maslov map. In the case of singularities, there
in fact exists a canonical relative Maslov map, which will always be used
implicitly in what follows. We put a tilde to indicate that a Lagrangian is
graded.
• The category is directed, which means that we choose morphisms such that,
(1)

 0
C · id
Hom(L̃i , L˜j ) =

CF ∗ (L̃i , L˜j )
if i > j
if i = j
if i < j.
• The composition defined this way is not associative. Therefore Fuk→ (f ) is
not a category in the usual sense. There are higher order multiplications
µi and these satisfy what’s called A∞ relations. [7].
The category Fuk→ (f ) is not an invariant of the singularity, notably it depends
on the choice of basis of vanishing cycles (and on various other auxiliary choices
in setting up the Floer theor). In order to produce an invariant, we follow the
general procedure to extend an A∞ -category to a (usual) triangulated category, as
suggested by Kontsevich [11]. This is called the derived A∞ category, and denoted
by Db Fuk→ (f ). This indeed is an invariant of the singularity, i.e. different choices
produce categories that are equivalent as triangulated categories.
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We will not go into this construction, but let us make a final remark on its
structure. There exists objects (X1 , . . . , Xµ ) in Db Fuk→ (f ) corresponding naturally
to (L˜1 , . . . , L˜µ ), which form a full exceptional sequence. The morphisms between
them are given by the corresponding Lagrangian Floer cohomology groups.
2.2. Computational techniques. The reason that the case p = 2 is easier is the
stabilization property of singularities, which allows one to reduce the problem to a
more managable one. Here we give a categorified version of the classical statement.
Proposition 1. Let f : Cn → C have an isolated singularity at the origin. Then
the function g : Cn × Cy → C, given by g(x, y) = f (x) + y 2 also has an isolated
singularity at the origin. Moreover, Db Fuk→ (f ) is isomorphic to Db Fuk→ (g).
This was conjectured by Seidel in [14]. A proof can be found in the more general
statements of [17] or [15].
Computing the intersection form of an isolated singularity is a hard problem in
general. But this problem is essentially solved in the case where the function is
in two variables, by the A’Campo/Gusein-Zade method (see [1],[2],[8],[9] ). This
method in fact tells us more than just the intersection form. One can draw a
complete picture of the Milnor fiber with a distinguished basis of vanishing cycles
[3]. Consequently, the derived Fukaya category admits a nice description.
Proposition 2. Let f : C2 → C be a real polynomial with an isolated singularity at
the origin, and let f˜ be a real Morsification of it. We denote the intersection of the
Milnor ball for this deformation with R2 by N . Then, Db Fuk→ (f ) is isomorphic to
Db Morse(N, f˜).
Seidel is the first one to observe and prove this statement [14], using the results
in [3]. For a more general framework for which a similar description exists see [10].
The last ingredient in the proof is the theory of mutations in triangulated categories as developed by Bondal, Gorodentsev and others. The reader is referred
to [12], for definitions and proofs of the statements. All quivers in this paper are
ordered, so we omit the word ordered from now on.
Proposition 3. Let (E1 , . . . , En ) be a full exceptional sequence in a triangulated
category. Then, any (left or right) mutation of (E1 , . . . , En ) also is a full exceptional
sequence.
Proposition 4. Let (E0 , . . . , En ) be a full strongly exceptional sequence
L in a triangulated category C. Let A be the C-algebra HomC (E, E), where E = i Ei . Then,
C is equivalent to the bounded derived category Db (mod-A) of right modules over
A.
Remark 1. The algebra A can be represented as the path algebra of a quiver with
relations in the obvious way. We refer to this quiver with relations as the associated
quiver of the strong exceptional collection.
3. Proof of the Theorem
We have the following equality,
(1)
x2 + y q + z r + λxyz = y q + z r −
λ2 y 2 z 2
λyz 2
+ (x +
) .
4
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Figure 2
Figure 3
Hence, by a biholomorphic change of coordinates and Proposition 1, we have
that Db Fuk→ T (2, q, r) is isomorphic to Db Fuk→ A(q, r), where A(q, r) represents
the singularity at the origin of the function xq + y r − λx2 y 2 .
Consider the deformation xq + y r − λx2 y 2 − tλ−1 xq−2 y r−2 , for t ∈ [0, 1]. It
can be checked easily that for sufficiently large λ this is a Morsification of A(q, r).
Clearly at t = 1 function factors into two An singularities. Using the standard real
Morsifications of An for n > 1 (the divides are shown in Figure 2), we get a real
Morsification of A(q, r), of which divide looks like the two examples in Figure 3. It
is easy to check that this is indeed a real Morsification by counting the number of
intersection points. Note that we can arrange the intersection of the two pieces to
be as in the pictures by admissable homotopies and this will not do any harm in
what follows by [14], Section 7. The same is true for picking one of the two ways
for choosing positive and negative regions. As realized by the author later on, this
construction essentially exists in the examples of [9].
Now, using Proposition 2, we can easily deduce2 the following.
2after possibly changing the sign of a generator, which clearly makes no change to the derived
category.
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Figure 4
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Figure 5
Proposition 5. Let B be the path algebra (over C) of the quiver in Figure 4, with
relations aY − bX = cX − dZ = 0.
Then, Db Fuk→ T (2, q, r) is isomorphic to the bounded derived category Db (mod-B)
of right modules over B in the triangulated sense.
It now suffices to prove that Db (mod-B) is isomorphic to Db (mod-A). Let
P1 , . . . , Pµ be the projective modules of B, which form a strongly exceptional collection in Db (mod-B). Consider the following sequence of mutations,
(P1 , . . . , P5 , . . . , Pµ ) → ({P1 → P2 }[−1], P1 , P3 , P4 , P5 , . . . , Pµ ) →
({P1 → P2 }[−1], P3 , P4 , P5 , {P1 → P3 ⊕ P4 ⊕ P5 }, . . . , Pµ ).
We did three mutations at once in the last arrow. It is trivial to check that
once we shift the first element, the last collection is strongly exceptional, and the
associated quiver is as in Figure 5 with relations aX + bY + cZ = 0. Using Proposition 3 and 4, we can replace B with the path algebra of this quiver in the previous
paragraph and apply mutations again as follows,
(P1 , . . . , Pµ ) → (P1 , P5 , {P2 → P5 }, {P3 → P5 }, {P4 → P5 }, . . . , Pµ ).
This is again three mutations at once. It is clear that this last collection is
also strongly exceptional and the associated quiver is the one in Figure 1 with the
relations as in Theorem 1 except the two tails.
Recall that a mutation of two objects with no maps between them has only the
effect of changing the ordering of the two elements. Keeping this in mind, the
tails can be fixed using the following move (and analogous dual move) which is a
combination of two mutations and changing the quiver right after, as we did above.
Assume that Pi−1 has maps to only Pi and Pi+1 , and none of the other Pj ’s map
to Pi−1 , then the move is
(. . . , Pi−1 , Pi , Pi+1 , . . .) → (. . . , {Pi−1 → Pi }, {Pi−1 → Pi+1 }, Pi−1 , . . .).
It is clear that the end result is again a strong exceptional sequence after shifting
Pi−1 . A simple induction finishes the proof.
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