Koszul resolution

We begin by reviewing the Koszul resolution. Suppose that is a smooth variety and is a holomorphic vector bundle of rank with a holomorphic section . Let denote the zero-set of the section and suppose this is a smooth subvariety of codimension . We have the short exact sequence of sheaves

where denotes the sheaf on whose sections over are the sections of the structure sheaf of over and where denotes the ideal sheaf of , whose sections over an open set are just -multiples of . There is a morphism (where denotes the sheaf of sections of the dual vector bundle to ) given by sending to (‘interior product’). The image of this morphism is precisely the ideal sheaf of . We can continue to the left as follows:

where each map contracts a ‘-form’ in with the section . It’s an easy exercise to check that this is an exact sequence. Each term from leftwards is a locally free sheaf, i.e. a holomorphic vector bundle. This is called the Koszul resolution of the sheaf .

Beilinson resolution of the diagonal

For the case we are interested in we take and , the diagonal. We need to write this as the zero-set of a transversely vanishing section of some bundle. Consider as the space of complex lines in a vector space and recall that the line bundle has fibre over equal to the line . The dual bundle has as global sections . Remember the picture of the tangent bundle to the 2-sphere, where the tangent space at an oriented line is the quotient of (in this case ) by ? This translates into our sheafy language as an exact sequence (the Euler sequence)

where is the tangent bundle to . We will take as our bundle

on . Pick a basis of coordinates on (considered as sections of ) and let denote the corresponding vector fields (considered as sections of $T(-1)$). Then

is a section of and doesn’t depend on the choice of basis (here and are the corresponding coordinates on the first and second factors). The key observation now is that the vanishing set of the section is precisely the diagonal: . With a bit of thought this follows straight from the Euler sequence above.

The corresponding Koszul resolution above is called the Beilinson resolution of the diagonal.

Derived category of

We will now make use of the various operations one has at hand in the derived category (derived tensor product, pushforward, pullback) and the following projection formula

where is a morphism of varieties and are elements of the derived categories of respectively.

For an element we define the transform of to be . By the projection formula,

since by symmetry . But the Beilinson resolution gives us short exact sequences (which we interpret as triangles in the derived category)

…

Therefore is in the triangulated subcategory generated by for . However,

and so

which gives (using the projection formula)

However, pushing forward along is just the derived pushforward to a point, i.e. taking cohomology. Therefore we get . Therefore the bundles are generators for the derived category (doing this the other way around (i.e. reversing the roles of and ) would give as generators).

Derived category of smooth projective variety

We will now show [Seidel, Lemma 5.4] that if is a generating collection for the derived category of then the derived pullbacks split-generate the derived category of an embedded smooth projective variety . First notice that you can use the generators to get for all (do it ambiently then pull back). Let me spout some algebraic geometry: a if is a locally free coherent sheaf on a projective variety over a Noetherian ring then is generated by global sections for some and the kernel of the projection map (given by picking a basis of global sections) is again coherent. One can therefore inductively form a resolution

for arbitrarily large. When , but this group is the only obstruction to finding a splitting of such an exact sequence. Therefore is isomorphic to in the derived category and one can split generate any locally free coherent sheaf. Now the locally free coherent sheaves generate the whole derived category, which is itself split-closed (two facts we won’t trouble ourselves with).

[References to Seidel's paper will henceforth be made in square brackets.]

What is mirror symmetry? Mirror symmetry is a wide-ranging web of mathematical conjectures motivated by physics. Roughly speaking it posits the existence of ‘mirror pairs’ , of manifolds equipped with geometrical structures and such that certain geometrical invariants of are related to (different) geometrical invariants of . The predictions made by mirror symmetry have been resoundingly confirmed in many examples.

More specifically? The manifolds and are Kähler manifolds. This means that there are two compatible geometric structures: a symplectic structure (a nondegenerate, closed 2-form) and a complex structure (an endomorphism of with , arising from an atlas of complex charts with holomorphic transition functions). Compatibility means that the 2-tensor is a positive-definite metric for which the endomorphism is orthogonal.

Mirror symmetry relates the symplectic geometry of with the complex geometry of (and vice versa). In particular it relates the “Gromov-Witten invariants” of (which are symplectic invariants, counting pseudoholomorphic curves in different homology classes passing through a collection of cycles) to certain “Yukawa couplings” determined by the Hodge theory of .

Homological mirror symmetry is one extremely promising refinement of the mirror symmetry conjecture due to Kontsevich (http://arxiv.org/abs/alg-geom/9411018) which asserts that, for a mirror pair, the split-closed derived Fukaya category of is equivalent to the derived category of coherent sheaves on . These two categories are complicated algebraic gadgets cooked up from the symplectic (respectively complex) geometry of (respectively ). The “closed string” version of the conjecture (in terms of Gromov-Witten invariants and Yukawa couplings) should be recovered from this categorical version by taking the Hochschild cohomology of the category (the Hochschild cohomology of the Fukaya category is supposed to give us the quantum cohomology of which encodes the information about the Gromov-Witten invariants).

SYZ. Of course, the conjecture in this form asserts nothing at all until you specify some mirror pairs. Another version of mirror symmetry (known as SYZ after its conjecturers, Strominger-Yau-Zaslow, later developed by Gross-Siebert and others) gives a conjectural construction of out of by dualising a ‘special Lagrangian torus fibration’ (part of the conjecture is that such fibrations exist (at least in large parts of the manifolds)).

In light of the general picture I have barely even sketched, the aims of Seidel’s paper are relatively modest: to verify the conjecture in the case when is a quartic K3 surface. To me, what is impressive about this paper is the array of ideas in symplectic geometry and techniques for working with the Fukaya category that Seidel had to develop in order to give his proof. These ideas and techniques pervade much of modern symplectic geometry and it is for this reason, rather than a consuming desire to understand mirror symmetry, that I want to read this paper. With that introduction, let me say something about K3 surfaces.

K3 surfaces: For us a K3 surface is a simply-connected complex surface with first Chern class zero.

The following examples can be checked by the adjunction formula: quartic surfaces in , complete intersections of a quadric and a cubic 3-fold in or of three quadric 4-folds in . These are somehow atypical, because most K3 surfaces are not projective varieties: of the 20-(complex)-dimensional moduli space of complex K3 surfaces, only a codimension 1 submoduli space consists of projective K3s. However, all K3s are Kähler, and moreover they all admit hyperKähler metrics. A hyperKähler metric admits a triple of three compatible complex structures I, J, K which obey commutation relations like the quaternions. These give rise to Kähler forms . With respect to , a - or -holomorphic curve is special Lagrangian. Since there are certainly K3 surfaces which are fibred by holomorphic tori (elliptically fibred K3s) there are K3s which admit the kind of special Lagrangian torus fibrations required by SYZ mirror symmetry, so things look to be in good shape.

We are interested in the symplectic manifold underlying the projective quartic surface (note that this is uniquely characterised by its volume: symplectic parallel transport in the family of quartic surfaces allows us to identify different quartics symplectically). This symplectic manifold we will denote by .

Exercise: Is this symplectomorphic to one of the other complete intersection K3s described above?

Some combination of SYZ/physics wizardry allows Seidel to guess the mirror. To define the mirror he introduces a strange coefficient ring (the rational Novikov field) and defines the mirror as a quartic surface in . The field consists of formal series

where is some integer (different are allowed for different formal series) and for all sufficiently large . The mirror quartic is the minimal resolution of the orbifold

where is the group of 4-by-4 diagonal matrices where and . I’ll explain more about minimal resolutions of orbifolds in a later blog post (about [Section 5]).

This mirror looks misleadingly non-complex! Of course, forgetting the algebraic nonsense and just looking at the equation, it is immediately apparent that it just describes a family of complex quartics. The choice of ground field actually makes a lot of sense: the use of formal power series is natural from the symplectic points of view, where we need a Novikov ring to encode areas of discs. The q-parameter we introduce corresponds to rescaling the symplectic form – remember the quartic is only determined symplectically once we know its volume! So we have a FAMILY of symplectic quartic surfaces , where is a formal parameter. Mirror to that we need a family of complex manifolds over the formal punctured disc. That’s what this coefficient field deals with: it’s just the (algebraic closure of the) localisation of at 0. Since you always have a map from a variety to Spec of the coefficient field we can think of Seidel’s mirror as a family over the (universal cover of the) formal punctured disc.

Something which confuses me is that the the limit as goes to infinity is generally called the ‘large radius limit’ and is supposed to correspond to the ‘large complex structure limit’ under mirror symmetry. But a more natural candidate for the latter seems to be the limit, which is a union of coordinate hyperplanes… any explanations would be welcome!

The ‘mirror map’ identifying the Novikov parameter q with the coordinate in the base of this family is not specified by Seidel: he shows that there is some change of q-coordinates for which mirror symmetry holds but not what the change of coordinates is. This is an artefact of his proof: he calculates the two sides of the mirror and shows that for general algebraic reasons they are isomorphic without giving an isomorphism. Presumably this would follow from a deeper geometrical understanding of mirror symmetry. There is even a conjectural candidate for the mirror map which is a power series encoding dimensions of representations of the Monster group (see for example http://arxiv.org/abs/1101.4601).

We see that our approach will necessarily be highly phenomenological! We will not gain any deep understanding of why mirror symmetry holds, but we will see that it does hold in this case.

Proving HMS. Let’s now dissect what it means for homological mirror symmetry to hold. What is the split-closed derived Fukaya category? What is the derived category of coherent sheaves? How do we calculate with these things or show they are the same?

Triangulated categories. What is a derived category? Deriving is something you do to one category to produce a triangulated category. On the complex side you start with a dg-category of vector bundles (which is trivially an category with zero higher compositions), on the symplectic side you start with the Fukaya category. The process of taking the derived category goes by looking at a category of twisted complexes and then taking homology. This is showed in [Section 5] to recover the usual “inverting quiso” approach to the derived category of coherent sheaves.

Fukaya category. We will postpone any discussion of the complex side (‘B-model’) for now and just talk about the Fukaya category. The Fukaya category takes as its objects a collection of Lagrangian submanifolds of . Recall that these are submanifolds on which the symplectic form restricts to zero. These objects are supposed to be the boundary conditions of strings in string theory.

Motivation from physics: The whole setting of mirror symmetry is superconformal field theory (whatever that might mean). Open strings need boundary conditions and in order for the corresponding sigma-model to have supersymmetry these boundary conditions need to satisfy certain conditions themselves. In the case of the A-model, we need our boundary conditions to be special Lagrangian. Explicit special Lagrangian submanifolds are sometimes hard to come-by (they are the solutions of a highly nonlinear variational problem), though in the case of K3s they can be constructed using the hyperKähler structure as above. In general, symplectic geometers like to avoid solving nonlinear problems other than pseudoholomorphic curves so in keeping with this general philosophy we will consider a wider class of Lagrangian submanifolds (discussed below). We still won’t consider all Lagrangians, but then physicists never said we could.

Motivation from Floer theory: Floer theory tells us that Lagrangians are ‘natural candidates for the endpoints of strings’: Floer theory, roughly speaking, takes the space of paths between two Lagrangians and looks at the Morse theory of a certain action (the symplectic action of paths). The critical points are constant paths (necessarily at intersections between the Lagrangians) and the gradient trajectories trace out pseudoholomorphic strips with boundary on the Lagrangians. Gromov’s theory of pseudoholomorphic curves tells us that Lagrangians provide very natural boundary conditions for pseudoholomorphic strips ( being totally real for some almost complex structure means that the equation for such strips is a nonlinear Fredholm problem; being totally real for all -compatible almost complex structures is the same as being Lagrangian).

Fukaya et al (http://www.math.kyoto-u.ac.jp/~fukaya/fooo.dvi) introduced an algebro-combinatorial gadget which encodes the data of Lagrangian intersections and the connecting Floer trajectories, called a filtered -category. Fortunately for us, the ‘filtered’ part of this theory is redundant in the case of K3 surfaces (for elementary dimension-counting reasons) as we will see in [Lemma 8.3]. To each object of the Fukaya category we will associate an -algebra and to each pair we associate a morphism space which is an -bimodule – Luis will talk about the underlying algebra next time.

In fact the objects are more complicated than just Lagrangians: an object is an oriented Lagrangian equipped with extra data:

• A ‘grading’ on .
• A spin structure on .
• A “covariantly constant multisection” of of some degree .
• A choice of almost complex structure for which there are no -holomorphic discs with boundary on .

These data require some explanation.

Gradings. On a symplectic -manifold with one can find a nonvanishing volume form of type with respect to some compatible almost complex . This defines a circle-valued phase function  on the Lagrangian Grassmannian bundle (i.e. the bundle over whose fibre at consists of the Lagrangian planes in )

where is a basis of . For example, take the torus and the volume form . This gives the circle of slope a phase of . A grading is a lift of to a real-valued function.

The shift in the triangulated category we get by deriving the Fukaya category will be a shift in the choice of grading. Notice that the existence of a grading imposes restrictions on . For example, if is a disc with boundary on then one can define the Maslov index of the disc to be the winding number of the phase restricted to the boundary of the disc. If there exists a grading then this winding number must vanish because the restriction of the phase to the boundary of the disc must factor through the universal cover of . Hence all discs with boundary on a graded Lagrangian have Maslov index zero.

Note that special Lagrangian submanifolds are precisely those for which the phase function is identically equal t0 1 (in particular they have a grading). Therefore this is a good topological notion with which replace the more stringent special Lagrangian condition.

Spin structure. A spin structure is just a topological choice required to coherently orient moduli spaces of Floer trajectories. The less said, the better (for the moment).

The covariantly constant multisection is a bit of a mystery to me still to be honest,  but it seems to be used to say something about the symplectic action. To make it less of a mystery, we will see in [Section 6] that if we cut an ample divisor out of , leaving behind an affine quartic with an exact symplectic form , and consider an exact Lagrangian (i.e. one for which for some function ) then we can cook up a natural covariantly constant section out of and . So having such a multisection is some kind of ‘exactness’ condition (though the ambient space is projective, hence not exact). It’s called rationality of the Lagrangian.

The non-existence of pseudoholomorphic discs for a generic almost complex structure gives a crucial simplification of the Floer theory. Why is it true? The expected dimension for the moduli space of Maslov discs with boundary on an -dimensional Lagrangian is

For our (2-dimensional) Lagrangians we know that all discs have Maslov index zero (by the existence of a grading) so this expected dimension is -1. Transversality can be achieved in the usual way for somewhere-injective discs (by perturbing the almost complex structure at an injective point) so we can generically ensure there are no somewhere-injective discs. For such a generic almost complex structure, if there is any (possibly nowhere injective) holomorphic disc then Kwon-Oh and Lazzarini have shown that one can decompose it into subdiscs which are multiple covers of somewhere injective discs. But there are no somewhere injective discs! So there are no discs at all, for a generic choice of .

Morphisms. We will discuss morphism spaces properly when we come to them. Suffice it to say that the morphism space from a to is the vector space on the intersection points . There are also operations defined in terms of moduli spaces of pseudoholomorphic strips.

Split-closed, derived. Next we observe that this category is much much too big to be of any use to us. There is no symplectic manifold of dimension 4 or more where we know all the Lagrangians and for the quartic we know very little indeed (except that there are lots of Lagrangians). To make the category more tractable we take the split-closed derived category. ‘Derived’ roughly means that we add cones of morphisms to make it into a triangulated category. Split-closed means that we take as our objects the pairs where is an object of the derived category and is an idempotent morphism, i.e. . We can think of as a ‘summand’ of L, where I is projection onto the summand. Why do we do this? Pragmatically, the derived category of coherent sheaves turns out to be split-closed (i.e. equivalent to its split-closure), so our candidate mirror must also be split-closed. In certain examples (for example mirror symmetry for the 4-torus, http://arxiv.org/abs/hep-th/0109098) the objects introduced by split-closure actually account for a discrepancy between the K-theory of the derived category and the Fukaya category which was noticed by Kapustin and Orlov (http://arxiv.org/abs/hep-th/0109098) and attributed to the presence of extra geometric objects (coisotropic branes) in the Fukaya category. To my knowledge the precise geometric interpretation of split-closure is still unclear.

Deriving seems more reasonable for the following amazing reason (Seidel’s long exact sequence http://arxiv.org/abs/math/0105186): if and are two Lagrangians and is a sphere then the Lagrangian obtained by Dehn twisting around behaves Floer-theoretically like a cone. This means that the Floer homology of with any other Lagrangian fits into a long exact sequence

where the maps are defined by counts of pseudoholomorphic polygons with boundary on the Lagrangians in question. Cones on other morphisms don’t necessarily have the same nice geometric interpretation but at least there’s now a good motivation for enlarging the Fukaya category to make it triangulated.

Having thus enlarged the Fukaya category, it becomes easier to deal with. Most crucially we have a hope of ‘generating’ it with a finite amount of data. For example, the category of coherent sheaves of a smooth n-dimensional complex projective variety is something big and intractable, but its derived category is actually ‘generated’ by a finite collection of vector bundles (Beilinson, http://www.springerlink.com/content/n150p08171g063rk/). Here, we say a collection of objects of a split-closed triangulated category generate if the smallest split-closed triangulated subcategory of containing this collection.

Seidel gives a criterion [Lemma 9.2] for when a collection of Lagrangian spheres generate the whole (split-closed derived) Fukaya category of , an affine variety, in our case a quartic surface minus a genus 3 curve (hyperplane section). Suppose is a collection of Lagrangian spheres in such that the composition of (graded) Dehn twists in these spheres is negative (when extended to the projective variety X). Then these spheres generate the Fukaya category. Here, one can define a notion of grading for symplectomorphisms of Calabi-Yau symplectic manifolds like we did for Lagrangians: the derivative acts on the Lagrangian Grassmannian and one can define a phase which measures how much the result twists around the Maslov cycle. A grading is a lift of this to . The Dehn twist in a graded Lagrangian is naturally graded. When X itself is Calabi-Yau (like K3), not only does the symplectomorphism of M extend to X, but one can find a canonical grading on the extension. That the extension of the composition is negative means that for some integer , is negative.

This begs two questions:

• Why do we care about affine quartics?
• Is there a candidate collection of graded Lagrangians with negative monodromy?

Projective and affine. I should first say that the split-generation argument also works for the projective case (see [Section 9c]). However, in the affine case we have the advantage that we can explicitly compute the subcategory of the Fukaya category generated by the Lagrangian spheres in question. The relation between the projective case we’re interested in and the affine case we can compute is another of Seidel’s beautiful insights.

The Fukaya category of the projective variety is a deformation of the Fukaya category of the affine part .

This has a nice geometric interpretation. Remember that the q-parameter was introduced in the Fukaya category to compensate that we might have infinite sequences of holomorphic discs with boundary on a Lagrangian, with increasing area (note that if the area is bounded then by Gromov compactness we get finite counts if the expected dimension is zero and the almost complex structure is chosen generically). Another way to deal with this discs is to excise an ample divisor $latex D$ (whose Poincare dual is cohomologous to a multiple of the symplectic form) from : in our case, a hyperplane section. Let be an exact Lagrangian in the complement of the divisor. In the affine part, there can be no holomorphic discs since is exact. That means that all discs intersect . Moreover, by Poincare duality if a disc intersects with multiplicity then it has area .

This means that the subcategory of consisting of Lagrangians which are exact in the affine part is a deformation of the Fukaya category of exact Lagrangians in the affine part. But we will see below that there is a collection of such exact Lagrangians which split-generate both derived categories by the split-generation criterion! (There seems to be a technical reason, which I don’t yet appreciate, why the split-generation only works for the category of rational graded Lagrangians in .)

In the case in hand we will see [Corollary 8.21] that the deformation is nontrivial. This comes from a geometric phenomenon called wall-crossing. Recall that we could generically achieve a situation in which there were no holomorphic discs with boundary on our graded Lagrangians. Since the expected dimension for these discs is -1 we expect that in 1-parameter families of almost complex structures we will find isolated t for which there is a -holomorphic disc. It’s like crossing over a wall in the space of almost complex structures for which the -structure undergoes a quasiisomorphism, hence it’s called wall-crossing. In [Section 8] we will see why the occurrence of this phenomenon implies nontriviality of the deformation and exhibit a Lagrangian for which wall-crossing occurs.

Negative monodromy. Take a quartic 3-fold in $\mathbb{CP}^4$ and a generic (Lefschetz) pencil of hyperplanes . The hyperplanes cut out quartic K3 surfaces with a finite number of p giving singular quartics (the genericity assumption ensures that these have at worst one node). All the hyperplanes intersect along a which intersects each K3 in a common hyperplane section (the base locus), i.e. a genus 3 curve . If we excise a smooth member (WLOG of the family then we have a family of affine K3s () over . One can define symplectic parallel transport in this family (away from the singular fibres) and make sense of symplectic monodromy around any loop in which doesn’t hit any p for which is singular (critical points). In particular, you could take a large radius circle encircling all the critical points. It is well-known that this monodromy is isotopic to the identity (because it can dually be seen as the monodromy around the point at infinity) and that it can be expressed as a product of Dehn twists around a collection of Lagrangian spheres (vanishing cycles) coming from the singularities of the pencil. There is a graded version of this story [Lemma 6.11] which tells us that the global monodromy is actually a negative grading shift by degree 2. [Chapter 7] is then dedicated to the case when is not smooth but has normal crossings, the result [Proposition 7.22] being that the global monodromy is (as good as) negative (for the purposes of proving split-generation by vanishing cycles).

Why do we need this generalisation? Because Seidel uses the highly degenerate quartic at infinity. There are 64 other vanishing cycles and the subcategory of the Fukaya category they span is something we can compute. There is something magical about this choice.

Magic: There is a unique deformation of (up to automorphisms of the Novikov field). We denote it by .

Now sits inside and split-generates. The following lemma then implies that is equivalent to

[Lemma 2.4]: If A’ in A is a subcategory which split generates DA then DA’ and DA are quasiequivalent.

A last comment on this part of the story: to compute itself is a lot of hard work and not something we will focus on (mainly because the calculations seem to have been computerised and arduous). The technique is to represent the vanishing cycles as matching cycles for a pencil of (affine) genus 3 curves on the affine quartic and then reduce the computation to a combinatorial count of polygons on these. The clever part is contained in [Remark 10.2]: you only have to do finitely many calculations to determine the whole -algebra: you need to show that there are only two possible -structures on a particular vector space and one of them is formal (no higher products). Therefore if you can compute the homology of your -algebra and find a nonvanishing Massey product, you know which -algebra it is!

To prove mirror symmetry it suffices to find a collection of coherent sheaves on the member of the mirror family (which is a highly singular quartic) which span a subcategory equivalent to which split-generate. The fact that we’re working over the funny Novikov field means that our mirror category is actually a deformation of this (see [Sections 5 and 10]) and in [Lemma 10.9] we’ll see it’s a nontrivial deformation. By the magic we observed before, this tells us that the two categories are equivalent via a change of q-coordinate, which is Seidel’s mirror theorem.

1. Richard Thomas’s notes on derived categories: http://arxiv.org/abs/math/0001045
2. Andrei Caldararu’s notes on derived categories: http://arxiv.org/abs/math/0501094