Chern classes of pushforwards - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T03:34:42Z http://mathoverflow.net/feeds/question/49827 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49827/chern-classes-of-pushforwards Chern classes of pushforwards Charles Siegel 2010-12-18T21:19:17Z 2010-12-28T03:13:51Z <p>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.</p> <p>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 $Y$.</p> http://mathoverflow.net/questions/49827/chern-classes-of-pushforwards/49829#49829 Answer by Dave Anderson for Chern classes of pushforwards Dave Anderson 2010-12-18T22:34:59Z 2010-12-18T23:49:09Z <p>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).</p> <p>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 $\dim f(D)&lt; \dim D$. 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 $f_*[D]=0$.</p> <p>All this is in Fulton's <em>Intersection Theory</em>, Section 1.4.</p>