Homological Mirror Symmetry for the 4-punctured torus

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MSc Mathematics

Master Thesis

Homological Mirror Symmetry for the

4-punctured torus

Author: Supervisor:

Ronen Brilleslijper

dr. R.R.J. Bocklandt

Examination date:

June 24, 2020

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Abstract

In this thesis a specific occurrence of Homological Mirror Symmetry is proven. On the B-side, a curve over C[[q]] is considered that is a deformation of the wheel of four projective lines. Its mirror is a four-punctured torus, where the punctures are filled by formal expressions t1, . . . , t4 ∈ C[[q]]. The proof of this correspondence relies heavily on

two isomorphisms from [LP14]. The first one is an isomorphism eUsns

1,4 → A5 from a stack

of pointed curves to affine space. The second isomorphism eUsns

1,4 → M∞(E1,4) goes to

a moduli space of A∞-structures on a certain quiver algebra E1,4. These isomorphisms

are used to view the A- and B-model as points in A5. The appropriate expressions for the ti can then be found, by comparing these points in A5.

Title: Homological Mirror Symmetry for the 4-punctured torus Author: Ronen Brilleslijper, ronenzb1996@gmail.com, 10855041 Supervisor: dr. R.R.J. Bocklandt

Second Examiner: dr. H.B. Posthuma Examination date: June 24, 2020

Korteweg-de Vries Institute for Mathematics University of Amsterdam

Science Park 105-107, 1098 XG Amsterdam http://kdvi.uva.nl

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Introduction

“The apex of mathematical achievement occurs when two or more fields which were thought to be entirely unrelated turn out to be closely intertwined. Mathematicians have never de-cided whether they should feel excited or upset by such events.” — Gian-Carlo Rota, Indiscrete Thoughts

Around the year 1990, a group of geometers was asked to calculate the number of degree 3 curves in a certain space called the quintic. The number of degree 1 and 2 curves were already known to be 2875 and 609250 respectively. The problem was so hard, that the geometers had to write a complicated computer program to come up with an answer. When they gave their answer, a group of string theorists said that the number they came up with was wrong. They themselves had calculated the number as well. Not only did they give the right answer for degree 3 curves, they had an answer for the number of degree n curves for any n. The way they were able to do this, was by translating the problem into a different setting, where it became a matter of calculating just a single integral ([Dij17]). This idea became known as Mirror Symmetry.

Mathematicians quickly became interested in this phenomenon and started researching the details of such correspondences between problems. They conjectured that for a certain type of compact complex algebraic variety X there would exist a symplectic manifold X0, such that the algebraic geometry on X is equivalent to the symplectic geometry on X0in a sense that we will make precise later on. The first mathematician to formalize Mirror Symmetry in the way it is studied today was Maxim Kontsevich. In 1994 he formulated his Homological Mirror Symmetry (HMS) conjecture as an equivalence between two categories, capturing the symplectic and algebraic geometry of the spaces involved ([Kon94]).

Before getting into technicalities, we take a moment to appreciate the beauty of the connection between the two aforementioned types of geometry. Symplectic geometry – being inspired by Hamiltonian mechanics – is a very physical type of geometry, whereas algebraic geometry is set in a much more abstract framework. The fact that they are so closely intertwined in some cases, is not at all obvious and depicts an amazing serendipity of mathematics.

Since the first introduction of the HMS conjecture, a lot of work has been done on this concept and its applications. In a handful of cases, the original HMS conjecture has been proven (see for example [Sei08b] and [AS10]). Apart from that, a lot of generalizations

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of the original conjecture have been made. One example of a generalization comes from removing the compactness condition of the complex variety (see for example [AAE+11] and [Boc11]). Another generalization has been made by Yanki Lekili, Timothy Perutz and Alexander Polishchuk. In [LP12] and [LP16] they started working with deformations of the complex varieties. Their work is the main inspiration for this thesis and will be used to prove a version of HMS for a specific family of curves over C[[q]].

The family of curves that we will consider is a deformation of an elliptic curve. It is well-known that the mirror of an elliptic curve is a symplectic torus ([PZ98]), meaning that this pair of spaces satisfies the HMS conjecture. The deformation that we will be studying deforms the elliptic curve into a singular curve that is the wheel of four projective lines, as depicted on the left-hand side of figure 1. The wheel of four projective lines is known to be mirror to the four-punctured torus ([LP16]). Thus, the mirror manifolds of all the individual curves considered are already known, but the goal of this thesis is to find the family of manifolds that is mirror to the family of curves on the algebraic side. This will be achieved by assigning formal power series in q to each of the four punctures on the torus.

X X0

q 6= 0

q = 0

Figure 1.: The left-hand side of this figure depicts a deformation of an elliptic curve to the wheel of four projective lines. On the right-hand side its mirror manifolds are depicted.

The language in which we will formulate HMS is that of A∞-categories. This notion is

a generalization of ordinary categories. Besides taking into account the composition of morphisms, it also encompasses a higher product structure on morphism spaces. This will be made precise in section 1.1. On the symplectic side – also called the A-side – an A∞-category is used, called the Fukaya category. There are different versions of this

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category, of which we will introduce two. On the side of complex geometry – known as the B-side – we will look at the A∞-category of perfect complexes. This category is

obtained from the category of locally free sheaves of finite rank in a way that will be explained in section 1.4.

Objective

In this thesis, we consider the family of curves over C[[q]] described above, namely a deformation of an elliptic curve to the wheel of four projective lines over C. Our ultimate goal will be to find a model on the A-side to which W is mirror. The model that will be used on the A-side is the 2-torus with 4 marked points. In [LP12] a description of mirror symmetry for the torus with a single marked point was given and in [LP16] this was generalized to the torus with n marked points. In the latter article, a version of the Fukaya category – called the relative Fukaya category – is used, which is an A∞-category

that is linear over C[[t1, t2, t3, t4]]. As mentioned above, our model on the B-side, namely

the category Perf(W ) of perfect complexes over W , is an A∞-category that is linear over

C[[q]]. We will prove that choosing the appropriate expressions for the ti in terms of q

will yield an equivalence of these categories, thereby providing an A-model that is mirror to W . This is the main theorem of this thesis as formulated below.

Theorem A. There are t1, . . . , t4 ∈ C[[q]] such that DπFt1,...,t4(T, z1, . . . , z4) is

equiva-lent to Perf(W ) as A∞-categories.

Here T denotes the 2-torus R2_/Z2 _{and z}

1, . . . , z4 are marked points. The A∞-category

DπFt1,...,t4(T, z1, . . . , z4) contains the relevant symplectic information associated to the

torus with marked points. This will be explained more in section 1.3.

The work of this thesis will accumulate to proving this theorem in chapter 6. The idea is as follows. In section 2.2 a certain quiver algebra E1,4 is introduced. From the

A-side, an A∞-structure will be obtained on E1,4⊗ C[[t1, t2, t3, t4]] and from the B-side

such a structure will be obtained on E1,4⊗ C[[q]]. We will introduce an isomorphism

M∞(E1,4⊗ R) → A5, where R is a C-algebra and M∞(E1,4⊗ R) denotes the space of

A∞ structures on E1,4⊗ R (see theorems 2.1.2 and 2.2.2). Hence, both A∞-structures

coming from the A-side and the B-side can be interpreted as sets of coordinates in A5 over the appropriate base ring. Thus, equating these two sets of coordinates will give us expressions for the ti in terms of q (theorem 6.2.1). Reverting the isomorphism

mentioned above yields that, using these expressions for the ti, the A∞-structures on

E1,4⊗ C[[q]] coming from the A-side and B-side are equivalent. The A∞-structures on

the Fukaya category and the category of perfect complexes are completely determined by the structure they induce on E1,4⊗ C[[q]], hence they must be equivalent (see lemma

3.3.1. from [LP16] and lemma 4.2.2).

It is interesting to note that in the process of proving our main theorem, we encounter a phenomenon that is also relevant in a broader context. In chapter 5 we calculate the coordinates in A5 of the A-side. To do this, we need to find a way on the A-side

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to interpret the residue of a differential form on a complex variety, a notion from the B-side. We do this by explicitly calculating an isomorphism between a morphism space on the A-side and the corresponding morphism space on the B-side. In general, the equivalences of categories used in HMS are very hard to explicitly define, thus our way of doing this provides an interesting example of a case in which it is possible to do so and might inspire other ways of relating algebraic invariants to concepts on the A-side.

Overview of chapters

This thesis is devoted to proving theorem A. We will start by introducing the relevant concepts in chapter 1, like the notions of A∞-categories, Fukaya categories and the

category of perfect complexes. Chapters 2 and 3 treat the B-side of our calculations. Relevant results from [LP14] are reviewed in chapter 2 and the explicit coordinates in A5 of the B-side are calculated in chapter 3. The A-side of the thesis is covered in chapters 4 and 5. Chapter 4 will review the relevant results of [LP16] and chapter 5 calculates the coordinates in A5 of the A-side. Finally, in chapter 6 the two sets of coordinates will be compared in order to find expressions for the ti in terms of q. The thesis will be

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1. Preliminaries

This chapter will briefly cover some of the concepts needed to understand this thesis. References will be given to more extensive explanations. For this thesis, vector spaces are always defined over C.

1.1. The A

∞

-formalism

In this section the A∞-formalism will be introduced. As a reference for this section and

the next, see [Kel99]. We start by defining A∞-structures on algebras.

Definition 1.1.1. An A∞-structure on a Z-graded vector space A• is a sequence of

maps µi: A⊗i→ A of degree 2 − i, for i = 1, 2, . . ., that satisfy

X

r+s+t=n r,t≥0,s>0

(−1)µr+t+1(a1, . . . , ar, µs(ar+1, . . . , ar+s), ar+s+1, . . . , an) = 0

for all n and all a1, . . . , an ∈ A. Here = −t +Pti=1deg(ar+s+i).1 We call the pair

A = (A•, µ) an A∞-algebra.

Given A∞-algebras A and B, a morphism f : A → B is given by a sequence of maps

fi : A⊗i→ B of degree 1 − i such that for all n

X r+s+t=n r,t≥0,s>0 (−1)fr+t+1(1⊗r⊗ µs⊗1⊗t) = X 1≤r≤n i1+···+ır=n µr(fi1 ⊗ · · · ⊗ fir), (1.1)

where is defined the same as above after applying these expressions to elements a1, . . . , an∈ A.

One can think of the map µ1as a differential on A•, since it has degree 1, so that it makes

sense to look at the cohomology H∗A of this complex. We call a morphism f : A → B a quasi-isomorphism if f1 is a quasi-isomorphism of complexes. The next theorem asserts

a well-defined A∞-structure on H∗A.

Theorem 1.1.2. For an A∞-algebra A = (A•, µA) there is an A∞-structure on the

cohomology H∗A, such that µ1 = 0 and µ2([a]) = µA2(a) for all a ∈ A, where [a] denotes

the class of a in cohomology. Moreover, this structure can be chosen such that the map H∗A → A lifting the identity of H∗A is a quasi-isomorphism of A∞-algebras.

1

Note that the sign appearing here is different than the one used in [Kel99]. We use the sign convention of [Sei08a], because this convention is used more often in symplectic geometry.

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We call H∗A together with the A∞-structure defined in the theorem a minimal model

of A.

Now, we move on to A∞-structures on categories. The definition is as follows.

Definition 1.1.3. An A∞-category A is given by a class of objects Ob(A), a Z-graded

vector space HomA(A, B) for all A, B ∈ Ob(A) and linear maps

µi : HomA(Ai−1, Ai) ⊗ HomA(Ai−2, Ai−1) ⊗ · · · ⊗ HomA(A0, A1) → HomA(A0, Ai)

of degree 2 − i for all i > 0 and A0, . . . , Ai ∈ Ob(A), such that

X

r+s+t=n r,t≥0,s>0

(−1)µr+t+1◦ (1⊗r⊗ µs⊗1⊗t) = 0,

where = −t +Pt

i=1deg(ar+s+i) as before, when we apply this expression to elements

aj ∈ HomA(Aj−1, Aj). If moreover the morphism spaces are R-modules for some

C-algebra R and the multiplication maps µi respect this module structure, then we call A

an A∞-category that is linear over R. An A∞-functor F : A → B between A∞-categories

is given by a map of objects F : Ob(A) → Ob(B) and linear maps

Fi : HomA(Ai−1, Ai) ⊗ HomA(Ai−2, Ai−1) ⊗ · · · ⊗ HomA(A0, A1) → HomB(F A0, F Ai)

of degree 1 − i for all i > 0 and A0, . . . , Ai ∈ Ob(A), satisfying an analogous condition

to equation (1.1).

We will sometimes write elements of HomA(A, B) as morphisms a : A → B. Theorem

1.1.2 above can be generalized to define a minimal model for A∞-categories, which we

also denote by H∗A. Let A ∈ Ob(A). We call an element e ∈ Hom0

A(A, A) a strong unit,

if µ2(e, f ) = f and µ2(g, e) = g for all morphisms f : B → A and g : A → C in A and

all other products involving e are zero. Note that in particular e represents a morphism in a minimal model of A.

1.2. Twisted complexes and derived categories

When working with A∞-categories it is often useful to consider a larger A∞-category

that is closed under taking shifts and includes complexes of objects. This A∞-category

is called the twisted completion and is constructed below.

Let A be an A∞-category and A, A0 ∈ Ob(A). By A[n] we denote the object A shifted

n places in degree, so that a morphism a ∈ HommA(A, A0) has degree m + n − n0 seen

as a morphism A[n] → A0[n0] in the category of shifts of A, called ZA. Now we can define a direct sum of elements Ai[ni] of ZA as the formal expressionLiAi[ni] such that

the morphism space between two direct sums is given by Hom(L

iAi[ni],

L

jBj[mj]) :=

L

i,jHomZA(Ai[ni], Bj[mj]). An element of Hom(L_iAi[ni],L_jBj[mj]) can be seen as

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The A∞-products in A can be extended to direct sums using the matrix multiplication

conventions. For example

µ2(a, b)i,j :=

X

k

µ2(ai,k, bk,j).

Using direct sums we can define twisted complexes over A.

Definition 1.2.1. Let A be an A∞-category. Then a twisted complex over A is given by

a pair (T, δ), where T =L

iAi[ni] is direct sum of shifted objects and δ ∈ Hom1(T, T ) is

a degree one endomorphism of T . Moreover, we assume δ to be strictly upper triangular in the sense that δi,j = 0 for all i ≥ j, and δ must satisfy the so-called Maurer-Cartan

equation:

∞

X

t=1

(−1)t(t−1)2 µ_t(δ, . . . , δ) = 0.

We can define A∞-products between these twisted complexes, coming from the A∞

-structure on A. Given twisted complexes (T0, δ0), . . . , (Tn, δi) over A we define the

twisted product ˜µi : Hom(Ti−1, Ti) ⊗ · · · ⊗ Hom(T0, T1) → Hom(T0, Ti) by

˜ µi:= X m0,...,mi≥0 µ ◦ (δ⊗mi i ⊗1Hom(Ti−1,Ti)⊗ δ ⊗mi−1

i−1 ⊗1Hom(Ti−2,Ti−1)⊗ · · · ⊗1Hom(T0,T1)⊗ δ

⊗m0

0 )

Using these products we can define the A∞-category of twisted complexes over A.

Definition 1.2.2. Let A be an A∞-category. Define the A∞-category TwA by setting

its objects as the twisted complexes over A, morphism spaces as the morphism spaces between the underlying direct sums and the A∞-products as the twisted products defined

above. The derived category of A is defined as DA := H∗TwA.

Note that we can embed A in TwA by sending A ∈ Ob(A) to the twisted complex (A, 0). Hence, TwA is viewed as an enlargement of A. An important class of objects in TwA is given by taking so-called cones of morphisms, which we will define now.

Definition 1.2.3. Let A, B be two objects in TwA and f : B → A a degree one morphism. Then the cone of f is defined as

Cone(f ) := A ⊕ B,δA f 0 δB ∈ TwA.

We will sometimes denote the cone of a morphism f : B → A between two objects in A by

Cone(f ) = B f A.

Sometimes the A∞-category DA is still not big enough.2 In that case we define the split

derived category DπA. Its objects are pairs (D, e) where D ∈ Ob(DA) and u : D → D

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is an endomorphism of D satisfying u2 = u. The morphism from (D, u) to (D0, u0) in DπA are given by morphisms f ∈ HomDA(D, D0) satisfying f = u0 ◦ f ◦ u. Note that

DA embeds in DπA by sending an object D to (D,1_D). It will turn out in our case that the split derived category we will consider can be described by a finite amount of data. To this extent we define the concept of generators.

Definition 1.2.4. Let S0, . . . , Sn be objects in DπA. Then we define hS0, . . . , Sni ⊆

Dπ_{A as the full subcategory that contains all objects isomorphic to objects in D}π_{S

0, . . . , Sn}.

We say the objects S0, . . . , Sn split generate DπA if hS0, . . . , Sni = DπA.

1.3. Fukaya categories

In this section we will encounter our first examples of A∞-categories, called Fukaya

categories. They will be the models containing the information on the A-side of mirror symmetry. See [Abo06] and chapter 6 of [LP12] as a reference.

Exact Fukaya Category

We start with a surface Σ equipped with a symplectic form ω. Let z1, . . . , zm be distinct

points in Σ and denote by Σ0 the punctured surface Σ0 := Σ\{z1, . . . , zm}. Note that

ω|Σ0 is exact so that we might choose a primitive θ. We fix an unoriented line field

l ⊆ T Σ, called a grading and define the notion of a Lagrangian brane.

Definition 1.3.1. A grading on an embedded curve L ⊆ Σ0 is given by a homotopy

class of paths from l|L to T L in T Σ0|L. To specify a spin-structure on an embedded

closed curve L ⊆ Σ0 we have to specify two things: an orientation and whether or not

the structure is trivial. A non-trivial structure corresponds to choosing a marked point ∗ ∈ L. An exact Lagrangian brane is an embedded closed curve L ⊆ Σ₀ that is exact with respect to θ, i.e. satisfies R_Lθ = 0, equipped with a grading and spin structure. The exact Lagrangian branes will be the objects of our Fukaya category. To define the morphism spaces we look at the Floer complex. Given exact Lagrangian branes L and L0 that intersect transversally, we define

CF (L, L0) := span_C(L ∩ L0).

We specify a grading on this vector space as follows. For y ∈ L ∩ L0 denote by α the net rotation of the path TyL → ly → TyL0. The degree of y is given by bα/πc + 1. On this

Floor complex, a differential can be defined.

Definition 1.3.2. For two intersection points p, q ∈ L ∩ L0 define the space M(q, p) as the set of immersed discs u in Σ0 satisfying that p and q are convex corners of the disc

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from p to q along ∂u lies on L0. The the differential is defined as d(p) := X q∈L∩L0 X u∈M(q,p) (−1)s(u)· q.

Here the sign (−1)s(u) is defined by adding a factor of −1 for every marked point on the boundary of u and by adding another factor of −1 if the orientation of L0 disagrees with the standard orientation of the disc.

This differential allows us to consider the cohomology HF (L, L0) := H∗(CF (L, L0)). We can also define higher products on the Floor complexes.

Definition 1.3.3. Let L0, . . . , Ln be exact Lagrangian branes that intersect

transver-sally and do not have triple intersections. Let pi,i+1 be an intersection point between

Li and Li+1 and p0,n between L0 and Ln. The space M(p0,n, pn−1,n, . . . , p0,1)

con-tains all immersed polygons u in Σ0 with convex corners at all the pi,i+1 and p0,n,

such that ∂u is a union of segments [pi−1,i, pi,i+1] ⊆ Li and the following the

bound-ary of u counterclockwise increases i (see figure 1.1). Then we define the product µn: CF (Ln−1, Ln) ⊗ · · · ⊗ CF (L0, L1) → CF (L0, Ln) by µn(pn−1,n, . . . , p0,1) = X p0,n∈L0∩Ln X u∈M(p0,n,pn−1,n,...,p0,1) (−1)s(u)· p_0,n.

Here, similarly as before, the sign (−1)s(u) _{is determined by adding a factor of −1 for}

every marked point on the boundary of u and adding other factors of −1 whenever the orientation of Lj disagrees with the standard orientation of the disc for odd intersection

points pi,j, with i < j.

Figure 1.1.: An element of M(p0,6, p5,6, . . . , p0,1). Figure adapted from [Abo06].

Now the exact Fukaya category of Σ0, Fex(Σ0), is defined as the category that has the

exact Lagrangian branes as objects and the Floer complexes as the morphims spaces. Note that we have defined the A∞-products only for sets of transversally intersecting

curves. When we want to calculate products involving Lagrangians that do not inter-sect transversally, say L0, . . . , Ln, we choose exact Hamiltonian symplectomorphisms

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φn : Σ0 → Σ0 such that φ0(L0), . . . , φn(Ln) intersect transversally. Then the Floor

complex may be calculated using these Lagrangians. In particular this means that for any Hamiltionian symplectomorphism φ and exact Lagrangian brane L, the objects L and φ(L) will be quasi-isomorphic in Fex(Σ0). For more information on the relevant

symplectic geometry see [MS17].

Relative Fukaya Category

The relative Fukaya category of Σ with the marked points z1, . . . , zm, can be seen as a

deformation of the exact Fukaya category Fex(Σ0). It has the same objects as Fex(Σ0),

but the morphism space between L and L0 is given by CF (L, L0) ⊗ C[[t1, . . . , tm]]. The

A∞-products are defined again by counting polygons, but this time the polygons may

lie in Σ and we keep track of their intersection numbers with the zi. For transversally

intersecting exact Lagrangian branes L0, . . . , Ln and intersection points p0,1, . . . , pn−1,n

and p0,ndefineM(p0,n, pn−1,n, . . . , p0,1) similar to M(p0,n, pn−1,n, . . . , p0,1) only this time

with polygons immersed in Σ instead of Σ0. Then the product µn is given by

µn(pn−1,n, . . . , p0,1) = X p0,n∈L0∩Ln u∈M(p0,n,pn−1,n,...,p0,1) (−1)s(u)tu·z1 1 · · · t u·zm m · p0,n, (1.2)

where u · zi denotes the intersection number of u and zi. The sign (−1)s(u)is determined

similarly as before.

Definition 1.3.4. The relative Fukaya category, F (Σ, z1, . . . , zm), is defined as follows.

The objects are the same as the objects of Fex(Σ0), the morphism spaces are given by

HomF (Σ,z1,...,zm)(L, L

0_{) := CF (L, L}0_{) ⊗ C[[t}

1, . . . , tm]], and the A∞-products are given

by equation (1.2). Note again that products between Lagrangians that do not intersect transversally can be calculated by applying Hamiltonian symplectomorphisms.

Note that in equation (1.2) we could also fill in expressions for the ti in terms of another

formal variable, say q. Choosing expressions t1, . . . , t4 ∈ C[[q]] and constructing the

relative Fukaya category for these values of the ti, yields an A∞-category with morphism

spaces that are C[[q]]-modules. When the specific expressions used fo the tiare relevant,

we will denote the relative Fukaya category by Ft1,...,t4(Σ, z1, . . . , zm)

1.4. Ext rings and perfect complexes

In this section another example of an A∞-structure will be introduced. This time it

contains the information from the B-side. References for this section are [Sta20] and [LP12]. Let X be a seperated Noetherian scheme and F and G be two sheaves of OX

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to be the Z-graded sheaf given by Hi(F , G)(U ) :=M j H

om

(Rj, Si+j)(U ) =M j HomOU(R j_| U, Si+j|U),

with differential dH given by dHf := dS ◦ f + (−1)degff ◦ dR. For every term in the

resulting complex we can define the Cech complex with differential dC using an affine

cover U of X.3 The resulting complex has two gradings and two differentials and is therefore called a double complex (see [Sta20] section 12.18 for the complete definition). The associated total complex is given by

Tn(F , G) := M

p+q=n

M

i0<···<ip

Hq(F , G)(Ui0,··· ,ip),

with differential df := dCf + (−1)pdHf for an element f of Chern degree p. If E , F , G

are three sheaves of OX-modules with locally free resolutions, there is a multiplication

map T•(F , G) ⊗ T•(E , F ) → T•(E , G) coming from the composition of morphisms. For elements α and β of total degree n and m respectively, the product is defined as

(α ◦ β)|U_{i0,··· ,ip} = p

X

r=0

(−1)(p+r)n+pr+rα|U_{i0,··· ,ir} ◦ β|Uir ,··· ,ip. (1.3)

See section 20.25 of [Sta20] for the proof that this is well-defined.4

Definition 1.4.1. The Ext-group of F and G is defined as Ext•(F , G) := H•T (F , G). If F = G the Ext-ring Ext•(F , F ) is defined as the minimal model of T•(F , F ).

Using the previous definitions we can define the category of perfect complexes. Start by considering the category vect(X). Its objects are given by locally free sheaves of OX-modules of finite rank. For two such sheaves F and G the morphism space is given

by

Homvect(X)(F , G) := T•(F , G).

This defines a Z-graded category that comes equipped with a differential on morphism spaces. We can view it as an A∞-category with µi= 0 for i > 2.

Definition 1.4.2. The category of perfect complexes on X, Perf(X), is defined as Perf(X) = H0(Tw vect(X)).

Note that for sheaves of OX-modules with locally free resolutions, the Ext-groups

func-tion as the morphism spaces in this category and endomorphism rings are given by Ext-rings.

3_{See [Har13] chapter III.4 for the definition of the Cech complex.} 4

Note that in [Sta20] the cup product is defined for the tensor product of complexes. However, the case of morphism spaces between complexes works analogously.

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2. Moduli space of curves

In this chapter we will cover some of the results from [LP14] that will be important later on. First, a stack of pointed curves called U_1,4sns will be introduced and two important isomorphisms are discussed. The first one being an isomorphism with affine space and the second one with a certain moduli space of A∞-structures on a quiver algebra. We

will highlight some aspects of the proofs that make the statements more understandable and refer to the original article for the proofs in full generality.

Before giving a formal definition of U_1,4sns, we want to gain intuition for what it has to look like. A generic point of U_1,4sns is an elliptic curve with four marked points on it. Under the correspondence between Usns

1,4 and A5 that will be discussed in section 2.1, the point

(0, 0, 0, 0, 0) ∈ A5 corresponds to a singular curve called the elliptic 4-fold curve with four marked points. It can be seen as the union of four generic lines passing through one point in P3, with a marked point on each of these lines. Another element of U_1,4sns is the wheel of four projective lines with a marked point on every copy of P1. In section 3.3 the coordinates of this element in A5 will be determined. The formal definition of U_1,4sns is as follows.

Definition 2.0.1. Let S be a scheme and C → S an algebraic space1 over S with distinct smooth sections p1, . . . , p4. Then we define (C, p1, p2, p3, p4) to be an element of

Usns

1,4 , if the following conditions hold:

(1) the structure map C → S is a flat, proper, finitely presented morphism, (2) the fibers of C → S are reduced, connected curves of arithmetic genus 1, (3) h0(OC(pi)) = 1 for all i and,

(4) OC(p1+ p2+ p3+ p4) is ample.

Note that up to scalar multiplication there is a unique choice for a generator of the space H0(C, ωC), where ωC is the dualizing sheaf. We denote by eU1,4sns the set of elements of

Usns

1,4 together with a choice of generator ω ∈ H0(C, ωC), i.e. a typical element of eU1,4sns is

denoted by (C, p1, p2, p3, p4, ω).

1_{We use the term algebraic space here in accordance with [LP14] to keep the generality of the statements.}

However, to avoid unnecessary complexity, in our cases we will always work with C as if it were a scheme or even a variety.

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2.1. Curves as affine space

The first important result we will cover is that eUsns

1,4 is isomorphic to A5. A nice property

of this isomorphism is that we can explicitly construct the map eUsns

1,4 → A5and its inverse.

Let us start with the following lemma.

Lemma 2.1.1. For an element (C, p1, p2, p3, p4, ω) ∈ eU1,4sns and i = 2, 3, 4 there exist

functions h1i ∈ H0(OC(p1 + pi)) such that Resp1(h1iω) = 1 and Respi(h1iω) = −1.

Moreover, we can normalize these functions to satisfy h12(p3) = h13(p2) = h14(p3) = 0.

Sketch of proof. As the proof of this lemma uses the Riemann-Roch theorem and the Residue formula we will prove it only for smooth curves here. For an approach on how to work with Riemann-Roch on singular curves see exercise IV.1.9. from [Har13] and for a more general version of the Residue formula see lemma 1.1.2. from [LP14]. A direct consequence of condition (3) of definition 2.0.1 and Riemann-Rochs theorem is that H1(C, OC(p1)) = 0. As 0 ≤ h1(OC(p1+ pi)) ≤ h1(OC(p1)) for i = 2, 3, 4, we must have

H1_{(C, O}

C(p1+ pi)) = 0. This yields h0(OC(p1+ pi)) = 2 by applying Riemann-Roch

another time. As h0(OC(p1)) = h0(OC(pi)) = 1, there is a function h1i that has simple

poles precisely at p1 and pi. We know ω is regular and non-zero at p12, so we can rescale

h1iso that Resp1(h1iω) = 1. Now, by the Residue formula Resp1(h1iω)+Respi(h1iω) = 0,

hence Respi(h1iω) = −1. This defines h1i uniquely up to the addition of a constant.

Using the lemma above we can construct a map eUsns

1,4 → A5. Note that the normalisation

conditions imply that h12h13 is regular near p2 and p3, so h12h13∈ H0(OC(2p1)). Now

h12h213∈ H0(OC(3p1+ p3)) and h212h13∈ H0(OC(3p1+ p2)). By considering the residues

of h12 and h13 at p1, we see that both h12h132 and h212h13 can be written as u−31 plus

higher order terms around p1, where u1 is a uniformizer at p1. So their difference

has a pole of order at most two at p1: h12h213 − h212h13 ∈ H0(OC(2p1 + p2 + p3)).

The set {h12h13, h12, h13, 1} is clearly linearly independent in the 4-dimensional space

H0(OC(2p1+ p2+ p3)), and is therefore a basis. Hence we can find constants a, b, c and

d such that

h12h213− h212h13= ah12h13+ bh12+ ch13+ d. (2.1)

We introduce the notation

cij := h1i(pj) ci := c2i ¯ ci := ci2 ¯ c := b + c.

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Now we can define the map eUsns 1,4 → A5 by Ψ : eUsns 1,4 → A5 (C, p1, p2, p3, p4, ω) 7→ (a, c, ¯c, c4, ¯c4). (2.2) The following theorem is a weaker version of theorem 1.4.2 from [LP14].

Theorem 2.1.2. Assume the base scheme S is affine, given by S = Spec(R) for some C-algebra R. Then the map Ψ defined in equation (2.2) is invertible.

Sketch of proof. We construct an explicit inverse to Ψ. Given a point (a, c, ¯c, c4, ¯c4) ∈ A5

we consider the algebra A := R[X2, X3, X4]

(f1, f2), where the polynomials f1 and f2

are given by

f1 := X2X4− X2X3− c4X4− ¯c4X2− c

f2 := X3X4− X2X3− (a + c4+ ¯c4)(X4− ¯c4) − ¯c + c.

Denote xi = Xi+ (f1, f2) ∈ A. Define the following increasing filtration on A:

F0A := R · 1

F1A := R · 1 ⊕ R · x2⊕ R · x3⊕ R · x4

FmA := (F1A)m,

where m ≥ 2 in the last equation. Note that the quotients FmA_F

m−1A give the

ho-mogeneous elements of A of degree m. It can be easily seen that F2A

F1A is freely

generated by {x2x3, x22, x23, x24}, since the remaining monomials of degree 2, namely x2x4

and x3x4, are equal to x2x3 up to lower order terms. Similarly, for any m ≥ 2 one can

prove that the set {xm−1₂ x3, xm2 , x3m, xm4 } freely generates FmAF_m−1_{A. Now the curve}

C can be defined as

C = Proj(RA), where RA := L

m≥0FmA. This yields a curve over Spec(R) that can be explicitly

described as a closed subscheme of P3 given by the equations 0 = yw − yz − c4xw − ¯c4xy − cx2

0 = zw − yz − (a + c4+ ¯c4)(xw − ¯c4x2) − (¯c − c)x2,

where x, y, z, w are the homogeneous coordinates on P3. By setting x = 0 we find four points

p1= (0 : 1 : 1 : 1)

p2= (0 : 1 : 0 : 0)

p3= (0 : 0 : 1 : 0)

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It can easily be checked that these points are all smooth, by working with the explicit equations for C given above. For a point s ∈ Spec(R) the fiber Cs is given by the

same equations as C, but this time over the residue field k(s). Therefore the Hilbert function of Cs, as defined in section I.7 from [Har13], can be computed using the fact that

FmA_F

m−1A is freely generated by 4 elements for all m ≥ 2. This yields ϕCs(m) = 4m

for m ≥ 2, hence the Hilbert polynomial PCs is given by PCs(m) = m for all m. So the

fibers have arithmetic genus pa(Cs) = 1 − PCs(0) = 1. Using this fact we can define

ω to be the unique global 1-form satisfying Resp1(h12ω) = 1, where h12 :=

y x. The

map (a, c, ¯c, c4, ¯c4) 7→ (C, p1, p2, p3, p4, ω) gives an inverse to Ψ. We will not check this

completely, but notice for example that h12 defined above has simple poles only in p1

and p2 and similarly h13:= _xz and h14:= w_x have the desired poles. The fact that they

satisfy equation (2.1) follows from an explicit calculation that is left to the reader (for an idea of the approach see [LP14]).

2.2. Curves versus A

∞

-structures

In this section we will discuss how to get from a curve to an A∞-structure on a certain

quiver algebra E1,4. Starting from a pointed curve (C, p1, p2, p3, p4) we can consider the

direct sum of sheaves G := OC⊕ Op1⊕ · · · ⊕ Op4. It turns out that the algebra structure

of Ext•(G, G) is independent of the curve. The higher product structure however does depend on the pointed curve as will be discussed later. First we want to define the quiver algebra E1,4. The underlying quiver Q is given by

Q :=

1

4 0 2

3

where we label the arrows 0 → i by Ai and i → 0 by Bi. On the quiver algebra CQ we

define the ideal I generated by the elements BiAi− BjAj and AiBj for i 6= j. Now set

E1,4 := CQ/I. Moreover, we define a grading on the algebra E1,4by defining deg(Ai) = 0

and deg(Bi) = 1 for all i. This graded algebra turns out to describe the algebra structure

of the Ext-ring described above, as stated in lemma 2.1.1. from [LP14]. We review the statement and discuss the proof below.

Lemma 2.2.1. For a pointed curve (C, p1, p2, p3, p4, ω) ∈ eU1,4sns over Spec(C) and G as

above, there is a canonical isomorphism of graded C-algebras Ext•(G, G) ∼= E1,4.

Proof. We start by calculating Ext•(G, G) as a group. For this we have to determine the Ext-groups between the different summands of G. The easiest one is Ext•(O, O), as O is locally free and therefore its own resolution. This yields that H•(O, O) =H

om

(O, O) ∼=

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O concentrated in degree 0, so that Ext•(O, O) = H•(O) is the regular cohomology of the sturcture sheaf. As C is a curve of genus 1, we get

Ext•(O, O) ∼= C ⊕ C[1],

where C[1] means a copy of C concecntrated in degree 1. Now we look at Ext•(O, Op)

where p is one of the marked points. We assume the existence of an affine cover U = {U, V } of C such that p ∈ U and p /∈ V .3 _{As O}

p is not locally free, we take the locally

free resolution O(−p) → O, where the map is given by inclusion. AsH

om

(O, O(−p)) ∼= O(−p) the complex H•_{(O, O}

p) is given by O(−p) → O in degrees −1 and 0. Again the

map is given by inclusion. This yields the following total complex: T−1(O, Op) = O(−p)(U ) ⊕ O(−p)(V )

T0(O, Op) = O(U ) ⊕ O(V ) ⊕ O(−p)(U ∩ V )

T1(O, Op) = O(U ∩ V ).

The differentials are given by

d−1(fU, fV) = (fU, fV, fU|U ∩V − fV|U ∩V)

d0(gU, gV, h) = gU|U ∩V − gV|U ∩V − h.

Note that d−1 in injective and d0 is surjective so that the cohomology is concentrated in degree 0. We see (note that p /∈ V ):

ker d0=    (gU, gV, h) gU ∈ O(U ) gV ∈ O(V ) h = gU|U ∩V − gV|U ∩V    im d−1=    (fU, fV, h) fU ∈ O(−p)(U ) fV ∈ O(V ) h = fU|U ∩V − fV|U ∩V    . So _{im d}ker d−10 ∼= O(U )

O(−p)(U ) which is canonically isomorphic to C. This yields

Ext•(O, Op) ∼= C,

where a natural choice of generator is given by Ap := (1, 0, 1) ∈ T0(O, Op). The

cal-culation of Ext•(Op, O) is similar, when noting H

om

(O(−p), O) ∼= O(p). We get that

H•(Op, O) is given bu O → O(p) in degrees 0 and 1, where the map is the inclusion

map. The total complex is given by

T0(Op, O) = O(U ) ⊕ O(V )

T1(Op, O) = O(p)(U ) ⊕ O(p)(V ) ⊕ O(U ∩ V )

T2(Op, O) = O(p)(U ∩ V )

3

In case of a cover with more elements the proof is essentially the same, but the notation gets a lot more involved.

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with similar differentials as above:

d0(fU, fV) = (fU, fV, fU|U ∩V − fV|U ∩V)

d1(gU, gV, h) = gU|U ∩V − gV|U ∩V − h.

Again the cohomology is concentrated in the middle part, as d0 _{is injective and d}1 _is

surjective. Now ker d1=    (gU, gV, h) gU ∈ O(p)(U ) gV ∈ O(V ) h = gU|U ∩V − gV|U ∩V    im d0=    (fU, fV, h) fU ∈ O(U ) fV ∈ O(V ) h = fU|U ∩V − fV|U ∩V    . So ker d_{im d}01 ∼= O(p)(U )

O(U ) . The map f + O(U ) 7→ Resp(f ω) gives an isomorphism

O(p)(U ) O(U ) ∼= C,

as ω is non-zero at p4. Thus

Ext•(Op, O) = C[1],

where a natural generator is given by Bp := (fp, 0, fp|U ∩V) ∈ T1(Op, O) for a function

fp ∈ O(p)(U ) such that fpω has residue 1 at p. Next we turn to Ext•(Op, Op). We get

H−1(Op, Op) ∼= O(−p)

H0_(O

p, Op) ∼= eO ⊕ O

H1(Op, Op) ∼= O(p),

where we denote by eO the copy of O coming fromH

om

(O(−p), O(−p)) and by O the copy coming from H

om

(O, O). The differential on this complex is given by d−1_H f = (−f, f ) and d0_H(g, h) = g + h. Thus we get a total complex

T−1(Op, Op) = O(−p)(U ) ⊕ O(−p)(V )

T0(OpOp) = eO(U ) ⊕ O(U ) ⊕ eO(V ) ⊕ O(V ) ⊕ O(−p)(U ∩ V )

T1(Op, Op) = O(p)(U ) ⊕ O(p)(V ) ⊕ eO(U ∩ V ) ⊕ O(U ∩ V )

T2(Op, Op) = O(p)(U ∩ V ),

with differentials

d−1(fU, fV) = (−fU, fU, −fV, fV, fU|U ∩V − fV|U ∩V)

d0(˜gU, gU, ˜gV, gV, h) = (˜gU + gU, ˜gV + gV, (˜gU− ˜gV)|U ∩V + h, (gU − gV)|U ∩V − h)

d1(kU, kV, ˜l, l) = (kU− kV)|U ∩V − ˜l − l.

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As d−1 is injective and d1 is surjective, the cohomology is concentrated in degrees 0 and 1. By the same reasoning as above the zeroth cohomology is one-dimensional as

O(U )

O(−p)(U ) ∼= C and the first cohomology is one-dimensional as

O(p)(U )

O(U ) ∼= C. So

Ext•(Op, Op) = C ⊕ C[1],

generated by ep := (1, −1, 0, 0, −1) ∈ T0(Op, Op) and Yp := (fp, 0, 0, fp|U ∩V) ∈ T1(Op, Op),

where fp a function in O(p)(U ) such that fpω has residue 1 at p. The final term to

cal-culate is Ext•(Op, Oq) for p and q distinct marked points. This is similar to the above

computation, assuming p ∈ U ∩ Vcand q ∈ V ∩ Uc. We will not cover this computation in detail here, but it results in

Ext•(Op, Oq) = 0.

The above computations prove that at least as graded groups Ext•(G, G) ∼= E1,4, by

identifying Api ↔ Ai, Bpi ↔ Bi.

Now for the ring structure, we check the products Bpi◦ Api and Apj◦ Bpi for j 6= i. For

the latter product, note that Ext•(Opi, Opj) = 0, so this product must yield zero. For

the product Bpi◦ Api, we see that the result must be a degree one element of Ext

•_{(O, O).}

In other words Bpi ◦ Api ∈H

om

(O, O)(U ∩ V ). See equation (1.3) for the definition of

this product.

= fpi|U ∩V

We want to compare this product to Bpj◦ Apj, for j 6= i. Let i = 1 and j = 2 and assume

p1 ∈ U ∩ Vc and p2∈ Uc∩ V . For the function fp1 we may take fp1 := h12|U, where h12

is the function defined in lemma 2.1.1. Note that the calculations of Ap and Bp above

assumed p ∈ U . Since p2 ∈ V by assumption now, we get different generators of the

Ext-groups. In this case we may take Ap2 = (0, 1, −1) and Bp2 = (0, −h12|V, h12|U ∩V).

By the calculation above, we have Bp1◦ Ap1 = h12|U ∩V and a similar calculation shows

Bp2 ◦ Ap2 = h12|U ∩V as well. Similar calculations hold for other point, so this proves

that for i 6= j we have Bpi◦ Api = Bpj ◦ Apj.

Lemma 2.2.1 shows that the graded algebra structure of Ext•(G, G) is independent of the pointed curve that we started with. However, the higher products indeed do depend on the pointed curve. This amounts to a map

e

U_1,4sns(C) → M∞(E1,4)

(C, p1, p2, p3, p4, ω) 7→ Ext•(G, G),

where eU_1,4sns(C) denotes the curves in eU_1,4sns over Spec(C) and M∞(E1,4) denotes the set

of A∞-structures on E1,4 up to equivalence. For curves over a different affine scheme

Spec(R) with R a C-algebra we get an A∞-structure on E1,4⊗ R in a similar way so we

can actually define the map above more generally. One of the main results of [LP14] is the following theorem, which you can find as theorem 2.2.8. in the orignal article.

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Theorem 2.2.2. For a C-algebra R, the map eU_1,4sns(R) → M∞(E1,4⊗ R) that sends a

pointed curve to the Ext-ring of OC⊕ Op1 ⊕ · · · Op4 is an isomorphism.

The proof of this theorem uses Hochschild cohomology, which unfortunately goes outside the scope of this thesis. For an introduction to this topic, see [Wei95] chapter 9.

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3. The B-side

In this chapter we introduce a family of pointed curves (W, p1, p2, p3, p4) over C[[q]], that

can be seen as a deformation from an elliptic curve with four marked points to the wheel of four projective lines with a marked point on every copy of P1. We determine the image of its fibers under the map Ψ : eUsns

1,4 → A5 from equation (2.2) (see theorems 3.3.1

and 3.4.1). Before proving these theorems we introduce the relevant notation and check that the curve considered is actually an element of U_1,4sns.

3.1. Definitions and notation

On the B-side, the following curve over C[[q]] is considered:

W := {((x : y), (z : w)) ∈ P1× P1_{| q(x}2_{+ y}2_)(z2_{+ w}2_{) = xyzw} ⊆ P}1_{× P}1_.

Here, q is a deformation parameter so it is seen as infinitesimally small. We define the following marked points:

p1 := ((1 : −1), (1 : −a(q))) p2 := ((1 : −a(q)), (1 : −1)) p3 := ((1 : −1), (−a(q) : 1)) p4 := ((−a(q) : 1), (1 : −1)) , where a(q) := (₁₋√_1−16q2 4q q 6= 0 0 q = 0.

Note that a(q) = 2q + 8q3+ . . . is an element of C[[q]]. It is easily checked that these points indeed lie on W . We can cover W by four affine sets1:

U1 = D(xz) ∩ W with coordinates X1 = y x and Y2= w z U2 = D(xw) ∩ W with coordinates X2 = z w and Y3= y x U3 = D(yw) ∩ W with coordinates X3 =

x

y and Y4= z w U4 = D(yz) ∩ W with coordinates X4 =

w

z and Y1= x y.

1

The numbering of these sets might seem arbitrary, but is chosen this way so that the intersections satisfy Ui−1∩ Ui= {(Xi, Yi+1) ∈ Ui| Xiis invertible}, just like in [LP16].

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The Ui are given by Ui = {(Xi, Yi+1) | q(1 + Xi2)(1 + Yi+12 ) = XiYi+1} ⊆ A2, where we

interpret the indices i modulo 4. Also, pi ∈ Ui for all i = 1, . . . , 4. Note that the pi

are chosen in such a way, that they match the marked points on the central fiber of the 4-Tate curve used by [LP16] for q = 0.

There is another useful way of looking at W , by considering its image under the Segre embedding. Recall that the Segre embedding for P1× P1 _{is given by}

P1× P1→ P3

((x : y), (z : w)) 7→ (xz : xw : yz : yw). It identifies P1 _{× P}1 _{with the subspace {(u}

0 : u1 : u2 : u3) | u0u3 = u1u2} ⊆ P3. The

image of W under this identification is

W0:= {(u0 : u1 : u2 : u3) | u0u3 = u1u2 = q(u20+ u21+ u22+ u23)} ⊆ P3.

We can cover P3 by the standard affine opens D(ui) and look at their intersections

Vi := W0 ∩ D(ui) with W0. We will denote the coordinates on Vi by ai,j = u_uj_i. In

table 3.1 the above information is summarized. We will check that (W, p1, . . . , p4) is an

element of the moduli stack U_1,4sns as defined in definition 2.0.1 in the following section.

P1× P1 3 ((x : y), (z : w)) P3 3 (u0: u1: u2 : u3) W = {q(x2+ y2)(z2+ w2) = xyzw} W0= {u0u3= u1u2 = q(u20+ u21+ u22+ u23)} U1 = D(xz) ∩ W V0 = D(u0) ∩ W0 U2 = D(xw) ∩ W V1 = D(u1) ∩ W0 U3 = D(yw) ∩ W V3 = D(u3) ∩ W0 U4 = D(yz) ∩ W V2 = D(u2) ∩ W0

Table 3.1.: Two different ways of looking at the same curve. Both columns of the table are identified under the Segre embedding (except of course the top row).

3.2. Properties of W

Note that the central fiber of W , denoted W0, is the wheel of four projective lines and

W is smooth outside of the central fiber. All fibers are reduced and connected. In the subsequent paragraphs we will prove the following lemma.

Lemma 3.2.1. The pointed curve (W, p1, . . . , p4) is an element of U1,4sns. I.e. the structure

map W → Spec(C[[q]]) is flat, proper, and finitely presented and has geometric fibers that are reduced, connected curves of arithmetic genus 1. Moreover, the sections pi are

smooth and distinct and satisfy h0(OW(pi)) = 1 for all i and OW(p1+ p2+ p3+ p4) is

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Flat, proper and finitely presented

To check that the structure map W → Spec(C[[q]]) is flat, we have to check that all the maps

C[[q]] → C[[q]][Xi, Yi+1]_(X

iYi+1− q(1 + Xi2)(1 + Yi+12 )) =: Mi

are flat. By proposition III.9.1A from [Har13] this is equivalent to showing that the map

a⊗_C[[q]]Mi→ Mi

is injective for every finitely generated ideal a ⊆ C[[q]]. This is clear, as C[[q]] itself already embeds injectively into Mi. So W → Spec(C[[q]]) is flat. The structure map

is H-projective in the sense of [Sta20] definition 29.41.1, hence projective and proper. All the Mi are finitely generated C[[q]]-algebra’s, so W → Spec(C[[q]]) is locally finitely

presented. It is also quasi-compact as Spec(C[[q]]) is affine and W is a closed subset of the quasi-compact space P1× P1_{. Together with the structure map being proper (hence}

separated), this shows that W → Spec(C[[q]]) is finitely presented.

Arithmetic genus of the fibers

First we determine the arithmetic genus of W0. By the Segre embedding, W0 can be

identified with the set

Y := W₀0 = {(u0 : u1: u2: u3) ∈ P3 | u0u3= u1u2= 0} ⊆ P3.

So the homogeneous coordinate ring of Y is given by S(Y ) = C[u0, . . . , u3]/(u0u3, u1u2).

The Hilbert function of Y is given by ϕY(l) = dimCS(Y )l as defined in section I.7 of

[Har13]. It is a straightforward exercise in combinatorics to check that ϕY(l) = 4l for

l ≥ 1, since we are essentially counting in how many ways it is possible to choose l variables from the set {u0, u1, u2, u3} without picking both u0 and u3 or u1 and u2. So

we get that the Hilbert polynomial of Y is given by PY(l) = 4l, so the arithmetic genus

of Y is pa(Y ) = 1 − PY(0) = 1.

For q 6= 0, Wq is smooth, hence the arithmetic and geometric genus coincide (see remark

8.18.2 of [Har13]). We will calculate the geometric genus by calculating the canonical divisor and using the formula deg K = 2g − 2 for a smooth projective curve. Note that V0 is given by the equations a0,3= a0,1a0,2= q(1 + a20,1+ a20,2+ a20,3). By differentiating

these equations we get that ωq|V0 := da0,1 a0,1− 2q(a0,2+ a0,1a0,3) = − da0,2 a0,2− 2q(a0,1+ a0,2a0,3) . (3.1) Clearly this expression does not have any zeros on V0. To check for poles, assume that

a0,1= 2q(a0,2+ a0,1a0,3). By multiplying both sides with a0,2 we get that

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So either 1 + a2_0,1 = 0 or 2a2_0,2= 1 + a2_0,2. In the first case we get that a0,1 = 2qa0,2(1 +

a2_0,1) = 0 which contradicts 1 + a2_0,1 = 0. In the second case we get that a0,2= ±1 which

implies a0,2 = ±a(q) or a0,2 = ±_a(q)1 .2 It is easy to check that in these points we have

that a0,2− 2q(a0,1+ a0,2a0,3) = ±

p

1 − 16q2 _{6= 0, since q is an infinitesimal parameter.}

So the differential form in equation (3.1) does not have any poles on V0. We can do

similar calculations on the other Vi and define

ωq :=            da0,1 a0,1−2q(a0,2+a0,1a0,3) = − da0,2 a0,2−2q(a0,1+a0,2a0,3) on V0 da1,3 a1,3−2q(a1,3a1,2+a1,0) = − da1,0 a1,0−2q(a1,0a1,2+a1,3) on V1 da2,0 a2,0−2q(a2,0a2,1+a2,3) = − da2,3 a2,3−2q(a2,1a2,3+a2,0) on V2 da3,2 a3,2−2q(a3,1+a3,2a3,0) = − da3,1 a3,1−2q(a3,2+a3,1a3,0) on V3. (3.2)

To check if this is well defined, note that on V0∩V1we have a0,1a1,0= 1. By differentiating

this and using the formula ai,j = u_uj_i we can see that

da0,1

a0,1− 2q(a0,2+ a0,1a0,3)

= − da1,0

a1,0− 2q(a1,0a1,2+ a1,3)

on the intersection V0∩V1. In a similar ways we can see that ωqis well defined on V0∩V1,

V0 ∩ V2, V3∩ V1 and V3∩ V2. Note also that V0∩ V3 = V0∩ V3∩ V1∩ V2, so to check

that ωq is well-defined on V0∩ V3 we can go via V1 or V2. A similar argument holds for

proving ωqis well-defined on V1∩ V2. Thus we have found a global differential form with

no zeros or poles, yielding that K = 0 and therefore pa(Wq) = g(Wq) = 1.

Strongly non-special divisor

Note that the piare distinct smooth sections and define the divisor D := p1+p2+p3+p4.

If we assume q 6= 0, then by Riemann-Roch h0(OWq(pi)) − h

0_(O

Wq(−pi)) = 1,

as the canonical divisor KWq = 0 (see previous paragraph). Since line bundles of negative

degree have no global sections on projective curves we find that h0(OWq(pi)) = 1 for all

i. As Wq is non-singular, we find that OWq(p1+ p2+ p3+ p4) is ample by proposition

7.1a from [Har66]. Now for q = 0, suppose f ∈ H0(OW0(pi)) is not constant. We know

that W0 is the wheel of four projective lines which we will denote by W0 =

S4 j=1P1j. For j 6= i we get that f |_P1 j ∈ H 0_(O P1 j) = C, and as W0\P 1

i is connected f must be constant

on this set. So, f |_P1 i ∈ H

0_(O

P1i(pi)) and satisfies f (q1) = f (q2) =: c, where q1 and q2

are the points were P1i intersects the other projective lines. But then f − c has exactly

one pole of order 0 and at least 2 zeros on P1i, which is impossible. So we find that

h0_(O

W0(pi)) = 1. Moreover, as D has degree one on every irreducible component of W0,

OW0(D) is ample. 2_{Note that} 1 a(q) = 1+ √ 1−16q2 4q for q 6= 0.

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The previous paragraphs prove lemma 3.2.1. Moreover, we found a generator of H0(W, ωW),

where ωW is the dualizing sheaf, in equation (3.2). Thus (W, p1, p2, p3, p4, ωq) is an

ele-ment of eUsns 1,4 .

3.3. Coordinates of the central fiber

In this section we want to determine the image of (W0, p1, p2, p3, p4, ω0) under the map

e Usns

1,4 → A5 of equation (2.2). We will prove the following theorem.

Theorem 3.3.1. Under the map Ψ : eUsns

1,4 → A5 the pointed curve (W0, p1, p2, p3, p4, ω0)

gets mapped to the point (−1, 0, 0, 0, 0) ∈ A5.

In order to prove this theorem, we first have to determine the functions h1ifor i = 2, 3, 4.

Note that ω0|U1 = − dX1 X1 = dY2 Y2 , and similar expressions hold on the other opens in P1_{× P}1_.

Finding the functions h1i

Define h12:=            xz xz+yz+xw on U1 z z+w on U2 0 on U3 x x+y on U4.

First, we check that this is a well defined rational function on W0. On U1 ∩ U2 the

coordinates x, z and w are all non-zero, so y = 0 on this set. This implies xz

xz + yz + xw = z z + w

on U1∩ U2. Similarly, on U1∩ U4 we have w = 0 and all other coordinates are non-zero,

so

xz

xz + yz + xw = x x + y

on this set. On U2∩U3and U3∩U4respectively z and x are zero, so on both intersections

the function makes sense and finally U1∩ U3 and U2∩ U4 are empty. This proves the

well-definedness. Now we check for poles. On U1= Z(X1Y2) ⊆ A2 the function is given

by

h12|U1 =

1 1 + X1+ Y2

.

So the poles are in the points (X1, Y2) = (−1, 0) and (X1, Y2) = (0, −1), which correspond

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we see that on D(X1) the function is given by _u1₁, hence p1 is a simple pole of h12. We calculate h12ω0|U1∩D(X1)= − 1 X1+ 1 dX1 X1 = − du1 u1(u1− 1) = 1 u1 + 1 1 − u1 du1.

Here, the second term in the brackets is regular at p1, so we see that Resp1(h12ω0) = 1.

Similarly we can calculate Resp2(h12ω0) = −1, using the uniformizer u2 := Y2+ 1 at p2.

It is an easy exercise to check that h12 does not have any new poles on the other affine

opens. Also, clearly h12(p3) = 0 as p3 ∈ U3. So h12 ∈ H0(W0, O(p1+ p2)) satisfies all

the conditions from lemma 2.1.1. Similarly, we can define

h14:=            −_x+yy on U1 0 on U2 −_z+wz on U3 −_xz+yz+ywyz on U4.

One can argue by symmetry observations or by going through the same steps as above that h14 ∈ H0(W0, O(p1+ p4)) is well-defined and satisfies the desired properties (note

that h14(p3) = 0). Finally, let

h13:= −

y x + y

on W0. It has poles only at p1 and p3 and both are of order one. Also,

h13ω0|U1∩D(X1) = X1 X1+ 1 dX1 X1 = 1 u1 du1,

so Resp1(h13ω0) = 1 and similarly Resp3(h13ω0) = −1. Also note h13(p2) = 0. Hence,

h13∈ H0(W0, O(p1+ p3)) satisfies the conditions.

Expressing products in terms of a basis

We want to determine constants a, b, c and d, such that

h12h213− h212h13= ah12h13+ bh12+ ch13+ d. (3.3)

We do this, by looking at the values of these function in the points q1 := ((1 : 0), (1 : 0))

q2 := ((1 : 0), (0 : 1))

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Note that these points all lie on W0. Table 3.2 depicts the values of h12 and h13in these points. i h12(qi) h13(qi) 1 1 0 2 0 0 3 0 -1

Table 3.2.: The values of h12 and h13 in three points.

Filling q2 in in equation (3.3) gives d = 0. Then filling in q1 and q3 respectively gives

b = 0 and c = 0. On U4 h12|U4 = Y1 Y1+ 1 h13|U4 = − 1 Y1+ 1 , so equation (3.3) becomes Y1 (Y1+ 1)2 = − Y1 (Y1+ 1)2 a.

We get a = −1. Also h12(p4) = h14(p2) = 0. So we get that (W0, p1, p2, p3, p4, ω0) maps

to (−1, 0, 0, 0, 0) ∈ A5.

3.4. Coordinates of the smooth fibers

We repeat the steps above for q 6= 0. We will prove the following theorem. Theorem 3.4.1. Under the map Ψ : eUsns

1,4 → A5 the pointed curve (Wq, p1, p2, p3, p4, ωq)

gets mapped to the point (−4q − 1, −q, −q, 0, 0) ∈ A5.

In the next paragraphs we will use expressions for ωq on the Ui. For example,

ωq|U1 = dY2 Y2− 2qX1(1 + Y22) = − dX1 X1− 2qY2(1 + X12) . Similar expressions hold on the other opens.

Finding the functions h1i

Let us first set

b(q) := − a(q)

2_{− 1}

p

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Then define h12= 1 b(q)· a(q)y + x y + x · a(q)w + z w + z .

This can have poles when y = −x or w = −z. There are two points with y = −x, namely p1= ((1 : −1), (1 : −a(q))) and p3 = ((1 : −1), (1 : −_a(q)1 )). Both of these points

lie in U1. On U1 we have h12|U1 = 1 b(q)· a(q)X1+ 1 X1+ 1 ·a(q)Y2+ 1 Y2+ 1 . (3.4)

An easy check gives that u1 = X1+ 1 is a uniformizer around both p1 and p3, which

means that it has a simple zero in these points. The numerator of (3.4) is zero in p3 and

therefore dividing by a function with a simple zero gives a regular function around p3.

So the only point with y = −x in which h12 has a pole is p1. Similarly, the only point

with w = −z in which h12 has a pole is p2. Both are simple poles, as they are defined

as a regular function over the uniformizer around that point. To calculate the residues we write out h12ωq around p1. This amounts to

h12ωq=

ϕ X1+ 1

dX1

on the appropriate set, where ϕ := − 1

b(q)·

(a(q)X1+ 1)(a(q)Y2+ 1)

(Y2+ 1)(X1− 2qY2(1 + X12))

.

As ϕ is regular at p1 we can write ϕ = c +Pi>0aiui1 for some constants c and ai all

but finitely many of which are zero. Explicitly c = ϕ(p1) = 1. So around p1 we can

write h12 as the sum of _u1₁ and some function that is regular at p1 times du1. This yields

Resp1(h12ωq) = 1. Similarly one can calculate Resp2(h12ωq) = −1. To see if h12 indeed

satisfies the required conditions, it is still necessary to check that h12(p3) = 0. We can

not simply fill in the coordinates of p3 to see this, as h12 is a quotient of two functions

that are zero in p3. Recall that on U1 the relation q(1 + X12)(1 + Y22) = X1Y2 holds.

Given X1 there are two solutions for Y2:

Y2=

X1±pX12− 4q2(1 + X12)2

2q(1 + X2 1)

.

The solution with a plus-sign holds around p1 and the one with a minus-sign holds

around p3. Substituting Y2 by the latter solution in equation (3.4) gives

h12= 1 b(q)· a(q)X1+ 1 Y2+ 1 · ψ X1+ 1 , (3.5) around p3, where ψ = 1 +(1 − p 1 − 16q2_)(X 1−pX12− 4q2(1 + X12)2) 8q2_{(1 + X}2 1) .

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A straightforward calculation shows that ψ(p3) = 0 and _∂X∂ψ₁(p3) = 0. So ψ = (X1+1)2· ¯ψ,

for some ¯ψ that is regular at p3. Equation (3.5) now gives that h12 is X1 + 1 times a

function that is regular at p3, hence h12(p3) = 0. By symmetry arguments one can argue

that h14= − 1 b(q)· y + a(q)x y + x · a(q)w + z w + z . Define f13= p 1 − 16q2_x y + x .

This rational function has poles at p1 and p3, which both have order 1. Now around

p1 f13ωq = − p 1 − 16q2 X1+ 1 · 1 X1− 2qY2(1 + X12) dX1 = 1 u1 · ϕ0du1,

for some ϕ0 that is regular at p1. Similar as above we write ϕ0= c0+Pi>0a0iui1 for some

constants c0 and a0_i. Filling in p1 gives c0 = ϕ0(p1) = 1. Indeed we find Resp1(f13ωq) = 1

and similarly Resp3(f13ωq) = −1. This suggests f13 is the function we are looking for,

but it is not yet normalised. Filling in p2 gives f13(p2) =

√

1−16q2

1−a(q) . So the function that

actually satisfies all conditions in lemma 2.1.1 is h13= p 1 − 16q2 x y + x− 1 1 − a(q) .

Expressing products in terms of a basis

As before, we will determine constants α, β, γ and δ in C[[q]] such that

h12h213− h212h13= αh12h13+ βh12+ γh13+ δ. (3.6)

Consider the points

r1 := ((a(q) : 1), (1 : 1))

r2 := ((1 : a(q)), (1 : 1))

r3 := ((1 : 1), (a(q) : 1))

r4 := ((1 : 1), (1 : a(q)))

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i h12(ri) h13(ri) 1 2q -1 2 1₂ -4q 3 2q p1 − 16q21 2 − 1 1−a(q) 4 1₂ p1 − 16q21 2 − 1 1−a(q)

Table 3.3.: The values of h12 and h13 in four points.

in in equation (3.6) gives a system of 4 equations in 4 unknowns that can be explicitly solved. The results are

α = −4q − 1 β = 0 γ = −q

δ = −q(1 + 4q).

Remaining coordinates

To find the last two coordinates in theorem 3.4.1, we still have to determine h12(p4) and

h14(p2). This is done in a similar way that h13(p2) was calculated above. A calculation

of h12(p4) is included below for the sake of completeness. The value of h14(p2) follows

from this by symmetry arguments.

Note that p4 ∈ U1 is given by (X1, Y2) = (−_a(q)1 , −1) and Y2+ 1 is a uniformiser at p4.

We can write

h12= η ·

a(q)X1+ 1

Y2+ 1

, (3.7)

around p4, for some rational function η that is regular at p4. Also, around p4 we

have X1 = Y2−pY22− 4q2(1 + Y22)2 2q(1 + Y2 2 , so we can write out

κ := a(q)X1+ 1

purely as a function of Y2. It can be seen that

κ(p4) = 0

∂κ ∂Y2

(p4) = 0.

Hence, κ = (Y2+ 1)2· ¯κ for some ¯κ that is regular at p4. Going back to equation (3.7)

we see that h12(p4) = 0. By symmetry, this implies h14(p2) = 0 too. This concludes

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4. The torus with four marked points

In [LP16] Yanki Lekili and Alexander Polishchuk proved a version of mirror symmetry for the n-punctured torus. In this chapter, some of their main results will be reviewed. We will try to make the ideas behind these theorems clear by sketching the ideas of the proofs and refer to their original article for the complete proofs. For our purposes, we will only treat the case n = 4. Let us begin by introducing the relevant notation.

4.1. Notation and main theorem

Define T to be the 2-torus and fix a symplectic form ω. We choose 4 points z1, . . . , z4 ∈ T

and denote T0 := T \{z1, . . . , z4}. Note that (T0, ω|T0) is an exact symplectic manifold,

so that we can choose a primitive θ for ω|T0. Using these data and an unoriented line

field on T , we can define the relative Fukaya category F (T, z1, . . . , z4). This is the model

Lekili and Polishchuk use on the A-side. On the B-side they define a family of curves T over C[[t1, . . . , t4]], called the 4-Tate curve, together with marked points σ1, . . . , σ4. The

main result of [LP16] is the following (see theorem A in their original article).

Theorem 4.1.1. The 4-punctured torus and the 4-Tate curve are mirror to each other in the sense that Dπ_{F (T, z}

1, . . . , z4) and Perf(T ) are equivalent as C[[t1, . . . , t4]]-linear

A∞-categories.

4.2. Intermediate results

First of all, we can choose θ in such a way that the Lagrangians L0, . . . , L4 from figure

4.1 are all exact and that furthermore τ (L0) is exact, where τ = τL1 ◦ · · · ◦ τL4 is the

composition of the Dehn twists along L1, L2, L3 and L4.1 If we view the torus as R2/Z2,

then these Lagrangians are given by

L0: = {(x, 0) | x ∈ R/Z}

Li: = {(n − i)/n, y) | y ∈ R/Z},

where i = 1, . . . , 4. They are all endowed with a non-trivial spin structure, as indicated by the stars in figure 4.1. The precise location of the zi does not matter as long as each

connected component of T \S4

j=0Lj contains exactly one of these points. Furthermore,

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Figure 4.1.: This figure depicts the exact Lagrangians L1, . . . , Ln. Note that in our case

n = 4. The stars on the Lagrangians indicate a non-trivial spin structure. Source: [LP16].

we choose the gradings on these Lagrangians, such that the unique intersection point between L0 and Li has degree zero, seen as a morphism L0 → Li.

As we are going to work a lot with Dehn twists of Lagrangians, it is useful to describe them in a different way. First we define that if L and L0 both have non-trivial spin-structures, τL(L0) has a non-trivial spin-structure as well. If L0has a trivial spin-structure

and L does not, then define τL(L0) to have a trivial spin-structure. The reason for

doing this, is so that the following proposition holds (see lemma 5.4 of [Abo06] for the proof).

Proposition 4.2.1. Let L and L0be two objects of F (T, z1, . . . , z4) that intersect

transver-sally and have a minimal amount of intersection points. Let x ∈ L∩L0 be an intersection point that has degree 1 as a morphism L → L0. Then τL(L0) and Cone(x) are

quasi-isomorphic as objects in TwF (T, z1, . . . , z4), hence isomorphic in DF (T, z1, . . . , z4).

Using this proposition, we can describe the relative Fukaya category of the 4-punctured torus using only a finite amount of data. To that extent we consider the following lemma, which is lemma 3.1.1. from [LP16].

Lemma 4.2.2. The Lagrangian branes L0, . . . , L4 defined above split generate the split

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Sketch of proof. From the theory of mapping class groups it is known that all exact Lagrangians can be obtained from Dehn twists along the curves underlying L0, . . . , L4

(see [FM12] for more on the theory of mapping class groups). We have seen that the Dehn twist of a Lagrangian corresponds to taking the cone of intersection points in proposition 4.2.1. Thus, all Lagrangian branes with non-trivial spin structure are obtained this way as twisted complexes in L0, . . . , L4. Now, let L be an object of the Fukaya category

with trivial spin-structure and denote by L0the object coming from the same underlying curve, but with non-trivial spin structure. Also, choose an object L00 with non-trivial spin structure that intersects L0 in a unique point. Then the Dehn twist (τL0 ◦ τ_L00)6

maps the curve L to itself (see [Iva02] lemma 4.1G). However the grading of L shifts after applying this Dehn twist, by lemma 5.9 of [Sei00]. Hence, we get (τL0 ◦ τ_L”)6(L) = L[2].

This implies that L[2] is quasi-isomorphic to the cone of a morphism ϕ : L → C in TwF (T, z1, . . . , z4), where C is some twisted complex in L0, . . . , L4. Write Cone(ϕ) =

(C ⊕ L, δ). Then there exist f : L[2] → (C ⊕ L, δ) and g : (C ⊕ L, δ) → L[2] such that d(f ) = d(g) = 0

µ(f, g) =1C⊕L+ d(α)

µ(g, f ) =1_L[2],

(4.1)

for some α : (C ⊕ L, δ) → (C ⊕ L, δ). Note that all endomorphisms of L lie in degree zero and one, so that

f =f1 0 g = g1 0 . Denote α =α11 α12 α21 α22 and define h :=−α21 g1 : C → L ⊕ L[2] k := ϕ f1 : L ⊕ L[2] → C.

Then writing out equations (4.1) and noting that µ(α21, f1) = 0 since it is a morphism

of degree -1 from L[2] → L, yields that h and k are quasi-inverse to each other. As C is an object in Tw{L0, . . . , L4}, we get that L ⊕ L[2] is in D{L0, . . . , L4}. Then

L ⊕ L[2], 1L 0 0 0

is an object in Dπ{L₀, . . . , L4} that is isomorphic to L. This

proves that objects with trivial spin-structure are also split-generated by the Li.

Now that we know the Lagrangians that determine the struture of the Fukaya cat-egory, we should study the morphisms between them. To this extent, we calculate HEndF (T,z1,...,z4)(L0⊕ · · · ⊕ L4) (compare with proposition 3.2.1. from [LP16]).

Lemma 4.2.3. There is an isomorphism of graded algebras

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Proof. All the Lj are homeomorphic to S1, and applying a Hamiltonian perturbation

yields two intersection points: one of degree 0 and one of degree 1. The differential is trivial, hence both points represent morphisms in cohomology. As mentiones above, there is a unique intersection point between Li and L0 for i = 1, . . . , 4, which gives

a degree zero morphism L0 → Li and hence a degree one morphism in the opposite

direction. This shows that as graded vector spaces HEndF (T,z1,...,z4)(L0⊕ · · · ⊕ L4) ∼=

E1,4⊗ C[[t1, t2, t3, t4]]. Of course, the composite morphism Li → L0 → Lj is zero for

i 6= j both not zero, as there are no morphisms Li → Lj. It can be easily seen that

there is a unique holomorphic triangle contributing to the product L0→ Li → L0, that

does not intersect any of the points zi. Hence, this product yields the unique degree

one endomorphism of L0 for all i. Equivalently, one can show that the composition

Li → L0 → Li equals the unique degree one endomorphism of Li. This proves the

desired isomorphism of graded algebras.

The final result that we want to mention is one that is less explicitly stated in [LP16]. It gives us a way to interpret the Dehn twist of Lagrangian with an object on the B-side. The equivalence of categories mentioned in theorem 4.1.1 identifies

L0↔ OT

Li↔ Oσi.

So the question remains, what we want to identify τ (L0) with on the B-side. Hereto, we

prove the following lemma.

Lemma 4.2.4. Under the equivalance of A∞-categories from theorem 4.1.1, the object

τ (L0) gets identified with an object isomorphic to OT(D) and τ2(L0) with an object

isomorphic to OT(2D), where D = σ1+ · · · + σ4.

Sketch of proof. Denote O := OT. We start by proving that τLi(L0) corresponds to an

object isomorphic to O(σi). Consider the short exact sequence of OT-modules

0 → O → O(σi) → Oσi → 0.

This sequence realises O(σi) as an extension of Oσi by O and hence defines a degree one

morphism ϕ ∈ Ext1(Oσi, O), whose cone is isomorphic to O(σi).

2 _{As O}

σi corresponds

with Li and O with L0, we get that the cone of ϕ corresponds to the cone of a degree

one morphism Li→ L0. By proposition 4.2.1 this is isomorphic to τLi(L0) in the derived

relative Fukaya category. Hence, up to isomorphism, τLi(L0) and O(σi) get identified

with one another. We can proceed by considering the short exact sequence 0 → O(σi) → O(σi+ σj) → Oσj → 0,

to prove that O(σi+ σj) corresponds to τLj ◦ τLi(L0) up to isomorphism. The desired

result follows by repeating these steps.

Homological Mirror Symmetry for the 4-punctured torus

MSc Mathematics

Master Thesis