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Topics in the classification of local systems

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First French-Spanish Congress of Mathematics Zaragoza, July 9–13, 2007

Topics in the classification of local systems

Carlos Simpson

We discuss a joint result with K. Corlette classifying local systems of rank 2 on a complex quasiprojective variety [1]. We also discuss the basics of the classification of local systems over P

1

− {x

1

, . . . , x

k

} based on Katz’s algorithm [3]. Finally we discuss a simplified approach to local systems over a quasiprojective variety by compactify- ing with Deligne-Mumford stacks, and a related formula for the Chern classes of a parabolic bundle [2].

Keywords: Fundamental group, Representation, Quasiprojective variety, Middle convolution, Deligne-Mumford stack, Parabolic bundle

Mathematics Subject Classification 2000: 14F35

References

[1] K. Corlette and C. Simpson . On the classification of rank two representations of quasiprojective fundamental groups. Preprint arXiv:math/0702287.

[2] J. N. Iyer and C. Simpson . The Chern character of a parabolic bundle, and a parabolic Reznikov theorem in the case of finite order at infinity. Preprint arXiv:math/0612144.

[3] C. Simpson . Katz’s middle convolution algorithm. Preprint arXiv:math/0610526. To appear, Pure and Applied Math. Quarterly.

1

CNRS, Laboratoire J. A. Dieudonn´ e

Universit´ e de Nice-Sophia Antipolis

06108 Nice Cedex 02, France

carlos@math.unice.fr

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