This is a note about my long-lasting misperception of the concept of generic sections of vector bundles. A generic section is a global section of a vector bundle transversal to the zero section which is obviously a global section. A theorem relates the generic sections to Chern classes: Chern classes of a vector bundle can be expressed as the Poincare duality of the degeneracy loci, regarded as the homological classes, of the generic sections (simply put). However, there is a trap in this statement that can be easily overlooked and hence lead to contradiction and confusion: Generic sections do no always exist. It took me very long to arrive at this conclusion which is still counter-intuitive to me; therefore, I would like to explain it in a simple example in hope that it could be helpful for people struggling with the same issue.
Let us consider a rank-2 complex trivial vector bundle \(E\) over a 2-sphere \(S^2\) parameterized by the equation \(x^2+y^2+z^2=1\). Let \(\varphi: E \to E\) be a bundle map defined by the following matrix-valued function
\begin{equation} \varphi|_{(x,y,z)} = \begin{pmatrix} z & x + i y \\ x - i y & -z \end{pmatrix} \end{equation}with two distinct constant eigenvalues \(\lambda_\pm = \pm1\). Furthermore, each eigen-value defines a line bundle (corresponding dented as \(L_\pm\)). The fibres of the line bundle defined by the eigen-value \(\lambda_{+}\) are of the form \(F_{(x,y,z)} = \mathrm{ker}[\varphi|_{(x,y,z)} - \mathrm{Id}]\); and the fibres of the line bundle defined by the eigen-value \(\lambda_{-}\) are likewise defined as \(F_{(x,y,z)} = \mathrm{ker}[\varphi|_{(x,y,z)} + \mathrm{Id}]\).
Using Chern-Weil theory, one can easily check that each line bundle as is defined above is a non-trivial bundle. That is two say, each bundle admits a non-trivial first Chern class. In the meantime, the trivial bundle \(E\) is the direct sum of the two line bundles as the the bundle map \(\varphi\) does not admit non-trivial monodromy. It follows that \(c_1(L_-) = - c_1(L_+)\) as the first Chern class of \(E\) is trivial.
Now suppose that each line bundle admits a generic section. Then the degeneracy locus (or equivalently, the vanishing locus) of each, through Poincare duality, should define \(c_1(L_\pm)\). However, at this point, we must answer the following question: Where is the minus sign in the first Chern classes from? We note that the vanishing locus, viewed as a homological class, should be treated as the fundamental class of the submanifold consisting of the points in \(s^2\) where the generic section vanishes. Thus, the Chern class obtained through Poincare duality should always be "positive". This should hold true universally which then inevitably leads to a conflict between our assumption of the existence of generic sections and the constraint \(c_1(L_-) = -c_1(L_+)\). In other words, one must the line bundles corresponding to the two eigen-values of \(\varphi\) must admit no generic section (as a matter of fact, this conclusion applies to both of them).
This simple example tells us that generic sections are not generically available as we can easily find simple examples where they are non-existent.