2025/05/07 by Teng Huang, Huang, Teng, Qiang Tan +1
Mathematics · #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2505.04071
openalex publication_date 2025/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we investigate the topological properties of complex manifolds by studying Dolbeault-Morse-Novikov cohomology. By establishing an integral inequality, we obtain two main results: (1) When a closed complex manifold admits a nonzero parallel (0,1)-form, the Dolbeault-Morse-Novikov cohomology must be trivial, which implies that the Hirzebruch χy-genus vanishes. (2) When a closed complex manifold admits a nowhere vanishing (0,1)-form, we establish a vanishing theorem for a certain class of twisted Dirac operators, which also forces the Hirzebruch χy-genus to be zero. In particular, we prove that the Hirzebruch χy-genus of a closed complex manifold vanishes if and only if the manifold admits a nowhere vanishing real vector field. These results generalize some classical theorems from Riemannian manifolds to the complex setting. As a culminating application, we prove that the Hirzebruch χy-genus must vanish on closed Gauduchon manifolds admitting positive holomorphic scalar curvature.