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arXiv · 2507.17375

Lines in the space of K\"ahler metrics

Tamás Darvas·Nicholas McCleerey
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Abstract

We establish a Ross-Witt Nystr\"om correspondence for weak geodesic lines in the (completed) space of K\"ahler metrics. We construct a wide range of weak geodesic lines on arbitrary projective K\"ahler manifolds that are not generated by holomorphic vector fields, in the process disproving a folklore conjecture popularized by Berndtsson. Remarkably, some of these weak geodesic lines turn out to be smooth. In the case of Riemann surfaces, our results can be significantly sharpened. Finally, we investigate the validity of Euclid's fifth postulate for the space of K\"ahler metrics.

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Tamás Darvas, Nicholas McCleerey. 2025-07-23. Lines in the space of K\"ahler metrics. https://arxiv.org/abs/2507.17375

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