arXiv subjects

Shing Tak Lam

Publications and source records attributed to Shing Tak Lam.

2 recordsLinked to original sources

arXiv2026

Semistability and asymptotics of geometric flows

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.

math.DG↗

arXiv2025

Canonical metrics on families of vector bundles

We introduce a geometric partial differential equation for families of holomorphic vector bundles, generalising the theory of Hermite--Einstein metrics. We consider families of holomorphic vector bundles which each admit Hermite--Einstein metrics, together with a first order deformation. On such families, we define the family Hermite--Einstein equation for Hermitian metrics, which we view as a notion of a canonical metric in this setting. We prove two main results concerning family Hermite--Einstein metrics. Firstly, we construct Hermite--Einstein metrics in adiabatic classes on product manifolds, assuming the existence of a family Hermite--Einstein metric. Secondly, we prove that the associated parabolic flow admits a unique smooth solution for all time, and use this to show that the Dirichlet problem always admits a unique solution.

math.DG↗