Let $X$ be a complex projective variety, $E$ be a rang $n$ bundle with $n<dim X$ and $s$ be a (holomorphic) section of $E$.
There is a relatively straightforward criterium to check if the space $s=0$ is non-empty. Namely it is enough to know that $c_n(E)\ne 0$.
Question. I would like to know how one could check that $s=0$ is connected. In the case that is of interest to me $X$ is a homogenious variety (i.e. it admits a transitive group action) and $E$ is an equivariant bundle.
Maybe there is some kind of Lefshetz principle that says that $s=0$ is connected if $E$ is "sufficiently" positive?