Abstract

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.