Timeline for Chern classes in flat families

4 events

when toggle format	what		by	license	comment
Jan 21, 2011 at 9:42	vote	accept	TonyS
Jan 21, 2011 at 3:53	comment	added	Sasha		It is better to look at the Hilbert polynomial of $L$, i.e. pick up an ample divisor $H$ on $X$ and consider $f(k) = \chi(X,L_t(kH))$. On one hand it is a constant function since $L$ is flat. On the other, if you compute it by Riemann-Roch you will get $c_1(L_t)H^{n-1}\cdot g(k)$, where $g$ is a certain polynomial and $n = \dim X$. So, it follows that $c_1(L_t)H^{n-1}$ is constant. And this is true FOR ANY ample divisor $H$. It follows that $c_1(L_t)$ is constant.
Jan 20, 2011 at 22:18	comment	added	TonyS		Okay, $T$ connected and cohomology with rational coefficients is okay for my example. How does the Riemann-Roch argument work? Something like: Because $T$ is connected, the Euler characteristic $\chi(X,L_t)$ of the line bundles is a constant function of $t\in T$. But for all $t\in T$ we have $\chi(X,L_t)=ch(L_t)td(X)$. So $ch(L_t)$ must be constant?
Jan 20, 2011 at 21:21	history	answered	Sasha	CC BY-SA 2.5