4
$\begingroup$

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?

$\endgroup$
5
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$
1
$\begingroup$

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 $\mathcal{F}$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Thanks for your answer. I was hoping there is a way to find the cycle of the zero locus using some chern class. Is this possible? $\endgroup$ – user2520938 May 21 '18 at 8:03
  • $\begingroup$ What do you mean? There are many possibilities (one of them is $X$ itself). For instance, imagine that your vector bundle is a sum of line bundles. Then you can take several components of your section to be zero, and the other components general. In this way you will get a general complete intersection of intermediate multidegree as the zero locus. $\endgroup$ – Sasha May 21 '18 at 8:21
  • $\begingroup$ I was hoping that for generic choices the cycle is well-defined, is this not the case? $\endgroup$ – user2520938 May 21 '18 at 8:25
  • $\begingroup$ @user2520938: Sorry, I misread your question. See below another answer. $\endgroup$ – Sasha May 21 '18 at 9:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.