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The HcscK equations in symplectic coordinates

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Abstract

The Donaldson–Fujiki Kähler reduction of the space of compatible almost complex structures, leading to the interpretation of the scalar curvature of Kähler metrics as a moment map, can be lifted canonically to a hyperkähler reduction. Donaldson proposed to consider the corresponding vanishing moment map conditions as (fully nonlinear) analogues of Hitchin’s equations, for which the underlying bundle is replaced by a polarised manifold. However this construction is well understood only in the case of complex curves. In this paper we study Donaldson’s hyperkähler reduction on abelian varieties and toric manifolds. We obtain a decoupling result, a variational characterisation, a relation to K-stability in the toric case, and prove existence and uniqueness under suitable assumptions on the “Higgs tensor”. We also discuss some aspects of the analogy with Higgs bundles.

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Notes

  1. Notice that there is a sign misprint in the statement; the correct sign can be found in [16, equation (3.3.6)].

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Acknowledgements

Part of this work was written while the first author was visiting the University of Illinois at Chicago. The first author wishes to thank the Department of Mathematics at UIC for kind hospitality, and particularly Julius Ross for many helpful discussions related to this work. We are grateful to Julien Keller and Vestislav Apostolov for some useful comments on an earlier version of these results. We are very grateful to the reviewer for the valuable suggestions, in particular regarding our stability result, Theorem 1.7. We would also like to thank all the participants in the Kähler geometry seminars at IGAP, Trieste.

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    Carlo Scarpa & Jacopo Stoppa

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Correspondence to Carlo Scarpa.

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Scarpa, C., Stoppa, J. The HcscK equations in symplectic coordinates. Math. Z. 301, 75–113 (2022). https://doi.org/10.1007/s00209-021-02902-8

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