differential geometry

Almost-Hermitian and Hermitian metrics

arXiv:2607.26673

summary

The notes discuss three problems in Hermitian geometry concerning the scalar curvature of the Chern connection and explore ways to construct canonical almost‑Hermitian and Hermitian metrics using PDE techniques, momentum maps, and variational methods.

Abstract

We describe three problems in Hermitian geometry related to the scalar curvature of the Chern connection, aimed at identifying and constructing canonical metrics. They are chosen as toy examples where different techniques naturally interact: analytic methods for partial differential equations, the notion of momentum map, and variational methods.

These notes were prepared for the course delivered by the author at the CIME Summer School "Cohomological properties of almost complex manifolds and special structures", held in Cetraro (CS), July 13--17, 2026

Topics & keywords

#hermitian geometry#almost-hermitian metrics#scalar curvature#chern connection#canonical metrics#momentum mapscalar curvatureChern connectionHermitian metricmomentum mapvariational methodpartial differential equations
Almost-Hermitian and Hermitian metrics · wovepaper