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? 



Connectedness of a section of an algebraic bundle
Let $Z = \{s = 0\}$. It is connected if and only if $H^0(Z)$ is 1-dimensional. You can compute $H^0(Z)$ by using Koszul resolution
$$
0 \to \Lambda^n E^\* \to \Lambda^{n-1}E^\* \to \dots \to E^\* \to O\_X \to O\_Y \to 0
$$
(which is indeed a resolution if $X$ is Cohen--Macaulay and $Y$ has codimension $n$ in $X$). So, if you know that $H^i(X,\Lambda^iE^\*) = 0$ for $i > 0$ the connectedness of $Y$ follows. Even if for some $i$ the cohomology is nontrivial, it can be killed in the spectral sequence, and you can check it.


