[For the entire collection see Zbl 0742.00081.]
The author considers the question whether there exists a solution of the matrix equation \(A_ 1\dots A_ k=I\) for matrices \(A_ i\) in specified conjugacy classes \(C_ i\subset SL(n,\mathbb{C})\) where one of the conjugacy classes \(C_ k\) is semisimple with distinct eigenvalues. P. Deligne described a phenomenon of rigidity, where if \(\sum\dim(C_ i)=2n^ 2-2\) then any irreducible solution is unique up to conjugacy.
The original purpose of the paper was to produce some examples of solutions when the rigidity condition holds. In fact the author produces two new families of rigid solutions, one for \(n\) odd and one for \(n\) even, and another solution with \(n=6\). He also proposes another way to think of the problem.
The fundamental group of \({\mathcal P}^ n-\{s_ 1,\dots,s_ k\}\), is generated by loops \(\gamma_ 1,\dots,\gamma_ k\) with the relation \(\gamma_ 1\dots\gamma_ k=I\). Thus a solution of the matrix equation is the same thing as a local system on \({\mathcal P}^ n-\{s_ 1,\dots,s_ k\}\), with monodromy transformations in specified conjugacy classes.
The author’s technique for getting existence of solutions is to use the results of “Harmonic bundles on noncompact curves” to construct local systems. He notes that the local system on \({\mathcal P}^ n-\{s_ 1,\dots,s_ k\}\) corresponding to a rigid irreducible solution has a number of additional structures.