2025/11/15 by Shu Shen, Shen, Shu, Jianqing Yu +1
Mathematics · #14F08 #18G80 #32S45 #35J05 #58J10 #Algebraic Geometry (math.AG) #Algebraic and Geometric Analysis #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2511.12211
openalex publication_date 2025/11/15 · openalex created_date 2025/11/19 · openalex updated_date 2026/07/28
We give a new differential-geometric proof of Grauert's theorem on the coherence of the higher direct image of a coherent sheaf under a proper holomorphic morphism between complex analytic spaces. In the smooth case, our approach is based on the antiholomorphic superconnection introduced by Block and further developed by Bismut-Shen-Wei. The required finiteness results follow from elliptic theory. In the singular case, we reduce the problem to the smooth setting using Hironaka's desingularization.