2012/03/30 by Matthias Stemmler
Mathematics · #Algebraic Geometry and Number Theory #Curvature #Divisor (algebraic geometry) #Einstein #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Hermitian manifold #Hermitian matrix #Holomorphic function #Mathematical analysis #Mathematical physics #Mathematics #Pure mathematics #Ricci curvature #Sheaf #Uniqueness #Vector bundle #math.CV #math.DG #msc:32L05 #msc:32Q20 #msc:53C07
paper · pdf · doi:10.1142/s0129167x12500917
published as Internat. J. Math. 23 (2012) 1250091 · 21 pages, International Journal of Mathematics (to appear)
arxiv created 2012/03/30 · openalex publication_date 2012/04/09 · arxiv updated 2012/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We adapt the notions of stability of holomorphic vector bundles in the sense of Mumford–Takemoto and Hermitian–Einstein metrics in holomorphic vector bundles for canonically polarized framed manifolds, i.e. compact complex manifolds X together with a smooth divisor D such that K X ⊗ [D] is ample. It turns out that the degree of a torsion-free coherent sheaf on X with respect to the polarization K X ⊗ [D] coincides with the degree with respect to the complete Kähler–Einstein metric g X\D on X\D. For stable holomorphic vector bundles, we prove the existence of a Hermitian–Einstein metric with respect to g X\D and also the uniqueness in an adapted sense.