# Can anything be said about the cohomology class defined by a section of a vector bundle if it is not of the expected dimension?

Let $$E$$ be a rank $$n$$ locally free sheaf on a smooth $$n$$ dimensional variety $$X$$, and $$s\in H^0(X,E)$$. If $$\dim Z(s)=0$$ (which is the expected dimension), then we can understand the cohomology class of $$Z(s)$$ as a chern class of $$E$$. Can anything be said about this class if $$\dim Z(s)>0$$?

• I don't know what $Z(s)$ is but if it is the zero locus of $Z$, your identification with Chern class seems to be incorrect if the intersection with the zero section is not transversal due to lack of information of multiplicity.
– Z. M
Aug 3 at 15:20
• @Z.M I meant for $Z(s)$ to denote the zero locus with its natural scheme structure, so including multiplicity information. Aug 3 at 15:56
• But in that case, how do you define the cohomology class of $Z(s)$ in that generality (i.e. for arbitrary subscheme)? Note that the usual intersection theory uses Chow's moving lemma to transform an arbitrary intersection into a proper intersection, in which case $\dim Z(s)=0$.
– Z. M
Aug 3 at 21:47
• Let $z:Z(s)\to X$ be the inclusion of the zero-scheme. Assume that $Z(s)$ is smooth. Let $N$ be the normal bundle of $Z(s)$ in $X$. There is a natural monomorphism $N\to z^*(E)$, whose quotient $V$ (the so-called excess normal bundle) is locally free. We then have the formula $$c^{\rm top}(E)=z_*(c^{\rm top}(V)).$$ This is a consequence of the "excess intersection formula" (see eg Fulton's book). Eg, if $s$ is the zero-section, the formula is tautological and if $Z(s)$ is discrete then you recover the fact that $c^{\rm top}(E)$ is the cycle class of the zero-scheme. Aug 5 at 14:52
• @DamianRössler Thanks, I did not know that! Should read up on my interestion theory I guess... Aug 5 at 16:34

Not much can be said. For instance, let $$X = \mathbb{P}^n$$ and $$E = \mathcal{O}(1)^{\oplus n}$$. Then the zero locus of a section might be any linear subspace in $$X$$. In particular, its cohomology class is $$H^k$$, where $$H$$ is the hyperplane class and $$k$$ may be any integer in the range $$0 \le k \le n$$.