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Vanishing conjecture for L2L^2 harmonic forms on Hitchin's moduli space

Let M{\cal M} be Hitchin's moduli space of rank-22 Higgs bundles with fixed determinant of degree 11 over a Riemann surface Σ\Sigma of genus g>1g>1. It is a simply connected, non-compact manifold of dimension 12g−1212g-12 with a complete hyperkähler metric. Vanishing conjecture. There are no non-trivial L2L^2 harmonic forms on Hitchin's moduli space of Higgs bundles.

The conjecture is posed as the analogue of Sen's conjecture for Hitchin's moduli space. The paper proves that the forgetful map Hcpt∗(M)→H∗(M)H^*_{cpt}({\cal M})\to H^*({\cal M}) is zero, showing that the topology supplies no L2L^2 harmonic forms; the conjecture itself is refuted for parabolic Higgs bundles by the toy example cited in the source.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The vanishing conjecture for L2L^2 harmonic forms on Hitchin's moduli space

    Let M\mathcal{M} be Hitchin's moduli space of Higgs bundles with fixed determinant of degree 11 over a Riemann surface Σ\Sigma of genus g>1g>1. It is a simply connected, non-compact hyperkähler manifold. The vanishing conjecture. There are no non-trivial L2L^2 harmonic forms on Hitchin's moduli space of Higgs bundles.

    This is the analogue for Hitchin's moduli space of Sen's conjecture for the universal covers of monopole moduli spaces. The source proves that the compactly supported-to-ordinary cohomology map vanishes, so topology supplies no L2L^2 harmonic forms; the stated conjecture concerns the absence of all such forms.

    source: Tamas Hausel, “Geometry of the moduli space of Higgs bundles”, arXiv:math/0107040 (2001).

Sources & referencesView supporting material

Primary source

Tamas Hausel, “Vanishing of intersection numbers on the moduli space of Higgs bundles”, arXiv:math/9805071 (1999).

harmonic-formshiggs-bundlesmoduli-spaces

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