Intuition behind a first Chern class computation On a complex smooth algebraic surface $X$, say we have a vector bundle $F$ which fits in an exact sequence
$$0\to F\to O_X^{r+1} \to A\to 0$$
with $A$ a torsion sheaf supported on a smooth curve $C\subset X$.
Locally, on the points of $C$, the second map $O_X^{r+1} \to O_C$ is given by the natural map $O_X\to O_C$ on one factor, plus the zero map on the remaining $r$ factors. Then $F$ is the kernel of this map.
Now what I would like to understand is the fact that
$$c_1(F)=-[C]$$
The point is that I am able to compute this by applying  Whitney formula to the above sequence. What I would like to understand is an apparently much simpler/intuitive argument:
The vector bundle map $F\to O_X^{r+1}$ drops rank along $C$, hence $\det F = O_X(-C)$.
(this is at p.533 in this article)
I would very much appreciate some intuitive explanations on the behavior of Chern classes.
 A: If you have a rank $n$ complex vector bundle $E$ on $X$, where $X$ is a compact topological manifold, and if $s_1$,...,$s_n$ are $n$ generic continuous sections of $E$, then the locus of points $x \in X$ at which the dimension of the complex span of $s_1(x)$,...,$s_n(x)$ drops (i.e. this dimension at $x$ is strictly less than $n$) forms a real cycle of codimension $2$ in $X$, whose Poincare dual is $c_1(E)$.
This can be reformulated as follows. If $\epsilon^n$ denotes the trivial complex vector bundle on $X$ of rank $n$, then given a bundle map $f: \epsilon^n \to E$ which is an isomorphism at a generic point of $X$, the locus of points $x \in X$ at which the rank of $f$ drops is the Poincare dual of $c_1(E)$. But the rank of $f$ is the same as the rank of $f^T: E^* \to (\epsilon^n)^*$, with of course $(\epsilon^n)^*$ isomorphic to $\epsilon^n$. This essentially explains the appearance of a minus sign in the formula $c_1(F) = -[C]$.
More precisely, if one has a bundle map $f$ from $E$ to $\epsilon^n$, where $E$ has rank $n$, which is an isomorphism at a generic point of $X$, and if $Z$ is the locus where the rank of $f$ drops, then $c_1(E)$ is minus the Poincare dual of the homology class of $[Z]$.
More generally, if $E$ has rank $n$ and $F$ has rank $N \geq n$ are complex vector bundles over $X$, and if $f$ is a bundle map from $E$ to $F$ which is generically injective, with $Z$ the locus of points of $X$ at which $f$ fails to be injective, then the Poincare dual of the homology class of $Z$ is $-c_1(E)+c_1(F)$.
What about the case where $X$ is a compact complex manifold, and $E$ and $F$ are holomorphic vector bundles of rank $n$ and $N \geq n$ respectively? Assume that there is a holomorphic bundle map from $E$ to $F$ which is generically injective. Can we say that the Poincare dual of the homology class of $Z$ is $-c_1(E)+c_1(F)$? In other words, is the degeneracy locus of a holomorphic bundle map $f$ which is generically injective, in the same homology class as the degeneracy locus of a continuous bundle map which is generically injective? I suspect that in general, the answer to that question is no (please check the literature though). However, if the sheaves of germs of holomorphic sections of $E$ and $F$ are locally generated by global holomorphic sections of $E$ and $F$ respectively, then I believe the same relationship holds between $Z$ and the first Chern classes of $E$ and $F$ (one place to look for a more careful statement and proof of the above would be the books on positivity in algebraic geometry by the same author, Prof. Lazarsfeld). 
