Locally standard torus actions and h'-vectors of simplicial posets

Abstract

We consider the orbit type filtration on a manifold $X$ with locally standard action of a compact torus and the corresponding homological spectral sequence $(E_X{)}_{\ast ,\ast }^r$. If all proper faces of the orbit space $Q=X/T$ are acyclic, and the free part of the action is trivial, this spectral sequence can be described in full. The ranks of diagonal terms are equal to the $h^{\prime }$-numbers of the Buchsbaum simplicial poset $S_Q$ dual to $Q$. Betti numbers of $X$ depend only on the orbit space $Q$ but not on the characteristic function. If $X$ is a slightly different object, namely the model space $X=(P\times T^n)/\sim$ where $P$ is a cone over Buchsbaum simplicial poset $S$, we prove that $\dim (E_X{)}_{p,p}^{\infty }=h_p^{\prime \prime }(S)$. This gives a topological evidence for the fact that $h^{\prime \prime }$-numbers of Buchsbaum simplicial posets are nonnegative.

21 pages, 3 figures + 1 inline figure

References in corpus (3)

• The tropical momentum map: a classification of toric log symplectic manifolds
• Locally standard torus actions and sheaves over Buchsbaum posets
• Homology of torus spaces with acyclic proper faces of the orbit space