Suppose $p$ is a homogeneous polynomial in $n$ complex variables. Let S be the hypersurface defined by $p(z)=0$. Then is the 1form $dp/p$ always nonexact on the complement $C^n\setminus S$?
Any answer or reference is appreciated.
Ron Yang
Suppose $p$ is a homogeneous polynomial in $n$ complex variables. Let S be the hypersurface defined by $p(z)=0$. Then is the 1form $dp/p$ always nonexact on the complement $C^n\setminus S$? Any answer or reference is appreciated. Ron Yang 


Assume that $p$ is nonzero. If the form $dp/p$ were exact, then locally a primitive would be $log(p)+const$; this is easily seen not to work as soon as you can "loop around" $S$ (e.g. restrict everything to a line intersecting $S$ and see what happens there). Thus the form $dp/p$ is exact if and only if $S$ is empty, and hence if and only if $p$ is constant. 

