K\"ahler thresholds

Gabriella Clemente; Carlos Simpson

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arXiv · 2606.08224

K\"ahler thresholds

Gabriella Clemente·Carlos Simpson

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Abstract

The topology of K\"ahler manifolds is largely determined by the geometry due to its rigidity. In particular, the cohomology and Hodge theory of compact K\"ahler manifolds is quite restricted. We prove that within a certain threshold, the odd Betti numbers of any compact almost-hermitian manifold satisfying a degenerate K\"ahler condition are even, and the even Betti numbers are strictly positive. We call this new type of degenerated K\"ahler manifold acK. Our approach to proving these results makes use of a compactness theorem for acK manifolds, and a new version of Hodge theory for compact manifolds endowed with a Sobolev regular K\"ahler strucutre. In addition, we lay out a program to pursue the study of acK geometry that accommodates not only the classical viewpoint, but also constructive and finitary proofs, as well as formalization with proof assistants.

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BibTeX RIS

Gabriella Clemente, Carlos Simpson. 2026-06-06. K\"ahler thresholds. https://arxiv.org/abs/2606.08224

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