The local structure of compactified Jacobians

Casalaina-Martin, Sebastian; Kass, Jesse Leo; Viviani, Filippo

doi:10.1112/plms/pdu063

Cited by 16 publications

(12 citation statements)

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“…A common feature of these compactifications is that they are constructed using geometric invariant theory (GIT), hence they only give coarse moduli spaces for their corresponding moduli functors. We refer to 1 and 16 for an account on the way the different coarse compactified Jacobians for nodal curves relate to one another.…”

Section: Introductionmentioning

confidence: 99%

Fine compactified Jacobians

Melo

Viviani

2012

Mathematische Nachrichten

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Abstract. To every singular reduced projective curve X one can associate many fine compactified Jacobians, depending on the choice of a polarization on X, each of which yields a modular compactification of a disjoint union of a finite number of copies of the generalized Jacobian of X. We investigate the geometric properties of fine compactified Jacobians focusing on curves having locally planar singularities. We give examples of nodal curves admitting non isomorphic (and even non homeomorphic over the field of complex numbers) fine compactified Jacobians. We study universal fine compactified Jacobians, which are relative fine compactified Jacobians over the semiuniversal deformation space of the curve X. Finally, we investigate the existence of twisted Abel maps with values in suitable fine compactified Jacobians.

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Section: Introductionmentioning

confidence: 99%

Fine compactified Jacobians

Melo

Viviani

2012

Mathematische Nachrichten

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“…for some uniformizer π ∈ R and some integers n, m ∈ N. This is [18,Lemma 6.2], a result proven using the deformation theory techniques used in [11]. When J /S is a family of coarse compactified Jacobians, the Luna slice argument used in loc.…”

Section: The Singularities Of a Compactified Jacobianmentioning

confidence: 99%

Autoduality holds for a degenerating abelian variety

Kass

2017

Res Math Sci

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We prove that certain degenerate abelian varieties that include compactified Jacobians, namely stable semiabelic varieties, satisfy autoduality. We establish this result by proving a comparison theorem that relates the associated family of Picard schemes to the Néron model, a result of independent interest. In our proof, a key fact is that the total space of a suitable family of stable semiabelic varieties has rational singularities.Keywords: Autoduality, Compactified Jacobians, Stable abelic variety Mathematics Subject Classification: Primary 14H40; Secondary 14K30, 14D20In this paper, we prove that certain degenerate abelian varieties satisfy autoduality, a result we now recall. The classical statement of autoduality is a statement about an abelian variety J 0 with an ample divisor 0 that defines a principal polarization. If τ x 0 : J 0 → J 0 denotes translation by a point x 0 , then the morphismfrom J 0 to the (identity component of the) Picard scheme is an isomorphism. The Picard scheme Pic 0 (J 0 /k) is the dual abelian variety, so the fact that (1) is an isomorphism implies that the autoduality theorem holds, i.e., that J 0 is self-dual. Here we construct an isomorphism analogous to (1) in which the abelian variety is replaced by a singular projective variety, namely a stable semiabelic variety. Stable semiabelic varieties are degenerate abelian varieties studied in the moduli theory of abelian varieties in the work of, e.g., Alexeev, Nakamura, and Olsson. An important example of these varieties is the compactified Jacobian of a nodal curve, or moduli space of degree d rank 1, torsion-free sheaves which are required to satisfy a semistability condition when X 0 is reducible. A stable semiabelic variety is acted upon by a natural semiabelian variety J 0 which equals the moduli space of multidegree 0 line bundles when J is a compactified Jacobian.Because J 0 is a (possibly reducible) projective variety, we can form the (identity component of the) Picard scheme Pic 0 (J 0 /k), and Main Theorem of this paper is that there is an isomorphism from J 0 0 to Pic 0 (J 0 /k) that is analogous to (1):Main Theorem (Autoduality) The autoduality theorem holds for stable semiabelic varieties.

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“…The inequality (2.5) was used to define stable rank-1 torsion-free sheaves on nodal curves in [MV12] and in [CMKV15].…”

Section: Fine Compactified Jacobiansmentioning

confidence: 99%

Fourier–Mukai and autoduality for compactified Jacobians. I

Melo

Rapagnetta

Viviani

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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To every singular reduced projective curve X one can associate, following E. Esteves, many fine compactified Jacobians, depending on the choice of a polarization on X, each of which yields a modular compactification of a disjoint union of the generalized Jacobian of X. We prove that, for a reduced curve with locally planar singularities, the integral (or Fourier-Mukai) transform with kernel the Poincaré sheaf from the derived category of the generalized Jacobian of X to the derived category of any fine compactified Jacobian of X is fully faithful, generalizing a previous result of D. Arinkin in the case of integral curves. As a consequence, we prove that there is a canonical isomorphism (called autoduality) between the generalized Jacobian of X and the connected component of the identity of the Picard scheme of any fine compactified Jacobian of X and that algebraic equivalence and numerical equivalence of line bundles coincide on any fine compactified Jacobian, generalizing previous results of Arinkin, Esteves, Gagné, Kleiman, Rocha, Sawon.The paper contains an Appendix in which we explain how our work can be interpreted in view of the Langlands duality for the Higgs bundles as proposed by Donagi-Pantev.Key words and phrases. Compactified Jacobians, Fourier-Mukai transform, autoduality, Poincaré bundle, Abel map. 1 More generally, for an arbitrary abelian variety A with dual abelian variety A ∨ , Mukai proved that the Fourier-Mukai transform associated to the Poincaré line bundle on A × A ∨ gives an equivalence between the bounded derived category of A and that of A ∨ . 1 are separating nodes (e.g. nodal curves of compact type). Finally, while this paper was under the referee process, two related papers have appeared on arXiv: D. Arinkin and R. Fedorov established in [AF16] a partial Fourier-Mukai transform for degenerate abelian schemes (in characteristic zero); J. L. Kass proved in [Kas] that autoduality holds true for (possibly coarse) compactified Jacobians of nodal curves and stable quasiabelian varieties (in characteristic zero).The main motivation for this work comes from the Langlands duality conjecture for Hitchin systems proposed by Donagi-Pantev in [DP12] as a classical limit of the conjectural geometric Langlands correspondence (which we review in more details in the Appendix ). In the special case of the Langlands self-dual linear group GL r , the Langlands duality conjecture predicts an autoequivalence Φ :of the bounded derived category of quasi-coherent sheaves of the moduli stack M of Higgs bundles of rank r on a fixed smooth projective curve C, which should intertwine the classical limit tensorization functors with the classical limit Hecke functors (see [DP12, Sec. 2] for more details). The moduli stack M of Higgs bundles admits a morphism H : M → A, called the Hitchin morphism, to an affine space A parametrizing certain degree-r singular covers of C, called spectral curves (see (10.1)). According to the so-called spectral correspondence (see Fact 10.3), the fiber of H −1 ( C) over a given spect...

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The local structure of compactified Jacobians

Cited by 16 publications

References 40 publications

Fine compactified Jacobians

Fine compactified Jacobians

Autoduality holds for a degenerating abelian variety

Fourier–Mukai and autoduality for compactified Jacobians. I

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