Lagrangian fibrations by Prym varieties

Sawon, Justin

doi:10.21711/231766362020/rmc479

Cited by 3 publications

(9 citation statements)

References 31 publications

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“…These Prymian integrable systems on K3 surfaces were first considered by Markushevich-Tikhomirov [40], and extended by Arbarello-Saccà-Ferretti [1], Matteini [41] and Sawon-Shen [50,52]. In [49], Sawon conjectured that these Prymian integrable systems degenerate into integrable systems related to the Hitchin system, leaving open the description of these conjectural systems.…”

Section: Context and Motivationmentioning

confidence: 99%

Degeneration of natural Lagrangians and Prymian integrable systems

Franco

2022

Math. Z.

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Starting from an anti-symplectic involution on a K3 surface, one can consider a natural Lagrangian subvariety inside the moduli space of sheaves over the K3. One can also construct a Prymian integrable system following a construction of Markushevich–Tikhomirov, extended by Arbarello–Saccà–Ferretti, Matteini and Sawon–Shen. In this article we address a question of Sawon, showing that these integrable systems and their associated natural Lagrangians degenerate, respectively, into fix loci of involutions considered by Heller–Schaposnik, García-Prada–Wilkin and Basu–García-Prada. Along the way we find interesting results such as the proof that the Donagi–Ein–Lazarsfeld degeneration is a degeneration of symplectic varieties, a generalization of this degeneration, originally described for K3 surfaces, to the case of an arbitrary smooth projective surface, and a description of the behaviour of certain involutions under this degeneration.

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Section: Context and Motivationmentioning

confidence: 99%

Degeneration of natural Lagrangians and Prymian integrable systems

Franco

2022

Math. Z.

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“…The class that we consider here arises from an anti-symplectic involution on a K3 surface. These systems were first constructed by Markushevich and Tikhomirov [MT] and later generalized by Arbarell, Saccà and Ferretti [ASF], Matteini [Mn], Sawon and Shen [SS2,Sh] and others. The case considered in [MT] corresponds to an anti-symplectic involution on a smooth K3 surface, whose associated quotient is a del Pezzo surface of degree 2.…”

Section: The Dualizing Involution and Prymian Fibrationsmentioning

confidence: 99%

“…Matteini [Mn] generalized the construction of [MT] to the case of antisymplectic involutions on K3 associated to a del Pezzo surface of degree 3. Sawon and Shen [SS2,Sh] studied the case of an antisymplectic involution on a K3 whose quotient is a del Pezzo surface of degree 1.…”

Section: The Dualizing Involution and Prymian Fibrationsmentioning

confidence: 99%

“…with the structure of an integrable system, that we call the Prymian integrable system associated to the anti-symplectic involution ζ − X . This construction has been considered by several authors [MT,ASF,Mn,SS2], although in all these cases the first Chern class is chosen to be primitive. As we are interested in the non-primitive case, we include a proof of it instead of just citing the previous articles.…”

Section: The Dualizing Involution and Prymian Fibrationsmentioning

confidence: 99%

“…Degenerations of natural Lagrangians and Prymian integrable systems. Sawon conjectured in [Sw] that the Prymian integrable systems constructed by Markushevich-Tikhomirov [MT], Arbarello-Saccà-Ferretti [ASF] and Matteini [Mn], degenerate into integrable systems related to the Hitchin system, leaving open the description of these conjectural systems• In this section we provide an explicit description of the degenerations of Section 4.3 in the cases studied in Section 2.4, focusing on the cases specified in [MT,ASF,Mn,SS2,Sh], which are constructed using primitive first Chern classes. We also review the degeneration given by Sawon-Shen [SS1] into the Sp(2m, C)-Higgs moduli space, studying as well the degeneration of the associated natural Lagrangian.…”

Section: Since ζ +mentioning

confidence: 99%

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Degeneration of natural Lagrangians and Prymian integrable systems

Franco¹

2021

Preprint

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Starting from an anti-symplectic involution on a K3 surface, one can consider a natural Lagrangian subvariety inside the moduli space of sheaves over the K3. One can also construct a Prymian integrable system following a construction of Markushevich-Tikhomirov, extended by Arbarello-Saccà-Ferretti, Matteini and Sawon-Chen. In this article we address a question of Sawon, showing that these integrable systems and their associated natural Lagrangians degenerate, respectively, into fix loci of involutions considered by Heller-Schaposnik, Garcia-Prada-Wilkins and Basu-Garcia-Prada.Along the way we find interesting results such as the proof that the Donagi-Ein-Lazarsfeled degeneration is a degeneration of symplectic varieties, a generalization of this degeneration, originally described for K3 surfaces, to the case of an arbitrary smooth projective surface, and a description of the behaviour of certain involutions under this degeneration.

show abstract