Cycle class of zeroes of a global section

Question

Let $\mathcal{F}$ be a locally free sheaf of rank $n$ on an $n$ dimensional complex manifold $X$. If the zero locus of a generic global section of $\mathcal{F}$ is $0$ dimensional, then its cycle class is equal to $c_n(\mathcal{F})$. What can be said if the generic section has a zero locus of positive dimension?

Answer 1

Let $V = H^0(X,\mathcal{F})$ be the space of global sections of $\mathcal{F}$ and let $$ V \otimes \mathcal{O}_X \to \mathcal{F} $$ be the evaluation morphism. If it is surjective (i.e., $\mathcal{F}$ is globally generated), then the zero locus of a general section is zero-dimensional, and has class $c_n(\mathcal{F})$.

So, assume that the evaluation map is not surjective. Assume, moreover, it has a constant rank, so that its image $\mathcal{F}' \subset \mathcal{F}$ is a vector subbundle of rank $r$. Then any global section of $\mathcal{F}$ comes from a global section of a globally generated vector bundle $\mathcal{F}'$, and their zero loci coincide. Therefore, the class of the zero locus of a general section of $\mathcal{F}$ is equal to $c_r(\mathcal{F}')$.

When the rank of the evaluation morphism is non-constant, the class of a general zero locus is not so easy to find. To sketch a possible approach, one can first blowup $X$ in such a way that the rank is constant on the blowup, and then pushforward $c_r(\mathcal{F}')$ with respect to this map.

Answer 2

Assume the zero locus has positive dimension, but is smooth (or is, at least, locally complete intersection). Then it comes with a vector bundle (so-called "excess intersection" bundle), whose top Chern class equals $c_n(\mathcal{F})$.

For instance, if the original section is just zero, its zero locus is the whole $X$, and the excess intersection bundle is $\mathcal{F}$.