Dolbeault-Morse-Novikov Cohomology on Complex manifolds and its applications

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 $\chi_{y}$-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 $\chi_{y}$-genus to be zero. In particular, we prove that the Hirzebruch $\chi_{y}$-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 $\chi_{y}$-genus must vanish on closed Gauduchon manifolds admitting positive holomorphic scalar curvature.