Exchange between perverse and weight filtration for the Hilbert schemes of points of two surfaces

Cataldo, Mark Andrea de; Hausel, Tamás; Migliorini, Luca

doi:10.5427/jsing.2013.7c

Cited by 15 publications

(19 citation statements)

References 12 publications

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“…In the paper [10] we prove that a result analogous to our main theorem P = W holds in a situation that is expected to be closely related to the moduli space of certain parabolic Higgs bundles of rank n on a genus one curve. Interestingly, in this case, the coincidence of the two filtrations holds, whereas the result (1) above, concerning the supports of the perverse sheaves being maximal on a large open set, fails, due to the fact that every new stratum contributes a new direct summand sheaf.…”

Section: )mentioning

confidence: 99%

Topology of Hitchin systems and Hodge theory of character varieties: the case A_1

2012

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For G = GL2, PGL2, SL2 we prove that the perverse filtration associated with the Hitchin map on the rational cohomology of the moduli space of twisted G-Higgs bundles on a compact Riemann surface C agrees with the weight filtration on the rational cohomology of the twisted G character variety of C when the cohomologies are identified via non-Abelian Hodge theory. The proof is accomplished by means of a study of the topology of the Hitchin map over the locus of integral spectral curves.

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Section: )mentioning

confidence: 99%

Topology of Hitchin systems and Hodge theory of character varieties: the case A_1

2012

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“…A striking phenomenon was discovered by de Cataldo, Hausel, and Migliorini in [deCHM12], namely that the canonical isomorphism…”

Section: 4mentioning

confidence: 96%

“…Such a phenomenon is referred to as "the P = W conjecture." The original P = W conjecture was verified for n = 2 in [deCHM12], while the n 3 cases are still open.…”

Section: 4mentioning

confidence: 96%

“…Our main motivation for the study of perverse filtrations for Hilbert schemes is the P = W conjecture [deCHM12] for (parabolic) Higgs bundles; see Section 1.4.…”

Section: Moduli Of Parabolic Higgs Bundlesmentioning

confidence: 99%

“…Following [deCHM12], we define P f k H d (X, Q) ⊂ H d (X, Q) to be the image of (1.2) (the shift [dim X − r] is to ensure that the perverse filtration is concentrated in the degrees [0, 2r]; see Section 2), and we call the increasing filtration…”

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confidence: 99%

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Perverse filtrations, Hilbert schemes, and the $P=W$ Conjecture for parabolic Higgs bundles

Shen¹,

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2021

Alg. Geom.

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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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Exchange between perverse and weight filtration for the Hilbert schemes of points of two surfaces

Cited by 15 publications

References 12 publications

Topology of Hitchin systems and Hodge theory of character varieties: the case A_1

Topology of Hitchin systems and Hodge theory of character varieties: the case A_1

Perverse filtrations, Hilbert schemes, and the $P=W$ Conjecture for parabolic Higgs bundles

Degeneration of natural Lagrangians and Prymian integrable systems

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