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Asymptotics of Hitchin’s metric on the Hitchin section. (English) Zbl 1417.53053

Summary: We consider Hitchin’s hyperkähler metric \(g\) on the moduli space \({\mathcal{M}}\) of degree zero SL(2)-Higgs bundles over a compact Riemann surface. It has been conjectured that, when one goes to infinity along a generic ray in \({\mathcal{M}}\), \(g\) converges to an explicit “semiflat” metric \(g^{\mathrm{sf}}\), with an exponential rate of convergence. We show that this is indeed the case for the restriction of \(g\) to the tangent bundle of the Hitchin section \({\mathcal{B} \subset \mathcal{M}}\).

MSC:

53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
32Q15 Kähler manifolds
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
14H15 Families, moduli of curves (analytic)
14H60 Vector bundles on curves and their moduli

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GitHub; blaschke
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References:

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