A variety of commuting triples (Q1826849)

Let \( {\mathcal C}(3,n)\) be the set of all triples of commuting \(n\times n \) matrices over an algebraically closed field \(F\) of characteristic zero and let \({\mathcal G}(3,n)\) be the subset of \({\mathcal C}(3,n)\) consisting of generic triples of matrices (i.e. having distinct eigenvalues). Several authors investigated the following problem: Given a triple of commuting \(n \times n \) complex matrices \(A\), \(B\), \(C\) and a positive real number \(\varepsilon\), is it possible to determine a triple \(A_\varepsilon\), \(B_\varepsilon\), \(C_\varepsilon\) of commuting generic \(n \times n \) complex matrices such that \(\| A - A_\varepsilon \|\), \( \| B - B_\varepsilon \| \), \( \| C - C_\varepsilon \| \)? The answer is known to be affirmative for dimensions not greater than \(5\) and negative for dimensions not smaller than \(30\). The author states the positive answer also for \(n=6\).