Lagrangian fibrations on hyperkähler varieties

May-July 2022, Université de Lille

This working group is centered around Lagrangian fibrations on hyperkähler varieties with a special focus on fibrations by Prym varieties.

In 2007, Markushevich and Tikhomirov proposed a construction of relative Prym varieties [6], which can be seen as a generalisation of the relative Jacobian varieties introduced by Beauville in [2]. Markushevich and Tikhomirov studied the relative Prym variety obtained from a K3 surface with a branched double cover to a del Pezzo surface of degree two.

The same construction can be applied to any pair (S, i), where S is a K3 surface and i an anti-symplectic involution, and by Nikulin's classification, there exist 75 deformation classes of such pairs [9]. Several examples are available in the literature: Arbarello–Ferretti and Saccà considered the case where the quotient S/i is an Enriques surface [1],, the cases where S/i is a del Pezzo surface of degree d were studied by Matteini

[7] for d = 3, and Sawon-Shen [10] for d = 1.

We study the Prymian fibrations obtained via this construction, primarily focusing on the examples of Markushevich-Tikhomorov and Arbarello-Saccà-Ferretti, the latter being our main reference. The goal of this working group is not only to understand the known examples in depth, but also to think about possible generalisations and applications.

The program consists of three 1-day meetings at the University of Lille, with three talks per day. The final session of each meeting is specifically devoted to open discussion among the participants. The scheduled duration of each talk is 1 hour and 15 minutes, with some flexibility. Each meeting starts at 10:30 and ends at 18:00.

Organizers: Corinne Bedussa & Grégoire Menet

First meeting: 9 May

Talk I: General construction of the relative Prym variety

Speaker: Grégoire Menet

The objective of this talk is to describe the construction of the relative Prym varieties *Prym*(D*/*C) arising from K3 surfaces with an anti-symplectic involution. After recalling the de nition of the Beauville-Mukai system I will explain the construction of the relative Prym variety starting from any K3 surface with an anti-symplectic involution. I will show how this construction depends on the choice of a polarization, prove that it admits a symplectic structure and endow it with a natural Lagrangian fibration. Afterwards, I will give an overview of examples seen in the literature ([6], [1], [7], [10]). The general construction can be deduced from [1] by changing the involution; another possible reference is [4] (see Sections 2.1-2.4). We will use the notation from [1].

Talk II: Singular fibers of the Markushevich Tikhomirov system

Speaker: Chenyu Bai

The objective of this talk is to describe the singular fibers of the Lagrangian fibration on the Markushevich Tikhomirov system as in [6, Section 4] by explaining, in particular, the proof of Proposition 4.3. It could be useful to include also the computations of Remark 5.9. As a nice application, we can compute the Euler characteristic of the relative Prym variety [6, Proposition 5.1], taking into account the correction explained in [7, Remark 5.5]. If time permits, we will describe the discriminant of the Markushevich-Tikhomirov system and compare it with the computation of the discriminant in [1, Proposition 9.1].

Talk III: Singular fibers of the Arbarello–Saccà Ferretti system

Speaker: Pietro Beri

In this talk we would like to understand the geometric structure of the Lagrangian fibration on the relative Prym variety *P *:= *Prym*(D*/*C) of [1]. As explained in Talk I, the support map *P *→ |*C*| endows *P *with a natural Lagrangian fibration. The general fibre, over the point corresponding to a curve *C**t*, is the Prym variety of the double cover of *C**t** *induced by the universal cover *S *→ *T *of the Enriques surface. The objective of this talk is to describe the fibres lying over the points corresponding to singular curves, the reference being [1, Section 3.4]. In particular one can show that when *C *has only one nodal singularity, the Prym variety *Prym*(*D/C*) is a rank one degeneration of an abelian variety. For the following talks, we will also need to understand the reducible case (see [1, Proposition 3.15]).

Open discussion

Second meeting: 28 June

Talk I: Local geometry of the relative Prym variety of Arbarello–Saccà–Ferretti

Speakers: Corinne Bedussa & Mauro Varesco

This talk should provide a description of the local structure of the relative Prym variety of [1]. The idea is to describe the local geometry of *Prym*(D*/*C) in terms of the ambient moduli space of sheaves *M *:= *M**v,H** *(*S*). The first step is to introduce the Kuranishi map *κ *and explain how to view a formal neighborhood of *M *at a polystable sheaf *F *as a quotient of *κ*−1(0). We can follow [1, Section 4.1], it could be interesting to check [3] or [5] to understand the general techniques.

We will then describe the tangent cone of the moduli space *M *at [*F *] in terms of the Kuranishi map. Note that in order to simplify the computation, one can assume *F *= *F*1 ⊕*F*2; we will explain why this is possible, after a change of the polarization *H *if necessary* *([1, Corollary 3.18 ]). To compute the tangent cone of *M *, one needs to understand the quadratic part of the Kuranishi map and the action of C∗ on its central fibre. Once this is done, it's easy to compute the tangent cone of the Prym variety *P *at [*F *], by identifying it with the fixed locus of the symplectic involution inside the tangent cone of *M *. This covers all [1, Section 4]. Similar techniques are also used in [6, Section 2 and Theorem 3.4].

Talk II: Singularities of the relative Prym varieties

Speakers: Corinne Bedussa & Mauro Varesco

By exploiting the geometry of the Kuranishi map [5], one can show that locally at a polystable sheaf *F *= *F*1 ⊕ *F*2, the relative Prym variety is isomorphic to its tangent cone (with an extra assumption on the support of the *F**i*'s [1, Proposition 5.1]). This property can be used to prove that, when the linear system |*C*| on the Enriques surface is non-hyperelliptic, the variety *Prym*(D*/*C) does not admit a symplectic desingularization. In [1], this is proved by showing that *P *is Q-factorial and terminal (non-smooth) at [*F *], therefore it does not admit a symplectic desingularization. Note that the terminality is a consequence of [8, Corollary 1] which shows that the symplectic singularities in codimension at least 4 are terminal. Similar methods allow to describe the analytic type of the 28 isolated singularities of the Markushevich-Tihomirov system; we explain this result ([6, Theorem 3.4]) and compare with [1].

Talk III: Symplectic desingularization in the hyperelliptic case

Speaker: Fabrizio Anella

When the linear system |*C*| is hyperelliptic, one can show that the variety *P *:= *Prym*(D*/*C) admits a symplectic desingularization. The key fact is that the Prym involution is induced, in this case, by a symplectic involution *κ *on the K3 surface *S*. By studying the walls of the ample cone of *S*, explain how to choose a polarization which makes the corresponding Prym variety both smooth and symplectic [1, Proposition 6.7]). Since changing the polarization preserves the birational type, it will follow that any such Prym variety has a symplectic desingularization. We then prove that *P *is bimeromorphic to a moduli space of sheaves on the minimal desingularization of the quotient *S/κ *and explain why this is a K3 surface. As a consquence, either *P *itself or its

desingularization are manifolds of *K*3-type. You can prove these results by following [1, Section 6].

Open discussion

Third meeting: 7 July

Open Discussion: Geometric description of the relative Prym varieties

Speaker: Pietro Beri

The relative Prym variety of Markushevich and Tikhomirov is bimeromorphic to the quotient *M *of the Hilbert scheme *S*[2] by the involution de ned in [6, Lemma 5.2]. The fixed locus of this involution can be explicitly described: it's the union of 28 isolated points and a smooth irreducible surface Σ. We analyze the partial resolution *M *′ of *M *obtained by blowing up the image of Σ, and explain how *P *can be related to *M *′ via a Mukai flop (see Corollary 5.7 in [6]). Compare with the constructions in Chapter 8 of [1] (see the proof of Theorem 8.1), and understand the geometry of the map Φ of [1]. Is it possible to prove an analogue of Lemma 5.2 in this setting? Can we use these methods to compute the second Betti number of this Prym variety?

Talk II: The hyperkähler SYZ-conjecture

Speaker: Dimitri Markushevich

We prove the conjecture for the hyperkähler manifolds of K3[n]-type. More precisely, we show that any manifold of -type endowed with a non-trivial nef divisor of Beauville Bogomolov square 0 admits a Lagrangian fibration.

Talk III: The non-archimedean SYZ fibration

Speaker: Johannes Nicaise

It was a fundamental insight of Kontsevich and Soibelman that the conjectural SYZ fibration should be more readily accessible through non-archimedean analytic geometry over the field of complex Laurent series. Together with Chenyang Xu and Tony Yue Yu, we have combined this idea with ingredients from the Minimal Model Programme (MMP) to construct non-archimedean SYZ fibrations for any maximal degeneration of Calabi-Yau varieties, and verify some of its expected properties. In particular, we proved that it is a torus fibration away from a codimension 2 subset of the base. I will explain this construction and illustrate the interplay with the MMP. No prior knowledge of non-archimedean geometry is required.

Possible further topics:

- Degeneration of Beauville-Mukai systems to a compactification of the GL-Hitchin system (Donagi-Ein-Lazarsfeld)
- Degeneration of Prym Lagrangian fibrations to a compactification of the
*Sp*(2*n,*C) Hitchin system (Sawon-Shen) - Non-Archimedean SYZ fibration (Nicaise-Xu-Yue Yu)
- Lagrangian fibrations by intermediate Jacobians and Prym varieties (Mumford, Casalaina Martin-Grushevsky-Hulek-Laza, Laza-Saccà-Voisin)
- Conjecture: every hyperkähler variety with a relative compactified Jacobian structure is a Beauville-Mukai system (Markushevich, Sawon)
- Lagrangian fibrations and Hitchin systems.

References

- E. Arbarello, G. Saccà, and A. Ferretti, Relative Prym varieties associated to the double cover of an Enriques surface, J. Differential Geom. 100 (2015), no. 2, 191-250.

- A. Beauville, Counting rational curves on K3 surfaces, Duke Math. J. 97 (1999), no. 1, 99-108.

- D. Kaledin, M. Lehn and C. Sorger, Singular symplectic moduli spaces, Invent. Math., 164(3):591-614, 2006.

- E. Franco, Degeneration of natural Lagrangians and Prymian integrable systems, arXiv:2103.05553.

- M. Lehn and C. Sorger, La singularité de O'Grady, J. Algebraic Geom., 15(4):753-770, 2006.

- D. Markushevich, A.S. Tikhomirov, New symplectic V-manifolds of dimension four via the relative compactified Prymian, International Journal of Mathematics, Vol. 18, No. 10 (2007) 1187-1224, World Scientific Publishing Company.

- T. Matteini, A singular symplectic variety of dimension 6 with a Lagrangian Prymian fibration, Manuscripta Math. 149 (2016), no. 1-2, 131-151.

- Y. Namikawa, A note on symplectic singularities, arXiv:0101028, 2001.

- V.V. Nikulin, Factor groups of groups of the automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections, Soviet Math. Dokl., 20 (1979), 1156-1158.

- C. Shen, Lagrangian Fibrations by Prym Varieties, PhD thesis available at https://cdr.lib.unc.edu/concern/dissertations/wm117t928