The spectral sequence of the canonical foliation of a Vaisman manifold (Q1749428)

The authors studies the spectral sequence of the canonical foliation \(\mathcal F\) of a compact Vaisman manifold. Let \(b_i^b\) denote the basic Betti number with respect to the foliation \(\mathcal F\) on \(M\) \((0\leq i\leq 2n)\). These \(b_i^b\) are uniquely determined by the ordinary Betti numbers \(b_i\) for a Vaisman manifold. Theorem. If \((M,\mathcal F)\) is the canonical foliation of a compact Vaisman manifold, then \[ \dim E^{u,1}_2\geq 2b^b_u \, 0\leq u\leq 2n. \] It is restated by using the classical results of Vaisman. Corollary. If \((M,\mathcal F)\) is the canonical foliation of a compact Vaisman manifold, then \[ \dim E^{u,1}_2\geq 2(-1)^u\sum_{i=0}^{[u/2]}\bigl(\bigl[\frac u2\bigr] -i+1\bigr)(b_{2i}-b_{{2i}-(-1)^u}) \;,0\leq u\leq n. \] Proposition. If a compact Vaisman manifold \((M,\mathcal F)\) admits a quasi-regular foliation (that is all leaves are compact), then equality holds in the inequality of the theorem. Finally the authors calculate spectral terms for the Hopf manifolds, solvmanifolds.