Semistability and asymptotics of geometric flows

Shing Tak Lam

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arXiv · 2606.09235

Semistability and asymptotics of geometric flows

Shing Tak Lam

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Abstract

We prove that the asymptotics of the Hermitian-Yang-Mills flow on a slope semistable holomorphic vector bundle over a compact K\"ahler manifold are determined algebro-geometrically, via the iterated filtration defined by Haiden-Katzarkov-Kontsevich-Pandit. This proves a conjecture of Haiden-Katzarkov-Kontsevich-Pandit in this setting. Moreover, we prove a non-linear analogue, relating the asymptotics of the Calabi flow near a cscK manifold to the iterated balancing filtration of the deformation space. In both settings, we reduce the infinite-dimensional flow to a finite-dimensional flow, following the foundational work of Chen-Sun. In finite dimensions, we prove that the asymptotics of the moment map flow are determined by the iterated balancing filtration, proving a conjecture of Ib\'a\~nez N\'u\~nez.

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Shing Tak Lam. 2026-06-08. Semistability and asymptotics of geometric flows. https://arxiv.org/abs/2606.09235

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