Higgs Bundles and Local Systems on Elliptic Curves

If S is an elliptic curve, the total space X of the cotangent bundle of S is the moduli space of rank one and degree zero Higgs bundles on S and the corresponding character variety Y is C* x C*. The punctual Hilbert scheme X^[n] of X can be identified with the moduli space of stable marked Higgs bundles on S and there is a natural isomorphism of graded vector spaces between the rational cohomology groups of the Hilbert schemes of X and Y that exchanges the perverse Leray filtration on X^[n] with the halved weight filtration on Y^[n]. We prove that there is a diffeomorphism between the Hilbert schemes that induces the given isomorphism in cohomology. We also give a complete description of Higgs bundles corresponding to subschemes of length n ≤ 3. Moreover, we discuss a conjecture by Simpson on the compactification of the moduli space of Higgs bundles and on the dual boundary complex of the character variety, proving a result going in the direction of Simpson’s conjecture.