  We investigate the hypercohomologies of truncated twisted holomorphic de Rham
complexes on (not necessarily compact) complex manifolds. In particular, we
generalize Leray-Hirsch, K\"{u}nneth and Poincar\'{e}-Serre duality theorems on
them. At last, a blowup formula is given, which affirmatively answers a
question posed by Chen, Y. and Yang, S. in \cite{CY}.
