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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Sheaves are of rank 2, contrary to the classical situation of line bundles.
- 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.
Throughout the monograph “vertical” means in the direction of the fibres of the projection π in (1.1).
- 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.
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.
References
A.A. Beĭlinson, J. Bernstein, P. Deligne, Faisceaux pervers. Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque. Soc. Math. France, Paris 100, 5–171 (1982)
R. Bott, S.S. Cern, Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections. Acta Math. 114, 71–112 (1965)
V.G. Drinfeld, Two-dimensional l-adic representations of the fundamental group of a curve over finite field and automorphic forms on GL(2). Am. J. Math. 105, 85–114 (1983)
E. Frenkel, Lectures on the Langlands program and conformal field theory, in Frontiers in Number Theory, Physics, and Geometry II (Springer, Berlin, 2007), pp. 387–533
V. Ginzburg, Perverse sheaves on a loop group and Langlands duality [arxiv: alg-geom/ 9511007]
L. Göttsche, W. Soergel, Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces. Math. Ann. 296(2), 235–245 (1993)
M.L. Green, Quadrics of rank four in the ideal of a canonical curve. Invent. Math. 75(1), 85–104 (1984)
P. Griffiths (ed.), in Topics in Transcendental Algebraic Geometry. Annals of Mathematical Studies, vol. 106 (Princeton University Press, Princeton, 1984)
M. Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture. J. Am. Math. Soc. 14(4), 941–1006 (2001)
I. Mirkovič, K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. Math. (2) 166(1), 95–143 (2007)
D. Mumford, Curves and Their Jacobians (The University of Michigan Press, Ann Arbor, 1975)
H. Nakajima, in Lectures on Hilbert Schemes of Points on Surfaces. University Lecture Series, vol. 18 (AMS, Providence, 1999)
I. Reider, Nonabelian Jacobian of smooth projective surfaces. J. Differ. Geom. 74, 425–505 (2006)
C. Simpson, Higgs Bundles and Local Systems, vol. 75 (Publ. Math. I.H.E.S., 1992), Springer, Berlin, pp. 5–95
A. Weil, Courbes Algébriques et Variétés Abéliennes (Hermann, Paris, 1971)
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.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-642-35662-9_1
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-35661-2
Online ISBN: 978-3-642-35662-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)