Automorphisms of generalised Kummer fourfolds
We classify non symplectic prime order automorphisms and all finite order symplectic automorphism groups of generalised Kummer fourfolds using lattice theory and recent results on ample cones and monodromy groups. We study various geometric realisations of the automorphisms obtained in the classification. In the case of higher dimensional generalised Kummers, we provide full results for finite symplectic automorphisms groups and we sketch the classification for prime order non-symplectic groups.
- 0 Introduction
- 1 Preliminaries
- 2 Non-symplectic automorphisms I: Classification for
- 3 Non-symplectic automorphisms II: Classification in dimension four
- 4 Examples of non-symplectic automorphisms
- 5 Symplectic automorphisms
- 6 Higher dimensions
In recent years a lot of attention was drawn to the study of hyperkähler manifolds and their automorphisms. After Beauville’s ([Beau83] and Huybrechts’ ([Huy99]) fundamental results, most of these developments studied various aspects of automorphisms of Hilbert schemes of points on K3 surfaces and their deformations ([BCMS14], [BCNS14], [BCS14], [BNS13], [BS12], [Boi12], [Cam12], [Mon13], [OW13]). The other famous series of deformation classes – the so-called generalised Kummer manifolds or hyperkähler manifolds of Kummer type – has been only been treated in very few articles: In [BNS11] the authors determined the kernel of the cohomological representation, which associates to every automorphism of a hyperkähler manifold the induced action on the second integral cohomology lattice. Oguiso ([Ogi12]) further studied the automorphisms in this kernel, proving that they, in fact, act non-trivially on the total cohomology of the underlying manifolds. In [OS11] examples of fixed point free automorphisms on generalised Kummer manifolds are presented. In [MW14], the so-called induced automorphisms have been introduced. This construction starts with a group homomorphism of a 2-torus, which then, under certain conditions, induces an automorphism of the Albanese fibre of certain moduli spaces of stable objects on the torus, which are hyperkähler manifolds of Kummer type.
One of the first tasks concerning classifications of automorphisms of hyperkähler manifolds is the classification of prime order automorphisms. Such an automorphism is either symplectic , i.e. preserving the symplectic structure of the manifold, or non-symplectic otherwise. After dividing prime order automorphisms into these two groups, the classification is usually done using lattice theory. The case of K3 surfaces has been done by [AST11] for the non-symplectic case and [GS07] in the symplectic case. In the case of K3-type manifolds we have the results of [BCS14] and [MW14] in the non-symplectic case and [Mon14] in the symplectic case.
The main aim of this article is to provide a first step towards a classification of automorphisms of prime order in the case of generalised Kummer fourfolds. We give a lattice theoretic classification in both, the symplectic and the non-symplectic case, using recent developments concerning ample cones and monodromy groups of generalised Kummer manifolds (cf. [Mon13b]). Furthermore we analyse the lattice theoretic result by studying the corresponding geometric realisations if available. We use the survey [MTW15] as a main source to study natural and induced automorphisms on Kummer fourfolds.
We would like to thank the Max Planck Institute in Mathematics, for hosting the three authors when this work was started. We are also grateful to Samuel Boissière and Bert Van Geemen for their comments and support. The first named author would also like to thank G. Höhn for his many insightful comments on this paper.
In this first section, we gather the required background material and fix some notation.
1.1 Generalised Kummer fourfolds
Let be a complex 2-torus and denote by the Hilbert scheme of three points on . The fibres of the isotrivial Albanese map are hyperkähler manifolds of dimension four known as generalised Kummer fourfolds. We call fourfolds of Kummer type all hyperkähler deformations of generalised Kummer fourfolds.
Let be a fourfold of Kummer type, then , where the pairing on is the Beauville-Bogomolov-Fujiki form.
By [BNS11], the cohomological representation
has kernel isomorphic to . We introduce the following convention: An automorphism of is said to be an automorphism of order if the induced action on has order .
The discriminant group of is isomorphic to and will be denoted by . For any lattice and any element we denote by the divisibility of in , i.e. the positive generator of the ideal .
1.2 Monodromy and ample cone
In order to study automorphisms, it is vital to understand the birational geometry of our manifolds. The latter is governed by parallel transport operators or monodromy operators, i.e. isometries of which are induced by parallel transport along a connected base. We have the following Hodge-theoretic Torelli Theorem due to Huybrechts, Markman and Verbitsky.
Theorem 1.1 ([Mar09, Thm. 1.3]).
Let and be hyperkähler manifolds. Then and are birational if and only if there is a parallel transport operator which is a Hodge isometry.
The group of parallel transport operators (denoted ) for Kummer fourfolds has been computed by Markman and Mehrotra [MM12, Corollary 4.8]:
Proposition 1.2 ([Markman-Mehrotra]).
Let be the group of orientation preserving isometries of . Let be the kernel of the map , where is the character of the action on . Then
We call a marking of a Kummer fourfold an isometry of with the reference lattice . A marked pair consists of a manifold and a marking. An isomorphism of marked pairs is an isomorphism respecting the markings. The above result implies that the moduli space of marked generalised Kummer fourfolds has four connected components. In a certain sense this is the smallest possible. Indeed, the moduli space of marked 2-tori also has four connected components (cf. [MTW15, Section 2]).
Within the set of birational maps, automorphism are exactly those which map the ample cone of a manifold to itself. Thus, the understanding of the ample cone is crucial for the study of automorphisms of any manifold. The ample cone for Kummer type fourfolds arising from moduli spaces of sheaves on abelian surfaces has been studied by Yoshioka [Yos12]. His results can be generalised to any manifold of Kummer type, using either [BHT13] or [Mon13b]. In particular, there are three kinds of faces of the ample cone, each of them orthogonal to one of the following:
Divisors of square and divisibility .
Divisors of square and divisibility .
Divisors of square and divisibility .
In the following, divisors of these kinds are called wall divisors. The first kind of divisors is not effective, however (or ) is a sum of reduced and irreducible uniruled divisors which can be contracted to a symplectic surface. The second kind of divisors is effective (or its opposite is effective) and again is a sum of reduced and irreducible uniruled divisors which can be contracted to a symplectic surface. The last kind of divisors are not effective and none of their multiples is. However they correspond to ’s inside our fourfold which induce Mukai flops. (In particular, is a class in which is the class of a line in such a .)
1.3 Moduli spaces
Besides generalised Kummer manifolds there is up to now only one other construction of manifolds of Kummer type: Let be an abelian surface and a so-called Mukai vector satisfying effective for and Assume that there exists a -invariant -generic stability condition. Then the Albanese fibre of the associated moduli space of stable objects is a fourfold of Kummer type.
In [MW14] the authors gave a lattice theoretic criterion to identify moduli spaces of stable objects.
Proposition 1.3 ([Mw14, Prop. 2.4]).
Let be a fourfold of Kummer type. If the algebraic part of the Hodge structure on (which is induced by the embedding ) contains a copy of as a direct summand, then is the Albanese fibre of a moduli space of stable objects.
Fix a Kummer fourfold and let be a finite group of automorphisms. Every element of acts on the holomorphic two-form of by homotheties. We call an automorphism symplectic if it preserves the symplectic form and non-symplectic otherwise. We denote by the invariant lattice and by the coinvariant lattice.
The fact that is not unimodular often makes things more complicated. It is therefore sometimes convenient to switch to a bigger unimodular lattice: We consider a primitive embedding – such an embedding is unique up to an isometry of – that sends identically into the first three copies of and sends , a generator of , to , where form a typical basis of the last copy of . The action of on is either trivial or , therefore there is a well defined extension of the action of on : if acts trivially on , then , otherwise . We keep calling and . The advantage in this setting is that if is of prime order , then and are -elementary lattices. This is by no means true for and . To prove this last statement we use the following lemma:
Let be a lattice, and let . Then the following hold:
contains for all .
contains for all and all .
is of -torsion.
It is obvious that is -invariant for all . For we have for all and all . Therefore is orthogonal to all -invariant vectors, hence it lies in . Let , we can write , where the first term lies in and the second in . ∎
Since and are -elementary lattices, we will use the classification results of Nikulin to find them. The first theorem deals with the case :
Theorem 1.5 ([Nik83]).
An even hyperbolic -elementary lattice of rank is uniquely determined by the invariants and exists if and only if the following conditions are satisfied:
And the second theorem deals with the case :
Theorem 1.6 ([Rs81, Section 1]).
An even hyperbolic -elementary lattice of rank with with invariants exists if and only if the following conditions are satisfied:
Such a lattice is uniquely determined by the invariants if .
Since the classification theorem of -elementary lattices with above deals only with hyperbolic lattices, we will need sometimes to split a lattice to study it. This is done by the following theorem:
Theorem 1.7 ([Nik79, Corollary 1.13.5]).
Let be an even lattice of rank . If is indefinite, and , then for some lattice .
The following Lemma gives some restrictions for a lattice to have an action of prime order:
Let be a lattice and a subgroup generated by of prime order . Then
is an integer and
Moreover, there are natural embeddings of into the discriminant groups and , and .
By [CR88, Theorem 74.3] we have the following isomorphism of -modules:
where the are ideals of with a primitive -th root of the unity, as -modules with , and the action of is defined by:
trivial on ,
by multiplication by on ,
and by on .
Since this decomposition respects the action of , we can look for invariant and coinvariant lattices on each term:
The action is trivial on so this term goes in the invariant part.
On , acts by . But and this morphism acts on by , hence all are coinvariant.
On , we compute so this can be zero only if . Moreover, the action restricts to by multiplication by , hence by the same argument as above, is the coinvariant part of . Then we look for invariant elements of , that is elements such that:
so divides . But , hence is fixed by the action and is in the invariant part of . Then we get:
where is isomorphic to or . But if it was , then we would have and the element of would decompose into , with and . By the Equation 1, this would mean that and this is impossible because . So , and .
Coming back to , what we did before implies that and , hence:
We put and, since and are primitive in and , by [Nik79, §1, 5] we get a natural embedding and the projections on and give the embeddings that we wanted.
Finally, for any we have , hence is an integer and . ∎
The following theorems will also be useful along the classification:
Theorem 1.9 ([Nik79, Theorem 1.13.3]).
Let be an even lattice with signature and discriminant form . If is indefinite and , then is the only lattice up to isometry with the invariants .
Proposition 1.10 ([Bcms14, Section 4]).
Let be a lattice with a non-trivial action of order , with rank and discriminant , then is a square in .
We recall briefly the notion of induced automorphisms on Kummer fourfolds. For a more detailed and systematic introduction we refer to [MW14].
Let be an abelian surface, a group automorphism of and a so-called Mukai vector satisfying effective for and as before. Suppose and assume that there exists a -invariant -generic stability condition. Then the Albanese fibre of the associated moduli space of stable objects is a fourfold of Kummer type by LABEL:propmod_spac and the pullback induces an automorphism on , called the induced automorphism of and .
The main result about induced automorphisms on fourfolds of Kummer type in [MW14] is the following lattice-theoretic characterisation:
Theorem 1.11 ([Mw14, Thm. 3.5]).
Let be a fourfold of Kummer type and an automorphism of . If the induced action of on is trivial and the isometry of (which is induced via the -equivariant embedding ) fixes a copy of , then is the Albanese fibre of a moduli space of stable objects and is an induced automorphism as introduced above.
Note that it is a straightforward calculation that every induced automorphism satisfies the conditions in the theorem above.
If we consider the special mukai vector , one of the associated moduli spaces is nothing but the generalised Kummer fourfold, i.e. the Albanese fibre of the Hilbert scheme . The induced automorphisms in this case are called natural automorphisms. They have been studied in detail in [Tar15].
Automorphisms from lattice isometries
To construct automorphisms directly is, in general, very difficult. Using the Hodge-theoretic Torelli theorem (LABEL:thmtorelli), we can show that certain lattice isometries of our reference lattice actually come from automorphisms of Kummer type fourfolds. Let us illustrate this by fixing an isometry of of order , which generates a cyclic subgroup (the group is obtained from using an arbitrary marking). Assume that the invariant lattice has signature . Now, let be a fourfold of Kummer type together with a marking such that the invariant lattice of the induced isometry group on is contained in . The group therefore consists of parallel transport operators and we obtain a group of birational self-maps of . If furthermore preserves all wall divisors in , then we actually constructed automorphisms of , which, in this case, can easily seen to be non-symplectic. Thus, in the following sections we seek to classify all such isometries of .
2 Non-symplectic automorphisms I: Classification for
Suppose is cyclic and generated by a non-symplectic element of maximal order .
Corollary 2.1 ([Oguiso-Schröer, [Os11]]).
We have and .
Moreover, in this case has signature (because it contains an invariant ample class) and has signature . Let now be a cyclic group of non-symplectic automorphisms on a fourfold of Kummer type. By the considerations at the beginning of LABEL:subseclat we obtain a -equivariant embedding of into . Conversely, can be considered as the orthogonal complement of a square vector , which will be referred to as the Mukai vector for . If the action of on is trivial, we have
Thus and have signature .
If acts as on – this can only happen for – we have
Thus in this case has signature and has signature .
It follows from LABEL:lemG_tors that all lattices and are -elementary, thus their descriminant group is isomorphic to for some integer .
Now, in order to classify non-symplectic automorphisms of fourfolds of Kummer type of prime order in the next section, we first study -elementary sublattices of that might occur as (co-)invariant lattices of isometries of of order . Note that this list, in principal, can be used to classify non-symplectic automorphisms of manifolds of Kummer type of any dimension (). In the case we add a column ’’ to indicate whether the quadratic form of the discriminant group of the lattices at hand is integer valued () or not ().
The following is a complete list of coinvariant lattices () and invariant lattices () of even rank and signature of order two isometries of .
We call an isometry induced if contains a copy of .
When , is positive definite, then we can find those lattices in the list [CS99, Table 15.1, p.360]. When or , we apply LABEL:thm2el to find the lattices. Since the list above is symmetric in and we proceed in the same way for .∎
Now we consider the case where has signature and signature .
This is a direct application of LABEL:thm2el. ∎
The following is a complete list of coinvariant lattices () and invariant lattices () of even rank and signature of order three isometries of . We call an isometry induced if contains a copy of .
When , is positive definite. We can then find those lattices in the list [CS99, Table 15.1, p.360] and is uniquely determined because we can write it as by LABEL:thmsplit with hyperbolic of rank , which is unique by LABEL:thmpel.
When , we can do the same thing as above, exchanging the roles of and .
When , we deal with the cases and by splitting a copy of again by LABEL:thmsplit and looking for the last piece in the list [CS99, Table 15.2a, p.362]. The only case left is by LABEL:lemaleqm. The isomorphism classes of -elementary discriminant forms of rank are those of and . But the signature of our lattice must be , hence the case of is impossible by [Nik79, Theorem 1.10.1], because . The only possible discriminant form is the one of . LABEL:thmuniclat applies here, so the lattice is isomorphic to . ∎
For the rank of must be four in order to obtain an isometry with determinant . Furthermore we exclude the cases , using Proposition 2.6 of [MTW15]. We are left with the following example:
We begin by looking for . By LABEL:lemaleqm, we always have (so ) and . Moreover, is odd by LABEL:propBCMS14p, so . We can then split a copy of by LABEL:thmsplit and look for the last piece in the list [CS99, Table 15.2a, p.362]. Since in this case , we deduce from LABEL:thmuniclat that . ∎
Also for there is a unique possibility:
We begin by looking for . By LABEL:lemaleqm, we always have (so ) and . Moreover, is odd by LABEL:propBCMS14p, so . We can then split a copy of by LABEL:thmsplit and look for the opposite of the last piece (for it to be positive definite) in the list [CS99, Table 15.1, p.360]. The lattice is then positive definite with rank and , we find it also in the list [CS99, Table 15.1, p.360]. ∎
For , in principle we could have several isometries of order which are induced by automorphisms. However, when the covariant lattice is isometric to the covariant lattice of a surface with an automorphism of order , we have a unique action on the lattice. Indeed, in [AST11, Prop. 9.3], it is proven that the connected components of the moduli space of surfaces with an automorphism of order is given by the isometry classes of the covariant lattice. If we had more than one isometry for each of them, this would be impossible, hence our claim. In particular, all covariant lattices in the above lists for occur as covariant lattices on surfaces.
3 Non-symplectic automorphisms II: Classification in dimension four
Now, we specialise to the case of fourfolds. Lattice-theoretically this is done by chosing a length vector in which will serve as a Mukai vector. The cohomology will then be isometric to . Some of the invariant or coinvariant lattices appearing in the previous section do not contain any such vector. But some lattices admit two non-conjugate choices. Keeping this in mind, we will give a list of all configurations of lattices occuring as invariant and co-invariant lattices of prime order non-symplectic automorphisms of Kummer 4-type manifolds. Note that for each such configuration there are four families in the moduli space of marked manifolds. Each two of them are geometrically identical, but the marking differs by . Finally the last family is obtained by composing with the ’dual’ isometry. The geometric meaning of this isometry is only understood for induced automorphisms, where we simply construct the ’dual manifold’ as the moduli space of objects on the dual abelian variety.
If we choose the vector to be in the invariant lattice , the induced action on has a trival action on the discriminant . In this situation the discriminant group of is isomorphic to for some integer .
If we choose to belong to the coinvariant lattice , we obtain a non-trivial action on . Note that since the action on must be given by (see LABEL:secprelim), this can only happen for . In this case the discriminant group of the invariant lattice is isomorphic to for some integer
In every table we add a column ’’ which gives the dimension of the family of manifolds with the corresponding automorphism.
We indicate by ’ind’ all entries corresponding to induced automorphisms and by ’nat’ those which are even natural.
The following classification is complete with respect to the possible lattices and , however to count the number of connected components of the moduli space of Kummer fourfolds with this prescribed action, one needs to count embeddings of (or in the appropriate cases) up to isometry.
3.1 , trivial action on
If the automorphism acts trivially on , we have and is obtained as the orthogonal complement of the square class . Note that is -elementary. We denote by and by .
3.2 , non-trivial action on
If the automorphism acts as on we have and is obtained as the orthogonal complement of a square element . Note that is -elementary. We denote by and by .
Note that none of these examples can be realised by an induced automorphism. The column ’MS’ indicates whether the manifolds in this family or moduli spaces of stable objects or not. We denote by ’LFwS’ those families of moduli spaces which are (more strongly) relative compactified Jacobians admitting a lagrangian fibration with section.
For the action on is trivial, hence and have signature and the latter is obtained as the orthogonal of a square class .
For Moreover, the determinant must be on to ensure that it is a monodromy operator, thus has even rank. There is only one family which is induced and where the divisor of is one:
4 Examples of non-symplectic automorphisms
In this section we collect all examples of non-symplectic automorphisms on fourfolds of Kummer type.
4.1 , trivial action on
Example 4.1 ([Invariant lattices , , and ]).
These are the natural automorphisms corresponding to the automorphisms of abelian surfaces in Section 4.1 of [MTW15].
Example 4.2 ([Invariant lattices and ]).
These are both induced automorphism which are not natural. The underlying abelian surfaces have invariant lattices (which generically is isomorphic to the Picard lattice) isometric to and respectively. The examples can be constructed as follows: Choose an ample class on the abelian surface of square which has divisibility inside the invariant lattice. The Albanese fibre of the moduli space of sheaves with Mukai vector is a fourfold of Kummer type and carries the induced automorphism from the surface. Note that the moduli spaces occurring here are relative compactified Jacobians over the linear system of genus .
4.2 , non-trivial action on
Example 4.3 ([Invariant lattices and ]).
Let be an abelian surface admitting a polarisation of type , i.e. there is an ample class of square yielding a -cover of . If we choose the Mukai vector the corresponding moduli space is the relative compactified Jacobian of degree over the genus linear system . The involution which is given by on the smooth fibres restricts to a lagrangian fibration with section on the Albanese fibre and yields thus an example of a non-symplectic involution on fourfolds of Kummer type. If has Picard rank one, a direct calculation shows that