On the connectedness of some degeneracy loci and of Ulrich subvarieties

We study connectedness of degeneracy loci $D_{r-k}(\phi )$ of morphisms $\phi :{\mathcal{O}}_X^{\oplus (r+1-k)}\to \mathcal{E}$, where $\mathcal{E}$ is a rank $r$ globally generated bundle on a smooth $n$-dimensional variety $X$ and $k\le 3$. For $k\le 2$ we give a characterization of connectedness in terms of vanishing of Chern classes. Moreover we prove that they are connected, for $k\le \min \{2,r-1,n-1\}$, if $\mathcal{E}$ is V-big. In the case of Ulrich bundles more precise results are given, both in general and in the case of surfaces.