Filtered lattice homology of surface singularities

Let $(X,o)$ be a complex analytic normal surface singularity with rational homology sphere link $M$. The `topological' lattice cohomology ${\mathbb H}^*=\oplus_{q\geq 0} {\mathbb H}^q$ associated with $M$ and with any of its spin$^c$ structures was introduced by the author. Each ${\mathbb H}^q$ is a graded ${\mathbb Z}[U]$--module. Here we consider its homological version ${\mathbb H}_*=\oplus_{q\geq 0}{\mathbb H}_q$. The construction uses a Riemann-Roch type weight function. A key intermediate product is a tower of spaces $\{S_n\}_{n\in {\mathbb Z}}$ such that ${\mathbb H}_q=\oplus_n H_q(S_n,{\mathbb Z})$. In this article we fix the embedded topological type of a reduced curve singularity $(C,o)$ embedded into $(X,o)$, that is, a 1-dimensional link $L_C\subset M$. Each component of $L_C$ will also carry a non-negative integral decoration. For any fixed $n$, the embedded link $L_C$ provides a natural filtration of the space $S_n$, which induces a homological spectral sequence converging to the homogeneous summand $H_q(S_n,{\mathbb Z})$ of the lattice homology. All the entries of all the pages of the spectral sequences are new invariants of the decorated $(M,L_C)$. Each page provides a triple graded ${\mathbb Z}[U]$-module. We provide several concrete computations of these pages and structure theorems for the corresponding multivariable Poincar\'e series associated with the entries of the spectral sequences. Connections with Jacobi theta series are also discussed.