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Néron models of intermediate Jacobians associated to moduli spaces

  • Ananyo DanEmail author
  • Inder Kaur
Article
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Abstract

Let \(\pi _1:\mathcal {X} \rightarrow \Delta \) be a flat family of smooth, projective curves of genus \(g \ge 2\), degenerating to an irreducible nodal curve \(X_0\) with exactly one node. Fix an invertible sheaf \(\mathcal {L}\) on \(\mathcal {X}\) of relative odd degree. Let \(\pi _2:\mathcal {G}(2,\mathcal {L}) \rightarrow \Delta \) be the relative Gieseker moduli space of rank 2 semi-stable vector bundles with determinant \(\mathcal {L}\) over \(\mathcal {X}\). Since \(\pi _2\) is smooth over \(\Delta ^*\), there exists a canonical family \(\widetilde{\rho }_i:\mathbf {J}^i_{\mathcal {G}(2, \mathcal {L})_{\Delta ^*}} \rightarrow \Delta ^{*}\) of i-th intermediate Jacobians i.e., for all \(t \in \Delta ^*\), \((\widetilde{\rho }_i)^{-1}(t)\) is the i-th intermediate Jacobian of \(\pi _2^{-1}(t)\). There exist different Néron models \(\overline{\rho }_i:\overline{\mathbf {J}}_{\mathcal {G}(2, \mathcal {L})}^i \rightarrow \Delta \) extending \(\widetilde{\rho }_i\) to the entire disc \(\Delta \), constructed by Clemens, Saito, Schnell, Zucker and Green–Griffiths–Kerr. In this article, we prove that in our setup, the Néron model \(\overline{\rho }_i\) is canonical in the sense that the different Néron models coincide and is an analytic fiber space which graphs admissible normal functions. We also show that for \(1 \le i \le \max \{2,g-1\}\), the central fiber of \(\overline{\rho }_i\) is a fibration over product of copies of \(J^k(\mathrm {Jac}(\widetilde{X}_0))\) for certain values of k, where \(\widetilde{X}_0\) is the normalization of \(X_0\). In particular, for \(g \ge 5\) and \(i=2, 3, 4\), the central fiber of \(\overline{\rho }_i\) is a semi-abelian variety. Furthermore, we prove that the i-th generalized intermediate Jacobian of the (singular) central fibre of \(\pi _2\) is a fibration over the central fibre of the Néron model \(\overline{\mathbf {J}}^i_{\mathcal {G}(2, \mathcal {L})}\). In fact, for \(i=2\) the fibration is an isomorphism.

Keywords

Torelli theorem Intermediate Jacobians Néron models Nodal curves Gieseker moduli space Limit mixed Hodge structures 

List of symbols

\(X_0,x_0\)

Irreducible nodal curve \(X_0\) with node at \(x_0\)

\(\pi : \widetilde{X}_0 \rightarrow X_0\)

Normalization of \(X_0\)

\(\Delta , \Delta ^*\)

Open, unit disc \(\Delta \) and \(\Delta ^*:=\Delta \backslash \{0\}\)

\(\rho : \mathcal {Y} \rightarrow \Delta \)

Family of projective varieties, smooth over \(\Delta ^*\)

\(\mathcal {Y}_t\)

The fiber \(\rho ^{-1}(t)\) for any \(t \in \Delta \)

\(\mathcal {Y}_\infty \)

The base change of the family \(\rho \) under the natural morphism \(\mathfrak {h} \rightarrow \Delta ^* \hookrightarrow \Delta \), where \(\mathfrak {h}\) is the universal covering of \(\Delta ^*\)

\(\mathcal {Y}_{\Delta ^*}\)

restriction of \(\mathcal {Y}\) to \(\Delta ^*\)

\(\mathcal {H}^i_{\mathcal {Y}_{\Delta ^*}}, F^p\mathcal {H}^i_{\mathcal {Y}_{\Delta ^*}}\)

Hodge bundles associated to the family \(\mathcal {Y}_{\Delta ^*}\)

\(\overline{\mathcal {H}}^i_{\mathcal {Y}_{\Delta ^*}}, F^p\overline{\mathcal {H}}^i_{\mathcal {Y}_{\Delta ^*}}\)

Canonical extensions of \(\mathcal {H}^i_{\mathcal {Y}_{\Delta ^*}}, F^p\mathcal {H}^i_{\mathcal {Y}_{\Delta ^*}}\), respectively

\(\widetilde{\rho }: \mathbf {J}^i_{\mathcal {Y}_{\Delta ^*}} \rightarrow \Delta ^*\)

Family of i-th intermediate Jacobians associated to \(\mathcal {Y}_{\Delta ^*}\)

\(\overline{\rho }: \overline{\mathbf {J}}^i_{\mathcal {Y}} \rightarrow \Delta \)

Néron model associated to \(\widetilde{\rho }\)

\(T_{s,i}, T_{s,i}^{\mathbb {Q}}\)

Local monodromy transformation associated to \(\rho \)

\(T_i:H^i(\mathcal {Y}_\infty , \mathbb {Q}) \rightarrow H^i(\mathcal {Y}_\infty , \mathbb {Q})\)

Limit monodromy transformation

\(N_i\)

\(\log (T_i)\)

\(\mathrm {sp}_i: H^i(\mathcal {Y}_0,\mathbb {Z}) \rightarrow H^i(\mathcal {Y}_\infty , \mathbb {Z})\)

Specialization morphism

\(M_Y(2,\mathcal {L}')\)

Moduli space of rank 2, semi-stable sheaves with determinant \(\mathcal {L}'\) over Y

\(\pi _1: \mathcal {X} \rightarrow \Delta \)

Family of projective curves with central fiber \(X_0\), smooth over \(\Delta ^*\)

\(\mathcal {L}, \mathcal {L}_0, \widetilde{\mathcal {L}}_0\)

Odd degree invertible sheaf \(\mathcal {L}\) on \(\mathcal {X}\), \(\mathcal {L}_0:=\mathcal {L}|_{X_0}\), \(\widetilde{\mathcal {L}}_0:=\pi ^*\mathcal {L}_0\)

\(\widetilde{\pi }_1: \widetilde{\mathcal {X}} \rightarrow \mathcal {X} \xrightarrow {\pi } \Delta \)

Blow-up of \(\mathcal {X}\) at \(x_0\)

\(\pi _2: \mathcal {G}(2,\mathcal {L}) \rightarrow \Delta \)

Relative Gieseker moduli space associated to \(\pi _1\)

\(\mathcal {G}_{X_0}(2,\mathcal {L}_0)\)

Central fiber of the moduli space \(\mathcal {G}(2,\mathcal {L})\)

\(\mathcal {G}_0, \mathcal {G}_1\)

The two irreducible components of \(\mathcal {G}_{X_0}(2,\mathcal {L}_0)\)

Mathematics Subject Classification

Primary 14C30 14C34 14D07 32G20 32S35 14D20 Secondary 14H40 

Notes

Acknowledgements

We thank Prof. J. F. de Bobadilla, Dr. B. Sigurdsson and Dr. S. Basu for numerous discussions. The first author is currently supported by ERCEA Consolidator Grant 615655-NMST and also by the Basque Government through the BERC \(2014-2017\) program and by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-\(2013-0323\). The second author is funded by CAPES-PNPD scholarship.

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Copyright information

© Universidad Complutense de Madrid 2019

Authors and Affiliations

  1. 1.BCAM - Basque Centre for Applied MathematicsBilbaoSpain
  2. 2.Pontifical Catholic University of Rio de Janeiro (PUC-Rio)Rio de JaneiroBrazil

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