Abstract
Let \({(\mathcal {X},\Omega)}\) be a closed polarized complex manifold, g be an extremal metric on \({\mathcal {X}}\) that represents the Kähler class Ω, and G be a compact connected subgroup of the isometry group Isom\({(\mathcal {X}, g)}\) . Assume that the Futaki invariant relative to G is nondegenerate at g. Consider a smooth family \({(\mathcal {M}\to B)}\) of polarized complex deformations of \({(\mathcal {X},\Omega)\simeq (\mathcal {M}_0,\Theta_0)}\) provided with a holomorphic action of G which is trivial on B. Then for every \({t\in B}\) sufficiently small, there exists an \({h^{1,1}(\mathcal {X})}\) -dimensional family of extremal Kähler metrics on \({\mathcal {M}_t}\) whose Kähler classes are arbitrarily close to Θ t . We apply this deformation theory to show that certain complex deformations of the Mukai–Umemura 3-fold admit Kähler–Einstein metrics.
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Rollin, Y., Simanca, S.R. & Tipler, C. Deformation of extremal metrics, complex manifolds and the relative Futaki invariant. Math. Z. 273, 547–568 (2013). https://doi.org/10.1007/s00209-012-1019-7
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DOI: https://doi.org/10.1007/s00209-012-1019-7