Chern classes of pushforwards

2010-12-18T21:19:17Z

Let $f:X\to Y$ be a proper morphism of normal varieties (smooth as DM stacks, but I only care about the coarse spaces). The map $f$ is generically finite, but not flat (so no hope of smoothness and Grothendieck-Riemann-Roch applying) and I have a fairly detailed understanding of the fiber over any point, images of $f$ restricted to divisors and so forth.

Now, take a divisor $D$, and identify it with an invertible sheaf in the standard way. I'm looking for a way to compute the first Chern class of $f_*D$ on $Y$.

Answer by Dave Anderson for Chern classes of pushforwards

2010-12-18T22:34:59Z

Do you know the (generic) degree of your map $f$? As you probably know, standard intersection theory says $f_*[D] = n[f(D)]$ as classes in $A_{d-1}Y$, where $n$ is the degree of $f$ (restricted to $D$) and $d=\dim X = \dim Y$. No flatness or smoothness hypotheses on $f$ are needed for this; the sticky point is in identifying these divisors with line bundles. But since you're dealing with smooth DM stacks, that should be ok (over ${\Bbb Q}$ at least).

EDIT (incorporating the comments): For a proper map $f$, there is a map defined at the cycle level by $$f_*[D] = n\cdot [f(D)],$$ where $n$ is the degree of $D$ over $f(D)$ (i.e., degree of the induced field extension) when these have equal dimension, and $n=0$ when $\dim f(D)< \dim D$. This passes to rational equivalence, so defines a map $A_{d-1}X \to A_{d-1}Y$. In particular, if $f$ collapses a divisor $D$, then $f_*[D]=0$.

All this is in Fulton's Intersection Theory, Section 1.4.

Chern classes of pushforwards - MathOverflow

Chern classes of pushforwards

Answer by Dave Anderson for Chern classes of pushforwards