https://neckpinch.com/authors/michael-hallam/New arXiv papers by Michael Hallam, from Neckpinch Tue, 03 Nov 2026 08:37:45 GMT https://neckpinch.com/papers/2408.06267/arxiv:2408.06267Mon, 12 Aug 2024 00:00:00 GMTMichael Hallam, Abdellah Lahdili — We introduce a new weighted version of the Hermite–Einstein equation, along with notions of weighted slope (semi/poly)stability, and prove that a vector bundle admits a weighted Hermite–Einstein metric if and only if it is weighted slope polystable. The new equation encompasses several well-known examples of canonical Hermitian metrics on vector bundles, including the usual Hermite–Einstein metrics, Kähler–Ricci solitons, and transversally Hermite–Einstein metrics on certain Sasaki manifolds. We prove that the equation arises naturally as a moment map, that solutions to the equation are unique up to scaling, and demonstrate a weighted Kobayashi–Lübke inequality satisfied by vector bundles admitting a weighted Hermite–Einstein metric. As an application of our techniques, we extend a bound of Tian on the Ricci curvature to a bound on a modified Ricci curvature, related to the existence of Kähler–Ricci solitons. Along the way, we introduce a new weighted vortex equation, as well as a weighted analogue of Gieseker stability. A key technical point is the application of a new extension of Inoue's equivariant intersection numbers to arbitrary weight functions on the moment polytope of a Kähler manifold with Hamiltonian torus action.Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryStability & uniquenessSasakian & contact geometry https://neckpinch.com/papers/2304.08338/arxiv:2304.08338Mon, 17 Apr 2023 00:00:00 GMTMichael Hallam — We show that if a compact Kähler manifold admits a weighted extremal metric for the action of a torus, so too does its blowup at a relatively stable point that is fixed by both the torus action and the extremal field. This generalises previous results on extremal metrics by Arezzo–Pacard–Singer and Székelyhidi to many other canonical metrics, including extremal Sasaki metrics, deformations of Kähler–Ricci solitons and μ-cscK metrics. In a sequel to this paper, we use this result to study the weighted K-stability of weighted extremal manifolds.Ricci solitonsKähler–Ricci solitonsSingularitiesKähler geometryStability & uniquenessKähler–Einstein & K-stabilitySasakian & contact geometry