  Let $M^n$ be a compact K$\ddot{a}$hler manifold with almost nonnegative Ricci
curvature and nonzero first Betti number. We show that the holomorphic Euler
number of $M^n$ vanishes, which gives a new obstruction for compact complex
manifolds admitting K$\ddot{a}$hler metrics with almost nonnegative Ricci
curvature. A crucial step in the proof is to show a vanishing theorem of
Dolbeault-Morse-Novikov cohomology.
