The homological projective dual of Sym^2 P(V)
We study the derived category of a complete intersection of bilinear divisors in the orbifold . Our results are in the spirit of Kuznetsov’s theory of homological projective duality, and we describe a homological projective duality relation between and a category of modules over a sheaf of Clifford algebras on .
The proof follows a recently developed strategy combining variation of GIT stability and categories of global matrix factorisations. We begin by translating into a derived category of factorisations on an LG model, and then apply VGIT to obtain a birational LG model. Finally, we interpret the derived factorisation category of the new LG model as a Clifford module category.
In some cases we can compute this Clifford module category as the derived category of a variety. As a corollary we get a new proof of a result of Hosono and Takagi, which says that a certain pair of nonbirational Calabi–Yau 3-folds have equivalent derived categories.
Let be a vector space, let be the quotient stack , and let be the morphism given by
Choose a vector subspace . We then get an orthogonal subspace . The main goal of this paper is to understand the derived category of the stack .
Our first result relates this category to a category of modules over a sheaf of Clifford algebras. We will define a certain -gerbe, , equipped with a locally free sheaf , whose rank is , and a section of . From this data we define a sheaf of Clifford algebras , where is the tensor algebra and is the 2-sided ideal generated by .
Let be the restriction of to , and keep the notation for the restriction . There is a derived category , whose objects are bounded complexes of coherent -modules. For such a complex and a point , the restriction is an -equivariant complex of sheaves on , hence splits as a shifted sum of -representations. We will define a subcategory of grade restricted objects, where is grade restricted if for all , only certain specified representations occur in the splitting of .
Let . We say has the expected dimension if its codimension in equals the codimension of in . Our first result is:
If has the expected dimension and is odd, then:
If , there is a fully faithful functor .
If , there is an equivalence .
If , there is a fully faithful functor .
If has the expected dimension and is even, then:
If , there is a fully faithful functor .
If , there is a non-trivial triangulated category which is a fully faithful subcategory of both and .
If , there is a fully faithful functor .
Explicit descriptions of the fully faithful functors and the subcategory will be given in the course of the proof. In the cases where includes into or when there is a subcategory common to both of them, Proposition 5.16 gives a description of the semiorthogonal complement to or inside .
Our second result is that for certain choices of , we can give a more geometric description of the category . The description will depend on the parity of .
Assume first that is odd. Interpreting the points of as symmetric matrices up to scale, we may stratify the space by the ranks of these matrices. We assume that does not intersect the locus of matrices of corank , and that the intersection of with the locus of corank matrices is nonsingular of the expected dimension for . This assumption holds for a general of dimension .
We define a nonsingular variety as a double cover of the corank 1 locus in , ramified in the corank 2 locus. At a corank 1 point , the 2 points of the fibre correspond to the 2 connected components of the variety of maximal isotropic subspaces of the quadratic space ; letting this description hold in families of in the natural way determines up to isomorphism (using e.g. Lemma 8.28).
Under the assumptions above, .
If we take with generic, then and are nonsingular Calabi–Yau 3-folds. Combining Theorem 1.1 and Proposition 1.2 gives . This result has been shown previously by Hosono and Takagi [HT13], using completely different methods. One interesting feature of this example is that and have fundamental groups and , respectively, hence are not birational.
Assume now that is even, that does not intersect the locus of matrices of corank , and that the intersection of with the locus of corank matrices is nonsingular of the expected dimension for . This assumption holds for a general of dimension . Define the variety as the double cover of the corank 0 locus in , ramified in the corank 1 locus.
Under the assumptions above, .
1.1 Proof of Theorem 1.1
The strategy of the proof will be to combine categories of matrix factorisations with variation of GIT stability. This approach was first described in [Seg11], inspired by the physics paper [HHP08]. See also [BDF13, ADS14, FK14] for other applications of this strategy.
Categories of matrix factorisations will be properly introduced in Section 3. For now it is enough to know that given a stack equipped with a function and some extra data, one can define the category of factorisations , which is a generalisation of the usual derived category .
The diagram below summarises the strategy.
The first step is to replace the category by an equivalent category of matrix factorisations. Let be the -equivariant sheaf on , equipped with the -action which leaves the restriction of to the diagonal of fixed. Then is cut out by a section of , where .
The space is a GIT quotient for a quotient stack , where . We show that there is a full “window” subcategory such that composing with the restriction functor we get an equivalence .
Having translated into a window category, we next cross the GIT wall.
Let be the projection. We can find a such that . The functor is then an equivalence, which restricts to give . This is Proposition 7.1.
We thus have
Theorem 1.1 now follows from these equivalences, because in each case it is obvious from the definitions that and are either contained one in the other as subcategories of or have non-trivial.
1.2 Related works
This project began as an attempt to understand and generalise Hosono and Takagi’s work in [HT13], which treats the special case where . They find an equivalence between two Calabi–Yau 3-folds and , and also conjecture that this equivalence generalises to a statement in homological projective duality. Ingalls and Kuznetsov have studied the case where in [IK15].
Our main theorem is inspired by Kuznetsov’s description of the derived categories of intersections of quadrics in terms of even Clifford algebras [Kuz08], which we informally recall in Section 2.4. Our category of Clifford modules is different from the one in that paper in two important ways. Firstly, our sheaf of Clifford algebras does not live on , but rather on the -gerbe . In particular, a module over is locally an -equivariant sheaf on . Secondly, the need to consider the subcategory of grade restricted modules is new to our case. Both of these features mean that the description in terms of Clifford modules is less useful than in the quadric case, and in proving Propositions 1.2 and 1.3 we work mostly with the equivalent category instead of with .
As the title indicates, our results are motivated by Kuznetsov’s theory of homological projective duality [Kuz07]. Our Theorem 1.1 is close to saying that the category is the homological projective dual of . We will explain this statement further in Section 2, which also contains background on homological projective duality.
As explained above, a crucial step in the proof of Theorem 1.1 is to relate the categories of factorisations on different GIT quotients. The techniques for doing this were introduced in this context by Segal in [Seg11], and have since been worked out in great generality by Ballard, Favero and Katzarkov [BFK12], and by Halpern-Leistner [HL15]. The main result of these two papers is that if is a GIT quotient, then there exists a full subcategory such that the restriction functor gives an equivalence . When is equipped with a superpotential , it is shown in [BFK12] that same results hold for factorisation categories, i.e. there is a such that .
To define , one first writes down a sequence of 1-parameter subgroups and a sequence of open subvarieties of the fix point loci . For any (or ), the restriction is then graded by -weights, and we define by saying if the -weights of are contained in a certain interval for all . Unfortunately, the precise results of [BFK12, HL15] are not applicable in our case, as for our GIT quotients , the subcategories of constructed by these papers are not comparable in the way we want. See Section 5.4 for a further discussion of this point.
We remedy this by giving an ad hoc definition of the subcategory . Since we only consider a quotient of an affine space, the technical details are considerably simpler than in the general case, and modifying the arguments of [BFK12, HL15] allows us to give a direct proof of the equivalence . A novel feature of our case is that it is necessary to consider weights with respect to a 2-dimensional subtorus of our group , instead of just to 1-parameter subgroups. The definition of the category follows [BFK12, HL15].
As mentioned above, the overall strategy of our proof has been applied successfully to several examples, beginning with [Seg11, Shi12]. Producing homological projective duals by this method was carried out in certain cases by Ballard et al. in [BDF13]. They apply this further to the example of degree hypersurfaces in [BDF14], in particular recovering Kuznetsov’s quadric example [Kuz08]. Our proof of the equivalence between the factorisation category and the Clifford module category goes along the same lines as parts of their proof in the case . See also [Dyc11], where a similar equivalence is shown for a single Clifford algebra.
The overall VGIT/LG model approach is also used in Addington, Donovan and Segal’s paper [ADS14], which reproves the Pfaffian–Grassmannian equivalence of Calabi–Yau 3-folds from [BC09, Kuz06]. The fact that we need to take a good subcategory has a parallel in their paper, as one of their gauged LG models is also an Artin stack. They speculate that this category corresponds to what physicists call the category of branes in an associated -model [ADS14, 4.1]. At present the choice of this subcategory is rather ad hoc, and it will be interesting to see to what extent it can be made in a general way.
We work over .
For objects in a triangulated category , we use the convention that is the space of maps in and is the graded space .
If is an algebraic group acting on and is a representation of , we write for the -equivariant sheaf . If is a -dimensional torus, we denote by the line bundle associated with the character . Finally, if , we write for , where is the character of which is on and which sends the generator of to .
This paper is a modified version of my Ph.D. thesis. I am very grateful to my Ph.D. supervisor Richard Thomas for suggesting this topic, for many interesting conversations, and for all the help, advice and encouragement he has provided over four years.
Thanks also to E. Segal, who explained to me many of the ideas and tools used here. I thank N. Addington, T. Bridgeland, T. Coates, D. Halpern-Leistner, K. Hori, S. Hosono, A. Kuznetsov and T. Pantev for useful conversations; Addington also jointly with Richard suggested the topic.
I thank M. Akhtar for thoroughly proofreading the thesis version of this paper; whatever mistakes or typos remain are of course entirely his fault.
2 Homological projective duality
Theorem 1.1 is motivated by Kuznetsov’s theory of homological projective duality, which we explain in this section. We first present the general definitions and results of the theory, taken from [Kuz07]. Next we discuss the example of HP duality for quadric hypersurfaces in . Finally we explain how our results are a form of HP duality for bilinear divisors in .
Note that the proofs of our propositions do not depend on the general results of HP duality, and so logically speaking this section is independent from the rest of the paper.
2.1 The base locus and the incidence variety
As a warm-up, we first treat a simple version of HP duality where the derived category results are clear from the geometry. Let be a smooth, projective variety with a morphism for some vector space , with not factoring through any linear subspace of , and let . Choose a linear subspace , which gives a linear system of divisors of class .
There are two natural schemes we can construct from this linear system. Firstly, we can intersect the divisors in the linear system to get the base locus . Secondly, we can construct the incidence variety , which consists of pairs such that .
Let us assume that has the expected dimension. The first step towards HP duality is the observation that then includes as a full subcategory of .
Consider first the case where .
Then is the blowup of in , and by [BO95, 3.4] we get a semiorthogonal decomposition
More generally, if , , then the projection has fibres over , which jump to over . This gives a semiorthogonal decomposition of with 1 piece isomorphic to and pieces isomorphic to . In general, the inclusion functor is given by with and as in the diagram
2.2 Lefschetz decompositions
Kuznetsov’s remarkable discovery is that this relation between the base locus and the universal hyperplane can be turned into something more interesting if we can put a certain extra structure on . Namely, assume that the derived category admits a semiorthogonal decomposition
where the are full subcategories of satisfying for all , and where denotes the full subcategory whose objects are , . Such a decomposition is called a Lefschetz decomposition.
For any hyperplane inducing a divisor , the functor
is full and faithful for . Furthermore, the image subcategories are semiorthogonal. Both of these facts are easy to show using our assumptions on and the exact triangle
We therefore have a full subcategory , and letting , we get a semiorthogonal decomposition
We see that decomposes into the parts inherited from , and the one new part . This motivates the term “Lefschetz decomposition”, cf. the Lefschetz hyperplane theorem.
More generally, let be a linear subspace, let , and let , which is the base locus of the linear system . We then get a semiorthogonal decomposition
One way of summarising HP duality is that if we know the category for all hyperplanes in the system , then we get a description of the category , in terms of the “homological projective dual” variety, which we now describe.
2.3 The homological projective dual
Let be a variety equipped with a map , and assume that for every point the fibre satisfies .
If satisfies a certain strengthening of this condition,
Here is analogous to the incidence variety , with the difference that the “categorical fibre” at each is now the interesting part rather than the whole of .
For any , let . Just as we saw above that includes into , we can now include into .
To state the precise result, we will need some notation. Let . Kuznetsov shows that admits a “dual” Lefschetz decomposition
where for . Let and be the dimension and codimension of , respectively.
Theorem 2.1 ([Kuz07], Thm. 1.1).
If and have the expected dimensions, then we have semiorthogonal decompositions
The most striking consequence of this theorem is that and have the semiorthogonal piece in common. The functor is obtained by composing the functor with a certain projection .
If the dimension of is sufficiently low (resp. high) we get a fully faithful inclusion (resp. ).
As an aside, we note that the notion of HP dual makes sense more generally than in the setting described here. In particular, one can drop the restriction of considering derived categories of varieties, and instead consider more general triangulated categories, linear over and . For some nice such categories the same results can be shown. The main results of this paper deal with HP duality in this extended sense; see also [BDF13] and the next section.
2.4 HP duality for quadrics
We will now explain the results of HP duality for the case of quadric hypersurfaces, worked out by Kuznetsov in [Kuz08]. This is both an instructive example of HP duality in general and formally quite similar to the case we treat in this paper.
In the terminology used above, we take , and let the map be the Veronese embedding with associated line bundle . Let . The semiorthogonal decomposition
gives rise to a Lefschetz decomposition with when is odd, and a Lefschetz decomposition , when is even.
Let us focus on the case where is odd; similar results hold for even . Consider a hyperplane , such that the associated quadric is nonsingular. One can then show that there is a semiorthogonal decomposition
where denotes (some twist of) the so-called spinor bundles.
The spinor bundles are natural bundles defined on all nonsingular quadrics; there are 2 spinor bundles on even-dimensional quadrics and 1 on odd-dimensional quadrics (see e.g. [Add09]). In low dimensions the spinor bundles are easily described: For a 2-dimensional quadric , they are and , and for the 4-dimensional quadric , they are the universal quotient bundle and the dual of the universal sub-bundle.
Let us write for the component of denoted by in the previous section. The decomposition (2.2) implies that . The spinor bundles satisfy , and , so we have . If we now deform the nonsingular quadric to a singular quadric of corank 1, the bundles and become isomorphic, and we can furthermore show that in this case .
Ignoring a technical issue which we will discuss shortly, this tells us exactly what the HP dual variety is over the locus in corresponding to nonsingular and corank 1 quadrics. Namely, we see that is a double cover of the locus of nonsingular quadrics, ramified in the locus of corank 1 quadrics.
The simplest application of Theorem 2.1 is now to the case of a general pencil generated by two quadrics . In this case the base locus , and is a double cover of , ramified in the points corresponding to singular quadrics in the pencil. Theorem 2.1 then gives an old result of Bondal and Orlov [BO95]:
The technical issue ignored above is the fact that our description of the HP dual category was only true point-wise and may fail in a global setting. To explain this complication, let us first give Kuznetsov’s general description of the HP dual in terms of Clifford algebras.
Let be a vector space with a quadratic form . The Clifford algebra is defined to be , where is the tensor algebra, and is the 2-sided ideal generated by . Taking gives the exterior algebra , and the Clifford algebras are in this sense deformations of . The natural grading on descends to a -grading on , and taking the degree 0 part we obtain the “even Clifford algebra” .
Now letting vary, one can fit these even Clifford algebras into a global family, i.e. there is a sheaf of algebras on such that the restriction to each is isomorphic to . Kuznetsov shows that the HP dual of is the category , i.e. the derived category of coherent -modules on . This means in particular that Theorem 2.1 holds when we interpret as .
Let us consider what this means for a single quadric. For any , if is the associated quadric, we find . If we assume that and hence is nonsingular, then it is a classical fact that for some . By Morita equivalence we then get . Thus we recover the statement that the fibre of the HP dual at is 2 points.
We can now explain the complication in the global description of the HP dual. Keeping to the locus of nonsingular , the above discussion shows that the centre of the algebra is a commutative algebra on , whose spectrum is a double cover . The algebra is then equivalent to an Azumaya algebra on , i.e. an algebra which is étale locally isomorphic to . The above results can be rephrased as saying that the HP dual is given by .
If there exists a locally free sheaf on such that , we can define an equivalence by the inverse functors and . This can always be done locally, but there is a global obstruction to the existence of such an , known as the Brauer class of , which lives in . In this example, the Brauer class does not always vanish, and in fact is not in general equivalent to .
2.5 HP duality for
The motivating problem for this paper is to construct the HP dual of , with respect to the natural map and a Lefschetz decomposition of which we describe as follows.
We think of sheaves on as -equivariant sheaves on . For any distinct , there is a unique -equivariant sheaf whose underlying sheaf on is . For any , there are two -equivariant structures on . We let be the -structure such that the -action is trivial along the diagonal in , and let be the other one. Note that then .
We take the initial piece in our Lefschetz decomposition of to be
If is even, we let . We remove 1 element from to get . We let
and then let .
By Proposition 5.16, this gives a Lefschetz decomposition
with , which is a fully faithful subcategory of . If were the HP dual of , this is in accordance with what Theorem 2.1 would give. In view of this and the similar results obtained in [BDF13], it seems very likely that is the correct HP dual, though strictly speaking we do not prove this here.
Geometric interpretation of the HP dual
Our Proposition 1.2 can be rephrased as saying that when is odd, then away from the corank locus the HP dual is a double cover of the corank 1 locus in , ramified in the corank 2 locus. Similarly, Proposition 1.3 says that when is even, the HP dual is a double cover ramified in the corank 1 locus.
Let us show concretely what this means in the case where is odd. First of all, if is such that is a nonsingular bilinear divisor, then we have
i.e. the interesting part is trivial.
Correspondingly, if is of corank 1, we would like to say that corresponds to the derived category of 2 points. This is almost correct, but must be modified slightly because the fibre of the double cover has higher dimension than expected. The correct statement is that is the derived category of the derived fibre product of and .
A somewhat surprising aspect of our description is that our HP dual is globally a variety, and that there is no need for an Azumaya algebra or Brauer class as in the case of quadrics. One way of thinking about this is that in the quadric case the spinor bundles, which are point-wise generators for the category of the HP dual, do not extend to globally defined bundles, and this can be explained by the presence of a Brauer twist. In our case, it turns out that we can write down an explicit global object which locally generates the HP dual category; this is the object called in Section 8.
3 Factorisation categories
We review some background material on derived categories of factorisations – further details can be found in [ADS14, BFK12, Shi12]. We first fix a definition of a gauged Landau–Ginzburg B-model (LG model for short).
A gauged LG model is the data of a smooth quasi-projective variety , equipped with:
An action of a reductive group .
An action of a 1-dimensional torus , commuting with the -action.
An element such that and fixes .
A function , which is -invariant and has weight 2 with respect to the -action, i.e. for and .
Let . The canonical character of induces a line bundle on , which we denote . For a sheaf on we write for . Note that is a section of .
By work of Positselski and Orlov [Pos11, EP15, Orl12], we can define a derived category of factorisations, , from the above data. An object of this category is a quasi-coherent sheaf on , equipped with a differential map , satisfying . We call such an object a factorisation. If we wish to emphasise the choice of differential, we denote this object by , otherwise we will simply write .
Consider the case where and is trivial. Then the action of on induces a grading on and makes it a dg algebra with vanishing differential. Since we require that acts trivially on , this grading will be even. Thus, is commutative as a dg algebra. A factorisation on is in this case the same thing as a graded -module with a differential squaring to . In particular, if , then a factorisation is the same thing as a dg module over .
If is a factorisation, we let be the factorisation whose underlying sheaf is and whose differential is . Given two factorisations we have a graded vector space
The differentials and give a differential on by the usual Leibniz rule. This differential squares to 0, and so is a dg vector space. We denote the homotopy category of the resulting dg category by .
The category is triangulated, with the shift functor as described above. The cone over is with an induced differential, in the same way as for the usual homotopy category of complexes.
In analogy with the definition of the ordinary derived category, we should now take the Verdier quotient of with respect to the subcategory of acyclic complexes. Since the differentials of factorisations do not square to 0, they do not have a notion of cohomology, and so the usual definition of acyclic does not make sense.
The correct definition of acyclic in this setting is the following: Consider a finite exact complex of factorisations
Exactness is here defined by considering the underlying sheaves, and we require the maps to be closed with respect to the differentials on . One can form the so-called totalisation of the above complex, which is a factorisation (see e.g. [Shi12, 2.12]). We declare to be acyclic, and let the category of acyclic objects be the thick triangulated subcategory of generated by such totalisations. Taking the Verdier quotient of with respect to the subcategory of acyclic objects gives the derived category .
3.1 Coherent and locally free factorisations
We say a factorisation is coherent if the underlying sheaf is. We define the category to be the full subcategory of objects isomorphic to coherent factorisations. The category is a generalisation of the usual bounded derived category, which is the special case where :
Proposition 3.3 ([Bdf13], 2.1.6).
If acts trivially on , then
We say a factorisation is locally free if the underlying sheaf is.
Proposition 3.4 ([Bfk14], 3.14).
Every factorisation on is isomorphic in to a locally free factorisation. Every coherent factorisation on is isomorphic in to a finite rank locally free factorisation.
We record the following lemma, which gives a useful criterion for checking that a complex is acyclic:
Lemma 3.5 ([Shi12], 2.12).
If and for all , then .
Suppose we are given a map of LG models , i.e. a morphism of stacks such that