arXiv subjects

Michael Hallam

Publications and source records attributed to Michael Hallam.

6 recordsLinked to original sources

arXiv2024

The weighted Hermite--Einstein equation

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, K\"ahler--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--L\"ubke 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 K\"ahler--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 K\"ahler manifold with Hamiltonian torus action.

math.DG↗

arXiv2023

Stability of weighted extremal manifolds through blowups

In a previous paper, we showed that the blowup of a weighted extremal K\"ahler manifold at a relatively stable fixed point admits a weighted extremal metric. Using this result, we prove that a weighted extremal manifold is relatively weighted K-polystable. In particular, a weighted cscK manifold is weighted K-polystable. This strengthens both the weighted K-semistability proved by Lahdili and Inoue, and the weighted K-polystability with respect to smooth degenerations by Apostolov--Jubert--Lahdili, allowing for possibly singular degenerations.

math.DG↗

arXiv2023

Weighted extremal metrics on blowups

We show that if a compact K\"ahler 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 Sz\'ekelyhidi to many other canonical metrics, including extremal Sasaki metrics, deformations of K\"ahler--Ricci solitons and $\mu$-cscK metrics. In a sequel to this paper, we use this result to study the weighted K-stability of weighted extremal manifolds.

math.DG↗

arXiv2023

The universal structure of moment maps in complex geometry

We introduce a geometric approach to the construction of moment maps in finite and infinite-dimensional complex geometry. We apply this to two settings: K\"ahler manifolds and holomorphic vector bundles. Our new approach exploits the existence of universal families and the theory of equivariant differential forms. For K\"ahler manifolds we give a new, geometric proof of Donaldson-Fujiki's moment map interpretation of the scalar curvature. Associated to arbitrary products of Chern classes of the manifold - namely to a central charge - we further introduce a geometric PDE determining a $Z$-critical K\"ahler metric, and show that these general equations also satisfy moment map properties. For holomorphic vector bundles, using a similar strategy we give a geometric proof of Atiyah-Bott's moment map interpretation of the Hermitian Yang-Mills condition. We then go on to give a new, geometric proof that the PDE determining a $Z$-critical connection - again associated to a choice of central charge - can be viewed as a moment map; deformed Hermitian Yang-Mills connections are a special case, in which our work gives a geometric proof of a result of Collins-Yau. Our main assertion is that this is the canonical way of producing moment maps in complex geometry - associated to any geometric problem along with a choice of stability condition - and hence that this accomplishes one of the main steps towards producing PDE counterparts to stability conditions in large generality.

math.DG↗

arXiv2020

Geodesics in the space of relatively K\"ahler metrics

We derive the geodesic equation for relatively K\"ahler metrics on fibrations and prove that any two such metrics with fibrewise constant scalar curvature are joined by a unique smooth geodesic. We then show convexity of the log-norm functional for this setting along geodesics, which yields simple proofs of Dervan and Sektnan's uniqueness result for optimal symplectic connections and a boundedness result for the log-norm functional. Next, we associate to a fibration degeneration a unique geodesic ray defined on a dense open subset. Calculating the limiting slope of the log-norm functional along a globally defined smooth geodesic ray, we prove that fibrations admitting optimal symplectic connections are polystable with respect to a large class of fibration degenerations that are smooth over the base. We give examples of such degenerations in the case of projectivised vector bundles and isotrivial fibrations.

math.DG↗

arXiv2017

Positive scalar curvature via end-periodic manifolds

We obtain two types of results on positive scalar curvature metrics for compact spin manifolds that are even dimensional. The first type of result are obstructions to the existence of positive scalar curvature metrics on such manifolds, expressed in terms of end-periodic eta invariants that were defined by Mrowka-Ruberman-Saveliev (MRS). These results are the even dimensional analogs of the results by Higson-Roe. The second type of result studies the number of path components of the space of positive scalar curvature metrics modulo diffeomorphism for compact spin manifolds that are even dimensional, whenever this space is non-empty. These extend and refine certain results in Botvinnik-Gilkey and also MRS. End-periodic analogs of K-homology and bordism theory are defined and are utilised to prove many of our results.

math.DG↗