Induced Automorphisms on Irreducible Symplectic Manifolds

Induced Automorphisms on Irreducible Symplectic Manifolds

Abstract

We introduce the notion of induced automorphisms in order to state a criterion to determine whether a given automorphism on a manifold of -type is, in fact, induced by an automorphism of a surface and the manifold is a moduli space of stable objects on the . This criterion is applied to the classification of non-symplectic prime order automorphisms on manifolds of -type and we prove that almost all cases are covered. Variations of this notion and the above criterion are introduced and discussed for the other known deformation types of irreducible symplectic manifolds. Furthermore we provide a description of the Picard lattice of several irreducible symplectic manifolds having a lagrangian fibration.

irreducible symplectic manifolds, automorphisms, moduli spaces of stable objects
:
Primary: 14J50 secondary: 14D06, 14F05 and 14K30
1

Introduction

The present paper deals with several questions concerning irreducible holomorphic symplectic manifolds and their automorphisms. All known examples of irreducible symplectic manifolds arise from symplectic surfaces, often as moduli spaces of sheaves on these surfaces. The easiest such example is the Hilbert scheme of points on a surface, constructed by Beauville in [Beau83]. This kind of construction allows to produce several examples of automorphisms on irreducible symplectic manifolds, simply by taking a surface with non-trivial automorphism group and considering the induced action on its Hilbert scheme. These kinds of automorphisms are called natural and were studied by Beauville [Beau83b], Boissière [Boi12] and many others. Very few examples of non-natural automorphisms are known, such as those constructed in [OW13], and a numerical criterion to distinguish between natural and non-natural automorphisms is available only in special cases, see [BS12] and [Mon13].

The main purpose of this paper is to provide a generalisation of the notion of natural automorphisms. This notion appeared the first time for moduli spaces of sheaves in the paper [OW13], a work inspired by the beautiful construction in [OS11, Sect. 5]. We extend the ideas drastically using recent developments in the theory of stability conditions by Bridgeland [Bri08] by Bayer and Macrì ([BM14] and [BM13]) and Yoshioka [Yos12]. This new notion of induced automorphisms includes all automorphisms on moduli spaces of stable objects, where the action is induced from an automorphism of the underlying surface. We provide a numerical criterion for the recognition of such examples and we apply this general framework to construct several new examples of induced, yet non-natural automorphisms. In particular, we prove the following:

Theorem.

Let be a manifold of -type, and let be a group of non-symplectic automorphisms. Assume that the action of fixes a copy of inside , then there exists a surface such that , is a moduli space of stable objects on and the action of on is induced by that on .

See LABEL:thmk3n_induced for the proof and a more precise statement. The same technique can be applied also to symplectic automorphisms, but in this case we obtain a more general result:

Theorem.

The moduli space of pairs with irreducible symplectic, symplectic and fixed up to conjugation has at most the same number of connected components as the moduli space of marked pairs .

See LABEL:thmsympl_def for details.

We mainly apply the new technique to automorphisms which do not preserve the symplectic form (i.e. non-symplectic automorphisms): In the recent work of Boissière, Camere and Sarti ([BCS14]) a lattice-theoretic classification of non-symplectic automorphisms of prime order (different from ) is given in the case of manifolds of -type. The case of involutions is of special interest since it provides a great number of automorphisms of irreducible symplectic manifolds, where, up to now, no geometric realisation is known. Using the notion of induced automorphisms we can realise almost all the unknown examples as automorphisms on moduli spaces of sheaves. Some examples are provided using degenerations of double EPW-sextics. We therefore have the following:

Corollary.

Aside from cases, there is a geometric realisation for every family of non-symplectic prime order (, ) automorphisms on manifolds of -type. Almost all families correspond to induced automorphisms.

A few more results concerning automorphisms are included, in particular we have the following for lagrangian fibrations with a section.

Proposition.

Let be a manifold of -type having a lagrangian fibration with a section. Then admits a primitive embedding of if is odd, and of otherwise.

See LABEL:proplagr_lattice for the proof and the following statements for analogous results on other irreducible symplectic manifolds. In particular we can exploit the above to prove that there exist lagrangian fibrations which can not be deformed to fibrations having a section.

The structure of the paper is as follows: In LABEL:secprel, we gather all preliminaries concerning lattice theory, irreducible symplectic manifolds and stability conditions. In LABEL:secperiods_mod_spac we give a numerical criterion to recognise a moduli space of stable objects on a (which was also recently proved in [Add14]) or abelian surface. In LABEL:secinduced_auto_group we provide the general framework to produce and distinguish groups of induced automorphisms on manifolds of -type. A conjectural setting for generalised Kummer manifolds is provided, which is proven to hold in the case that is a prime power. The case of symplectic automorphisms is treated in its full generality. In LABEL:seckieran_case, we discuss the notion of induced automorphisms on the two manifolds introduced by O’Grady. Finally, in LABEL:secappl we apply the theoretical construction to the recent work of Boissière, Camere and Sarti [BCS14] to explicitly construct all non-symplectic automorphisms of prime order (different from ) on manifolds of -type, apart from three cases. Moreover, we analyse lagrangian fibrations with a section together with the involution which is naturally induced by this section.

In this article all irreducible symplectic manifolds are assumed to be projective, unless stated otherwise.

Acknowledgements

We are grateful to Samuel Boissière, Chiara Camere and Alessandra Sarti for their comments and for letting us know about their work [BCS14]. We would also like to thank Paolo Stellari for telling us about [Yos12]. The first named author would like to thank Matthias Schütt and the Institute for Algebraic Geometry of the University of Hannover for their kind hospitality and for providing a nice working environment where this work was started. Moreover he would also like to thank Arvid Perego and Antonio Rapagnetta for useful discussions. The second named author wants to thank Gilberto Bini for his kind hospitality, the latter and Matthias Schütt and the Vigoni exchange program for supporting his visit to Milano. Finally, we are both grateful to the Max Planck Institut für Mathematik Bonn, for having partially supported the first named author and hosted the second named author.

1 Preliminaries

1.1 Lattice theory

Let be an even lattice, the group is called the discriminant group and the quadratic form of induces a form with values in . The length of is denoted by . If is an irreducible symplectic manifold, we denote by the discriminant group of the lattice . An overlattice of is any lattice such that is torsion. An embedding is primitive if the quotient has no torsion. The divisor of , denoted , is the positive integer such that .

If is a lattice, we denote with a lattice with the same structure as module but with quadratic form multiplied by . We denote with , and the positive definite lattices associated to the corresponding Dynkyn diagrams. Let be an integer, then we denote with a rank 1 lattice generated by an element of square and with the unique even unimodular lattice of signature .

Remark 1.

Overlattices of are in bijective correspondence with isotropic subgroups of . Moreover the discriminant group of an overlattice given by is

Lemma 2.

[GHS10, Lemma 3.5] Let be a lattice and let . Let be two elements such that the following holds:

  • .

  • in .

Then there exists an isometry of such that .

Corollary 3.

Let or . Then primitive embeddings of any corank lattice are determined, up to isometry, by the square of a generator of .

Lemma 4.

Let be a lattice and let . Let be a primitive embedding in an unimodular lattice. Let act trivially on . Then extends to a group of isometries of acting trivially on .

Definition 5.

Let be a lattice and . We denote by () the invariant (the co-invariant) lattice of . If acts on a manifold , we denote by () the invariant (the co-invariant) lattice of the induced action on .

Remark 6.

Let be a unimodular lattice and let . Then is of torsion.

Definition 7.

A lattice is called -elementary if . For such lattices, the invariant is defined to have value if the discriminant quadratic form is integer valued, otherwise.

Theorem 8.

[Nik80, Theorem 3.6.2] A -elementary indefinite lattice is uniquely determined by its rank, its signature, the length of its discriminant group and

Some basic building blocks for such lattices are , , , and for and , for .

1.2 Useful notions on irreducible symplectic manifolds

Here we gather several known results on irreducible symplectic manifolds. Many of these results are taken from the survey of Huybrechts [Huy01].

Definition 9.

A Kähler manifold is called an irreducible holomorphic symplectic manifold (short: an irreducible symplectic manifold) if the following hold:

  • is compact.

  • is simply connected.

  • , where is an everywhere non-degenerate symplectic 2-form.

There is the equivalent, but more differential geometric notion of hyperkähler manifold. In this article we keep with irreducible symplectic manifolds and refer the interested reader to the literature for a comparison of the two notions.

There are not many known examples of irreducible symplectic manifolds and for a long time the only known ones were surfaces (which are the only examples in dimension 2) and two families of examples given by Beauville [Beau83]:

Example 10.

Let be a surface and let be its -th symmetric product. There exists a minimal resolution of singularities (called the HilbertChow morphism)

where is the Douady space parametrizing zero dimensional analytic subsets of of length . Furthermore this resolution of singularities endows with a symplectic form induced by the symplectic form on . The manifold is an irreducible symplectic manifold of dimension and if , we have .
Whenever is an irreducible symplectic manifold deformation equivalent to one of these manifolds, we will call a manifold of -type.

Example 11.

Let be a complex -torus and let

be the minimal resolution of singularities of the symmetric product as in LABEL:exkntipo. Beauville proved that the symplectic form on induces a symplectic form on . However, this manifold is not irreducible symplectic since it is not simply connected. But if we consider the summation map

and set , we obtain a new irreducible symplectic manifold of dimension called generalised Kummer manifold of . If then is just the usual Kummer surface, otherwise it has . Note that the summation map is, in fact, the Albanese map of .
Whenever is an irreducible symplectic manifold deformation equivalent to one of these manifolds we will call
a manifold of Kummer -type.

Two more deformation types of irreducible symplectic manifolds are known and they were both discovered by O’Grady (see [OGr99] and [OGr03]). They are obtained as a symplectic resolution of singular moduli spaces of sheaves on or abelian surfaces. We will denote by the six-dimensional example and the ten-dimensional example. It is known that and . We call manifolds which are deformation equivalent to () manifolds of -type (-type).

Theorem 12.

Let be an irreducible symplectic manifold of dimension . Then there exists a canonically defined pairing on , the Beauville-Bogomolov pairing, and a constant (the Fujiki constant) such that the following holds:

Moreover and are deformation and birational invariants.

The Beauville-Bogomolov forms of known manifolds are the following:

Deformation class Lattice structure on
-type
Kummer -type
-type
-type

Let be an irreducible symplectic manifold, let us consider the natural map associating to a morphism its induced action on cohomology. Hassett and Tschinkel [HT13, Thm. 2.1] prove that the kernel of the map is a deformation invariant of . Some examples of such kernels are known: it is trivial if is the Hilbert scheme of points of a very general (see [Beau83b, Prop. 10] and it is generated by the group of points of order and the sign change on an abelian variety if (see [Beau83b, Prop. 9]).

1.3 Moduli of irreducible symplectic manifolds and the Torelli problem

Lemma 13.

Let be an irreducible symplectic manifold with Kähler class and symplectic form . Then there exists a family

called Twistor space such that with complex structure given by the Kähler class .

Definition 14.

Let be an irreducible symplectic manifold and let . An isometry is called a marking of . A pair is called a marked irreducible symplectic manifold.

Definition 15.

Let be a marked irreducible symplectic manifold and let . Let be the set of marked irreducible symplectic manifolds, where if and only if there exists an isomorphism such that .

Definition 16.

Let be an irreducible symplectic manifold and let be a lattice such that . Then we define the period domain as

Definition 17.

Let be a flat family of deformations of and let be a marking of into the lattice . Let moreover be a marking of compatible with . Then the period map is defined as follows:

The period map of the (flat) familiy of deformations of is called the local period map.

Theorem 18 (Local Torelli, Beauville [Beau83]).

Let and be as above and let moreover be a compatible marking of . Then the map is a local isomorphism.

This local isomorphism allows us to glue the various universal deformations giving the structure of a complex space. Another well known fact about the period map is the following:

Theorem 19 (Huybrechts, [Huy01]).

Let be a connected component of . Then the period map is surjective.

A weaker Global Torelli theorem holds, see [Huy10], [Mar11] and [Ver13].

Theorem 20 (Global Torelli, Huybrechts, Markman and Verbitsky).

Let and be two irreducible symplectic manifolds. Suppose is a parallel transport operator preserving the Hodge structure. Then there exists a birational map .

A characterization of parallel transport operators is needed to determine birational manifolds. In the case of manifolds of -type, this has been provided by Markman [Mar11]. Let be a lattice isometric to and let denote the set of primitive embeddings of into . Finally let be the orbit space of isometric primitive embeddings.

Theorem 21.

Let be a manifold of -type. Then there exists a canonically defined equivalence class of an embedding . A Hodge isometry is a parallel transport operator if and only if in

Note that the order of is , where is the number of different prime factors of .

Less is known about parallel transport for generalised Kummer manifolds, however there are some results due to Markman which appeared in [MM12, Corollary 4.8]. Let be a manifold of Kummer -type and let be the group of orientation preserving isometries of acting as on . Let be the kernel of the map , where is the character of the action on .

Proposition 22.

Keep notation as above, then and if is a prime power.

This implies that, if is a prime power, the moduli space of marked generalised kummer manifolds has connected components (corresponding to the four connected components of marked abelian surfaces).

1.4 Double EPW-Sextics

Double EPW-sextics were first introduced by O’Grady in [OGr06], they are in many ways a higher dimensional analogue to surfaces obtained as the double cover of ramified along a sextic curve.

Let be a six dimensional vector space with basis given by and let

be a volume form, giving a symplectic form on defined by

Let be the set of lagrangian subspaces of with respect to . Furthermore let be the vector bundle on with fibre

Let and let be the following composition

where the first map is the injection of as a subspace of and the second is the projection to the quotient with respect to . Therefore we define as the following locus:

Here is the EPW-sextic associated to and coincides with the degeneracy locus of if is general.

Definition 23.

Let be the open subset of lagrangian subspaces such that the following hold

  • ,

  • , where via the Plücker embedding.

Let us remark that contains the generic lagrangian subspace.

Theorem 24.

[OGr06, Theorem 1.1] For there exists a double cover ramified along such that is a manifold of -type.

A polarization of a Double EPW-sextic is given by the pullback of the hyperplane section of , a direct computation yields .

Remark 25.

Let us look a little into what can happen if the lagrangian contains some decomposable tensors. If contains an isolated decomposable tensor , then the double EPW-sextic is singular along a surface obtained as the double cover of the plane associated to , ramified along the sextic . There exists a symplectic resolution of singularities obtained by blowing up the singular surface. Moreover, as proven in [OGr12, Claim 3.8], the class of this exceptional divisor is orthogonal to the (now semiample) divisor of square 2 coming from the polarization of and has square and divisor .

We will be particularly interested in the following construction made by Ferretti [Fer12, Prop. 4.3]:

Proposition 26.

Let be a generic quartic with nodes and no other singularities. Then there exists a smooth complex contractible space of dimension with a distinguished point and a family such that

  • The central fibre is a symplectic resolution of ,

  • For generic the point is a symplectic resolution of a double EPW-sextic with singular surfaces.

1.5 Moduli spaces of stable objects

In this section we recall basic definitions and facts about moduli spaces of sheaves and Bridgeland stable objects on surfaces. For a more detailed treatment of the latter the interested reader is referred to [Bri08].

Let be a projective surface. Mukai defined a lattice structure on by setting

where and . This lattice is referred to as the Mukai lattice and we call vectors Mukai vectors. The Mukai lattice is isometric to .

Furthermore we may introduce a weight-two Hodge structure on by defining the -part to be

For an object we define the Mukai vector of by

It is of -type and satisfies

  • or

  • and effective or

  • and

Definition 27.

A non-zero vector satisfying and the conditions above is called a positive Mukai vector.

With this definition we can easily deduce:

Lemma 28.

Let be non-zero and of -type satisfying . Then either or its negative is a positive Mukai vector.

Let us now review some results on the birational geometry of moduli spaces of bridgeland stable objects on a surface. Let be a projective surface and fix two classes with ample. To this data Bridgeland associates a stability condition on the derived category The set of all such stability conditions is denoted by Next, we fix a primitive positive Mukai vector and assume that is generic with respect to The coarse moduli space of -stable objects of Mukai vector is a projective manifold of -type ([BM14, Thm. 5.9]) and we have an isometry of weight-two Hodge structures

Bayer and Macri studied the birational geometry of these moduli spaces; in particular, they introduced a chamber structure on We summarise their results:

Theorem 29.

[BM13, Thm. 1.1a) and Thm. 1.2]

  1. If and are generic stability conditions with respect to then and are birational.

  2. There is a surjective map

    mapping every chamber of onto a Kähler -type chamber such that for a generic the moduli space is the birational model of corresponding to the chamber containing

Note that for every positive Mukai vector at least one chamber in contains stability conditions whose stable objects are (up to a shift) stable sheaves in the sense of Gieseker.

Let be a marking and denote by the period map (restricted to a connected component of the moduli space of marked manifolds). The above theorem implies, in particular, that every manifold in the fibre is again a moduli space of stable objects on with the same Mukai vector

We even have the following stronger result:

Corollary 30.

Let and be Hodge isometric manifolds of -type. Then is a moduli space of stable objects on a surface if and only if the same holds for .

Proof.

Let be a moduli space of stable objects on a . We study the corresponding fibre of the period map. The number of connected components (neglecting the additional components that we can reach by composing with ) of the moduli space of marked manifolds is in one-to-one correspondence to the number of choices for Mukai vectors such that for generic stability conditions the moduli spaces are Hodge isometric. The corresponding embeddings of into are pairwise not conjugate, therefore the moduli spaces are pairwise not birational. Thus, by the theorem above, all manifolds Hodge isometric to are moduli spaces. ∎

Remark 31.

A very similar construction can be done in the case when is an abelian surface. Again, is a projective manifold and the fibre of the Albanese map is of Kummer -type ([Yos12, Thm. 1.9]). Again, the second cohomology of is Hodge isometric to and we have an analogous result as in LABEL:thmbmstabsurj. The only important difference is the following: By [Shi78, Lem. 3] for every -torus there is a Hodge isometry to its dual . Thus the moduli space of marked -tori has four connected components (corresponding to and , where is some marking of ). For every Mukai vector we define its dual as We see immediately that is positive if and only if is positive and the corresponding Albanese fibres and are Hodge isometric. (Here is the ’dual’ stability condition on defined in the obvious way.) Note that in general and are not birational ([Nam02]) but we, again, see that the moduli space of marked manifolds of Kummer -type has (at least) four components. Summarising, we can state that the above corollary holds also for manifolds of Kummer -type if is a prime power.

1.6 Induced Automorphisms on Moduli Spaces

We want to study automorphisms on the moduli spaces described above. Thus we include a direct generalisation of Section 3 in [OW13] to the case of moduli of Bridgeland stable objects. Let be a projective surface, be a positive Mukai vector, a -generic stability condition and an automorphism of .

Proposition 32.

Keep notation as above and assume that , and are -invariant. Then induces an automorphism of the moduli space If is not -invariant, we at least get a birational selfmap of .

Proof.

This is the analogue of [OW13, Prop. 3.1]. We only need to check that the pullback along induces an automorphism of the moduli functor. Indeed, from the definition of stability, it is clear that if an object is -stable, so is . ∎

Remark 33.

Note that we will typically consider a surface together with a non-symplectic automorphism fixing . In this case all stability conditions are -invariant.

In order to be able to study the induced action of on the second cohomology we need the following:

Lemma 34.

The isomorphism of Hodge structures

is -equivariant. In particular, we have

Proof.

Using Definition 5.4 of the above isomorphism in [BM14], we only need to check that for any curve we have

where is the FourierMukai transform associated with a quasi universal family on . But this follows easily from the fact that, by the definition of , is -invariant. ∎

As usual, the results can be translated in the appropriate sense to the case of abelian surfaces and their moduli. If we consider automorphisms of which preserve the origin (i.e. homomorphisms), a straightforward calculation shows that the induced automorphism respects the fibre over of the Albanese map and we obtain an induced action on the fibre. The other important class of automorphisms on are translations. By [BNS11] we know that we have an action of the group of -torsion points on , where .

2 Periods of Moduli Spaces

This section is devoted to answering the following question: How can we determine if a given manifold of -type is, in fact, a moduli space of stable objects on some K3 surface? We state a necessary and sufficient criterion entirely in terms of lattice theory.

Definition 1.

Let be a projective manifold of -type () and let be a primitive embedding of into the Mukai lattice . Endow the latter with the (unique) Hodge structure making an embedding of Hodge structures such that the complement of the image of is of type . We call a numerical moduli space if contains the hyperbolic plane as a direct summand.

Remark 2.

Note that, in general, there is more than one conjugacy class of embeddings as above. The condition to be a numerical moduli space, however, is independent of the choice of an embedding.

Proposition 3.

Let be a manifold of -type which is a numerical moduli space. Then there exists a projective surface such that is, in fact, a moduli space of stable objects of for some stability condition .

Proof.

The complement of the hyperbolic plane in is a lattice of signature and it inherits in a natural way a weight two Hodge structure. By the surjectivity of the period map for surfaces there exists a surface such that . From the hyperbolicity of it follows that is algebraic and furthermore we may choose an isometry respecting the given embedding of . The orthogonal complement of inside the Mukai lattice is a rank one lattice. The square of a generator has length . By LABEL:lemposvec we obtain a positive Mukai vector . Let be a -generic stability condition on and set . It is a manifold of -type which is, by construction, Hodge isometric to such that the embeddings of and into the Mukai lattice are equivalent. Thus by [Mar11, Cor. 9.9 (1)] and are birational. We may conclude using the results of [BM13], which we summarised in LABEL:ssecstab_obj. ∎

We remark that an independent proof of the above result has been given by Addington [Add14, Proposition 4] while this work was being prepared.
Copying the above definition to the case of Kummer -type manifolds (and replacing by ), we have the following result:

Proposition 4.

Let be a manifold of Kummer -type with a prime power. Assume that is a numerical moduli space. Then there exists an abelian surface such that is, in fact, the fibre of the Albanese map of a moduli space of stable objects of for some stability condition .

Proof.

It is the lacking of a precise description of the monodromy group of manifolds of Kummer -type for general which prevents us from proving the above theorem in general. However, the partial description given in LABEL:propmono_kum is enough to prove the theorem if is a prime power: there are four connected components of the moduli space of marked manifolds of Kummer -type. Now, as in the case of -type manifolds, we find an abelian surface and a Mukai vector such that the associated (Albanese fibre of the) moduli space is Hodge isometric to . Together with its birational models it covers all points in the fibre which belong to two out of the four connected components. (Here is some marking of . Note that we may always compose a marking with to end up in a second connected component.) The points in the fibre, which belong to the other two connected components, correspond to (Albanese fibres of) moduli spaces of stable objects on the dual abelian surface (cf. LABEL:rmkmod_stab_obj_ab). ∎

Conjecture 5.

The above statement holds, in fact, for all .

3 Induced Automorphism Groups

Next, let us consider what happens in the presence of automorphisms of the manifolds. We start by a definition.

Definition 1.

Let be a manifold of -type and let . We say that is an induced group of automorphisms if there exists a projective surface with , a -invariant Mukai vector and a -generic stability condition such that and the induced (maybe apriori birational) action on (cf. LABEL:propexist_induced) coincides with the given action of on .

The same definition can be given for manifolds of Kummer-type.

Definition 2.

Let be a manifold of -type and let . Let be a primitive embedding of inside the Mukai lattice . Then the group is called numerically induced if the following hold:

  • The group acts trivially on , so that its action can be extended to the Mukai lattice with as in LABEL:lemaut_extend.

  • The -part of the lattice contains the hyperbolic plane as a direct summand.

Remark 3.

The two definitions above can be phrased in a very similar fashion for manifolds of Kummer- and of - and -type (cf. LABEL:seckieran_case for the latter two). Furthermore note that it is easy to see that an induced group of automorphisms is numerically induced.

Theorem 4.

Let be a manifold of -type and let be a numerically induced group of automorphisms. Then there exists a projective surface with , a -invariant Mukai vector and a -generic stability condition such that and is induced.

Proof.

First of all, let us consider the case symplectic. Then we have and contains no elements of square . Since is numerically induced, let us write . We then have that embeds in the lattice and its orthogonal is , where the action of is trivial. Let us give this lattice the induced Hodge structure from and let be the corresponding surface. By LABEL:propk3n_moduli, is a moduli space of stable objects on . Now is a group of Hodge isometries on which contains no elements generated by reflections on elements of square , therefore and its induced action on is the one we started with. Since the representation of automorphisms on is faithful, we are done.
Now let us suppose that no non-trivial element of preserves the symplectic form. This implies . Without loss of generality we can suppose . As before we let and we let be the surface associated to the Hodge structure defined on inside . Again by LABEL:propk3n_moduli is a moduli space of stable objects on and is a group of Hodge isometries of preserving , therefore and its action on coincides with the induced one.
Finally, if is the symplectic part of , we obtain a surface as in the first step with . We can extend also the action of on by applying the second step to the group . ∎

Theorem 5.

Let be a manifold of Kummer-type with a prime power. Let be a numerically induced group of automorphism. Then there exists an abelian surface with , a -invariant Mukai vector and a stability condition such that is the Albanese fibre of and is induced.

Proof.

The proof is literally the same as the one of LABEL:thmk3n_induced. We remark here that all these manifolds have non-trivial automorphisms acting trivially on the second cohomology. However, these automorphisms deform smoothly on all manifolds of Kummer-type. ∎

In the special case of finite groups of automorphisms preserving the symplectic form, the notions of induced and natural morphisms coincide, as the following more general result shows:

Theorem 6.

Let be the moduli space of marked irreducible symplectic manifolds with . Let be a finite group of symplectic automorphisms acting on some and let contain all morphisms acting trivially on . Then the number of connected components of the moduli space of manifolds having the same -action as on is bounded by the number of connected components of .

Proof.

Let be another manifold such that acts on its second cohomology as it acts on , i.e. there exists an isometry which restricts to an isometry of and . In this way we can define a moduli space of marked manifolds having primitively embedded inside the Picard lattice. To prove our claim, it is enough to prove that can be deformed into if is a parallel transport operator, so we assume it is from now on.

Let be a -invariant Kähler class on and let be the twistor family associated to it. The action of extends fibrewise to , since a Kähler class is always preserved. If we let be the period domain associated to the lattice , we obtain that it has signature and is therefore connected by twistor lines (see [Huy10, Proposition 3.7]. These lines lift to twistor families given by -invariant Kähler classes on the preimage of the period map, as is done in [Mon13] in the -type case. In this way we can deform to a pair with the same period of . The final deformation step can be done using local -deformations. The very general point of such a deformation has and the positive cone coincides with the Kähler cone, therefore the local -deformations of and intersect and the action of on coincides on with the action deformed from . Now, since contains all morphisms acting trivially on , the action we deformed to coincides with the initial one. ∎

Example 7.

In this small example we want to illustrate that the first condition in LABEL:defnk3n_induced is necessary. Thus we are looking for an automorphism of a manifold of -type, such that the induced action on the discriminant of is non-trivial.

Let us first recall an example of a birational self-map that has been studied first by Beauville [Beau83b]. Let be a K3 of degree , that is, S is the intersection of a quadric and a cubic in . A general element in defines a -plane in and its intersection with is a scheme of length . Thus, taking the residual length subscheme defines a birational involution . Following Debarre ([Deb84, Théorème 4.1]), the action on is given by the reflection at the element . In particular, it sends to , where is the class of a hyperplane section. Thus the generator of the discriminant group is mapped to its negative.

A detailed analysis of the known results on the precise structure of the movable and the ample cone shows the following: The closure of the movable cone of is spanned by and . Thus it contains the class (which is invariant under the involution). However, is orthogonal to , which is a class of square and divisibility , hence a wall of the movable cone. This implies that this involution is not regular on any manifold birational to .
As it is often the case, we can obtain a regular involution by paying the price of dealing with a singular manifold (which is just the contraction of the giving the indeterminacy locus). This manifold can be seen as a moduli space of stable object on the surface with a bad choice of a polarisation, see [Mat14] for a closely related example.
Nevertheless, this example can be seen as a degeneration of a twenty dimensional family of manifolds with an involution acting as on the discriminant group. Indeed, if we take local deformations of where stays algebraic, the very general element of this family has ample cone which coincides with the positive cone and the birational involution deforms to a regular involution on it.

4 O’Grady’s Examples

In this section we introduce a notion of induced automorphisms and give a numerical criterion for manifolds of or -type. Part of our results are based on [MW14], where the kernel of the cohomological representation is determined for a manifold as above.

We fix a projective surface and consider a primitive Mukai vector of square . Set . The moduli space (we tacitly assume the choice of some -generic polarisation) is of dimension and its singular locus consists exactly of the points corresponding to -equivalence classes of sheaves of the form , where and are stable sheaves in . The blow up of along with its reduced scheme structure yields a symplectic resolution (cf. [LS06]). Perego and Rapagnetta showed that is a -factorial symplectic manifold admitting an appropriate analogue of a BeauvilleBogomolov form and a pure weight-two Hodge structure on . Furthermore they computed the Hodge and lattice structure on . They introduced the following notation: Set

By declaring to be of type and setting we obtain a pure weight-two Hodge structure and lattice structure on induced by the corresponding structures on . Note that, by definition, we have

Theorem 1.

There is a Hodge isometry of pure weight-two Hodge structures

(4.1)

where the lattice structure on the right hand side is the BeauvilleBogomolov form.

Proof.

[PR13, Theorem 3.4]

Now, assume we have an automorphism of the surface . It naturally induces an isometry of . If fixes the Mukai vector , we obtain an isometry of by demanding to be fixed.

Definition 2.

Let be a desingularised moduli space of -type and let . We call numerically induced if the following hold:

  • The induced action on fixes (cf. eq. (4.1 )).

  • The induced action on is numerically induced in the sense of definition LABEL:defnk3n_induced.

Proposition 3.

Let be an automorphism of the surface fixing the Mukai vector (and the chosen polarisation). It induces an automorphism on the desingularised moduli space such that the isometry (4.1) is -equivariant. In particular, the action of is numerically induced.

Proof.

By LABEL:propexist_induced we obtain an induced automorphism on the singular space and by LABEL:lemequivariant_induced_iso the Hodge isometry is -equivariant. Furthermore the singular locus is certainly -invariant and we have a well defined induced action on the normal bundle . Indeed, the fibre of over is isomorphic to and the map is the obvious pullback map. Thus we get an induced action of on the blow up . We immediately deduce that the class corresponding to the exceptional divisor of the blow up is fixed by the induced action. ∎

Definition 4.

Let be the symplectic resolution of the singular moduli space for a primitive Mukai vector of square on a projective surface . Let . We say that is an induced group of automorphisms if with and the induced action on coincides with the given one.

Proposition 5.

Let be a primitive Mukai vector of square and let be the symplectic resolution of singularities of for a projective surface . Let be a finite subgroup. Then and is induced if and only if it is numerically induced.

Proof.

The ‘only if’-part is the content of LABEL:propogrady_induced. For the other direction let be an automorphism of which is numerically induced. We thus obtain an induced isometry of which is induced in the sense of LABEL:defnk3n_induced. Thus we obtain an isomorphism on the underlying surface . Now, and the induced automorphism on have the same action on the second integral cohomology. By [MW14], the cohomological representation is injective, hence we are done. ∎

Remark 6.

Statements similar to LABEL:propk3n_moduli and LABEL:propkummer_moduli can be expected to hold true for O’Grady-type manifolds: Let be deformation equivalent to . Embed into and endow the latter with the induced Hodge structure. Suppose the -part of contains as a direct summand. Then we expect that is given as the desingularisation of a moduli space of stable objects on a surface.

Copying the appropriate definitions from the -case, we can conclude in a similar way in the abelian surface case:

Corollary 7.

Let be a primitive Mukai vector of square on a projective abelian surface and let