The divergence conjecture for CR QQ-curvature

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Let MM be a strictly pseudoconvex CR manifold of dimension 2n+12n+1, with integrable holomorphic tangent bundle T1,0MT^{1,0}M. Choose a contact form θ\theta, and let QθQ_\theta denote its CR QQ-curvature. Write ∇a\nabla_a and ∇a‾\nabla_{\overline{a}} for covariant derivatives associated with the Tanaka–Webster connection.

Divergence conjecture. On CR manifolds, the QQ-curvature is a divergence: there exist form-valued local invariants TaT_a and Ta‾T_{\overline{a}} of θ\theta such that

Qθ=∇a‾Ta+∇aTa‾.Q_\theta=\nabla_{\overline{a}}T_a+\nabla_aT_{\overline{a}}.

If true, this would provide a local-invariant divergence expression for the CR QQ-curvature, extending the explicit divergence formula known in dimension 33. The supplied text does not indicate whether the conjecture has been resolved.

References

Primary source

Spyros Alexakis and Kengo Hirachi, “Integral Kahler Invariants and the Bergman kernel asymptotics for line bundles”, arXiv:1501.02463 (2015).

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