Factor analysis: What to do when determinant is almost zero and when KMO for a variable is low?

I'm conducting a factor analysis on 40 interval-level variables, and have two immediate concerns:

1. The determinant is 6.608E-006, which is much lower than the cut-off of 0.00001. I went back and screened the correlation matrix to find significant, too-high correlations between variables; there's nothing even approaching 0.8. What now? Do I proceed anyway, just noting the determinant value, or is there something else I should be doing to adjust the data to exclude multicollinearity?

2. KMO is a respectable .846, but on examining the anti-correlation matrix, I found five variables whose individual KMO values were less than 0.5. Now what? Exclude? Ignore, as overall KMO is high enough already?