Projective Spectrum and Cyclic Cohomolgy

For a tuple $A=(A_1, A_2, ..., A_n)$ of elements in a unital algebra ${mathcal B}$ over $mathbb{C}$, its {em projective spectrum} $P(A)$ or $p(A)$ is the collection of $zin mathbb{C}^n$, or respectively $zin mathbb{P}^{n-1}$ such that the multi-parameter pencil $A(z)=z_1A_1+z_2A_2+cdots +z_nA_n$ is not invertible in ${mathcal B}$. ${mathcal B}$-valued $1$-form $A^{-1}(z)dA(z)$ contains much topological information about $P^c(A):=mathbb{C}^nsetminus P(A)$. In commutative cases, invariant multi-linear functionals are effective tools to extract that information. This paper shows that in non-commutative cases, the cyclic cohomology of ${mathcal B}$ does a similar job. In fact, a Chen-Weil type map $kappa$ from the cyclic cohomology of ${mathcal B}$ to the de Rham cohomology $H^*_d(P^c(A), mathbb{C})$ is established. As an example, we prove a closed high-order form of the classical Jacobis formula.