Talks.cam

The weighted Hermite--Einstein equation - Michael Hallam (Aarhus Universitet)
2024-05-09 10:00 - 11:00 UTC | Seminar Room 1, Newton Institute
In joint work with Abdellah Lahdili, we introduce a new weighted generalisation of the Hermite--Einstein equation for torus equivariant vector bundles over compact Kähler manifolds. The novel equation recovers various canonical Hermitian metrics on vector bundles in interesting geometric situations---examples include Kähler--Ricci solitons, as well as the transverse Hermite--Einstein metrics on Sasaki manifolds studied by Biswas--Schumacher and Baraglia--Hekmati. Extending the equivariant intersection theory introduced by Inoue to arbitrary weight functions on the moment polytope, we define the weighted slope of a vector bundle, extend the Kobayashi--Lübke inequality to the weighted setting, and give a proof of the moment map property of the weighted Hermite--Einstein equation via fibre integration of equivariant forms, following the approach of Dervan--Hallam. As a main result, we prove the weighted Kobayashi--Hitchin correspondence, namely that a T-equivariant vector bundle admits a weighted Hermite--Einstein metric if and only if the vector bundle is weighted slope polystable.
