Consider a holomorphic torus action on a complex manifold which lifts to a holomorphic vector bundle. When the connected components of the fixed-point set form a partially ordered set, we construct, using sheaf-theoretical techniques, two spectral sequences that converges to the twisted Dolbeault cohomology groups and those with compact support, respectively. These spectral sequences are the holomorphic counterparts of the instanton complex it standard Morse theory. The results proved imply holomorphic Morse inequalities arid fixed-point formulas on a possibly non-compact manifold. Finally, examples and applications are given.
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