The ∂-problem on Z(q)-domains

2024/03/07 by Chakrabarti, Debraj, Harrington, Phillip S., Raich, Andrew
#32F10 #32F32 #32W05 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2403.04756

Abstract

Given a complex manifold containing a relatively compact Z(q) domain, we give sufficient geometric conditions on the domain so that its L²-cohomology in degree (p,q) (known to be finite dimensional) vanishes. The condition consists of the existence of a smooth weight function in a neighborhood of the closure of the domain, where the complex Hessian of the weight has a prescribed number of eigenvalues of a particular sign, along with good interaction at the boundary of the Levi form with the complex Hessian, encoded in a subbundle of common positive directions for the two Hermitian forms.

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