2023/01/04 by Natsuo Miyatake, Miyatake, Natsuo
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2301.01485
openalex publication_date 2023/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a Higgs bundle over a compact Kähler manifold with a smooth, non-holomorphic Higgs field. We assume that the holomorphic vector bundle decomposes into a direct sum of holomorphic line bundles. Under an assumption on the zero set of the non-holomorphic Higgs field, we provide some necessary and sufficient conditions for Donaldson's functional which is restricted to the set of diagonal Hermitian metrics associated with a holomorphic decomposition of the vector bundle to attain a minimum. In particular, when the holomorphic vector bundle decomposes into a direct sum of holomorphic line bundles, we show that we can solve the Hermitian-Einstein equation under a strong assumption even if the Higgs field is non-holomorphic.