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We study basic Dolbeault cohomology and find new Weitzenb\"ock formulas on atransversely K\"ahler foliation. We investigate conditions on mean curvatureand Ricci curvature that impose restrictions on basic Dolbeault cohomology. Forexample, we prove that on a transversely K\"ahler foliation with positivetransversal Ricci curvature, there are no nonzero basic-harmonic forms of type$(r,0)$, among other results.

Basic Dolbeault cohomology and Weitzenb\"ock frmulas on transversely K\"ahler foliations