AFrölicher-type inequality forBott-Chern cohomology and its relationwith∂∂-lemma were introduced in[1].In this paper,we generalize these results to the cohomology groups with coefficients in flat complex vector bundles.
We propose a conjecture relevant to Galkin’s lower bound conjecture,and verify it for the blow-ups of a four-dimensional quadric at a point or along a projective plane.We also show that Conjecture O holds in these two cases.
Let N be a maximal discrete nest on an infinite-dimensional separable Hilbert space H,ξ=∑^(∞)_(n=1)en/2n be a separating vector for the commutant N',E_(ξ)be the projection from H onto the subspace[Cξ]spanned by the vectorξ,and Q be the projection from K=H⊕H⊕H onto the closed subspace{(η,η,η)^(T):η∈H}.Suppose that L is the projection lattice generated by the projections(E_(ξ) 0 0 0 0 0 0 0 0),{(E 0 0 0 0 0 0 0 0):E∈N},(I 0 0 0 I 0 0 0 0) and Q.We show that L is a Kadison-Singer lattice with the trivial commutant.Moreover,we prove that every n-th bounded cohomology group H~n(AlgL,B(K))with coefficients in B(K)is trivial for n≥1.
