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Introduction

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2072)

Abstract

It is hard to overestimate the role of the Jacobian in the theory of smooth complex projective curves. The celebrated theorem of Torelli says that a curve of genus ≥ 2 is determined, up to isomorphism, by its Jacobian and its theta-divisor. Virtually all projective geometric features of a curve can be extracted from its Jacobian. But the Jacobian of a curve has its intrinsic importance and beauty. It is enough to recall that it is a principally polarized abelian variety with an incredibly rich and beautiful theory of theta-functions.

Keywords

  • Chern Class
  • Hilbert Scheme
  • Nilpotent Orbit
  • Perverse Sheave
  • Smooth Projective Surface

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    Sheaves are of rank 2, contrary to the classical situation of line bundles.

  2. 2.

    In [R1], §4, this variety was called “nonabelian Albanese”. This terminology is not quite appropriate, since classically, the Albanese variety involves taking the dual of the space H 1, 0 of holomorphic 1-forms. The variety H parametrizing Higgs structures is certainly more like a direct analogue of the space of holomorphic 1-forms itself. Hence the change of terminology.

  3. 3.

    Throughout the monograph “vertical” means in the direction of the fibres of the projection π in (1.1).

  4. 4.

    A Springer fibre B λ is a fibre of the Springer resolution

    $$\sigma : \mbox{ $\tilde{\mathcal{N}}$}\rightarrow \mbox{ $\mathcal{N}$}(\mathbf{sl}_{d_{\Gamma }^{{\prime}}}(\mathbf{C}))$$

    of the nilpotent cone \(\mbox{ $\mathcal{N}$}(\mathbf{sl}_{d_{\Gamma }^{{\prime}}}(\mathbf{C}))\) of \(\mathbf{sl}_{d_{\Gamma }^{{\prime}}}(\mathbf{C})\) and where a fibre B λ is taken over the nilpotent orbit \(O_{\mbox{ $\lambda $}}\) in \(\mbox{ $\mathcal{N}$}(\mathbf{sl}_{d_{\Gamma }^{{\prime}}}(\mathbf{C}))\) corresponding to a partition λ of \(d_{\Gamma }^{{\prime}}\).

  5. 5.

    What we have in mind here is that correspondences in the middle dimension could be taken as a geometric substitute for the Galois side of the Langlands correspondence.

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Acknowledgements

It is a pleasure to thank Vladimir Roubtsov for his unflagging interest to this work. Our thanks go to the referee of [R1] who also suggested in his report a possible connection of our Jacobian with perverse sheaves.

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Reider, I. (2013). Introduction. In: Nonabelian Jacobian of Projective Surfaces. Lecture Notes in Mathematics, vol 2072. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35662-9_1

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