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Geometry of solutions to the c-projective metrizability equation

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Abstract

On an almost complex manifold, a quasi-Kähler metric, with canonical connection in the c-projective class of a given minimal complex connection, is equivalent to a nondegenerate solution of the c-projectively invariant metrizability equation. For this overdetermined equation, replacing this maximal rank condition on solutions with a nondegeneracy condition on the prolonged system yields a strictly wider class of solutions with non-vanishing (generalized) scalar curvature. We study the geometries induced by this class of solutions. For each solution, the strict point-wise signature partitions the underlying manifold into strata, in a manner that generalizes the model, a certain Lie group orbit decomposition of \(\mathbb{C}\mathbb{P}^m\). We describe the smooth nature and geometric structure of each strata component, generalizing the geometries of the embedded orbits in the model. This includes a quasi-Kähler metric on the open strata components that becomes singular at the strata boundary. The closed strata inherit almost CR-structures and can be viewed as a c-projective infinity for the given quasi-Kähler metric.

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Notes

  1. See the proof of Proposition 14 of [17] for a more details.

  2. For details see Proposition 4.1 of [13] or Proposition 7 of [17].

  3. This term and the method of constructing a special boundary scale was first done the setting of projective differential geometry by Sam Porath in [43].

  4. The prolongation connection is a natural modification of the tractor connection. See, e.g., [24] for a construction for general BGG operators.

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  1. Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic

    Keegan J. Flood

  2. Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, 1142, New Zealand

    A. Rod Gover

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K. J. Flood gratefully acknowledges support from the Czech Science Foundation (GAČR) Grant 20-11473S and A. R. Gover gratefully acknowledges support from the Royal Society of New Zealand via Marsden Grants 16-UOA-051 and 19-UOA-008.

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Flood, K.J., Gover, A.R. Geometry of solutions to the c-projective metrizability equation. Annali di Matematica 202, 1343–1368 (2023). https://doi.org/10.1007/s10231-022-01283-x

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