Geometric Quantization of Classical Metrics on the Moduli Space of Canonical Metrics

1 1 Deformation Theory of Complex Structures 2 1.1 Basic idea of analytic deformation theory . . . . . . . . . . . . . . . . 4 1.2 Existence of infinitesimal deformations . . . . . . . . . . . . . . . . . 7 1.3 Completeness of the analytic family . . . . . . . . . . . . . . . . . . . 11 2 Complex Deformation on Fano Kahler-Einstein Manifolds 17 2.1 Deformation of complex structures on Fano manifolds and gauge equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Deformation of the volume form and the Kahler-Einstein form . . . . 30 2.3 Deformation of plurianticanonical sections . . . . . . . . . . . . . . . 37 2.4 L-metric on the direct image sheaf . . . . . . . . . . . . . . . . . . . 44 2.5 Deformation of holomorphic vector fields . . . . . . . . . . . . . . . . 54 3 Pluri-subharmonicity of Harmonic Energy 59 3.1 Deformation of Kahler-Einstein manifolds of general type . . . . . . . 60 3.2 First variation of harmonic energy . . . . . . . . . . . . . . . . . . . . 63 3.3 Second variation of harmonic energy . . . . . . . . . . . . . . . . . . 66

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