Hodge theory and spectral sequences (Q1332829)

The author studies the cohomology of a differential complex \(V: 0\to V^ 0\to V^ 1\to V^ 2\to \dots\to V^ r\to 0\) defined by the operator \(d\) \((d^ 2=0)\), were \(V^ n\) are finite dimensional spaces, there exists a filtration of \(V\) by subcomplexes, i.e. \(V^ n= U^ n_ 0 U^ n_ 1\dots U^ n_ N= \dots= U_ \infty^ n\), \(d(U^ n_ i) U_ i^{n+1}\) and the spaces \(V^ n\) are endowed with inner products. Writing each \(V^ n\) as the sum of orthogonal subspaces and rescaling the inner products on these subspaces at different rates, one studies the behaviour of the eigenvalues of the associated Laplacians. In the limit, as some parameters become singular with respect to others, it is obtained a decoupling of the contributions from different subspaces which is provided just by the Leray spectral sequence.