On the Hitchin System

Details On the Hitchin System On the Hitchin System

1994-10-14

arxiv.org/abs/alg-geom/9410015v1

Mathematics

Algebraic Geometry

23 pages, LaTeX Version 2.09 <7 Dec 1989>

Scientific paper

The Hitchin system is a completely integrable hamiltonian system (CIHS) on the cotangent space to the moduli space of semi-stable vector bundles over a curve. We consider the case of rank-two vector bundles with trivial determinant. Such a bundle $E$ defines a divisor $D_E$ in the Jacobian of the curve and for any smooth point of $D_E$ we define a cotangent vector (a Higgs field). The Hitchin map on these Higgs fields is then determined in terms of the Gauss map on the divisor $D_E$. We apply the results to the $g=2$ case and show how Hitchin's system is related to classical line geometry in $\PP^3$.

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