The divergence conjecture for CR QQ-curvature

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 TaT_{\overline{a}} of θ\theta such that

Qθ=aTa+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.

Sources & referencesView supporting material

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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