chern class of bundle in Blochs “Cycles on arithmetic schemes”
$begingroup$
In Blochs "Cycles on arithmetic schemes and Euler characteristics of curves" he defines the bivariant chern class for bundles on a scheme.
Now I wanted to calculate a bivariant chern class with $n=1$ for what I need to calculate the first chern class $c_1(xi)$, where $$xi=pr_0^*xi_0-pr_1^*xi_1=Gtimes_{G_0} E_0-Gtimes_G xi_1=Gtimes_Y E_0-xi_1$$
for the canonical rank-$e_i$-subbundles $xi_0=E_0$ and $xi_1$.
Question: How can I find a more explicit (as explicit as possible) representation of $c_1(xi$)? I want to calculate $deg(c_1(xi))$.
Any hint can help!
I read parts of Fultons "Intersection theory", where Fulton defines the bivariant chern class in a slightly more general setting, but it did not help that much.
(Notation: He considers a scheme Y of finite type over a regular noetherian base, a closed subscheme $Xsubset Y$, a two-term-complex $E_1rightarrow E_0$ of vector bundles and $e_i=rk(E_i)$. $G=Grass_{e_1}(E_1oplus E_0)$ is the Grassmannian of rank-$e_1$-subbundles of $E_1oplus E_0$, $pr_0:Grightarrow G_0=Grass_{e_0}(E_0)=Y$ the structure map and $pr_1=id:Grightarrow G$.)
characteristic-classes algebraic-vector-bundles
asked Nov 13 '18 at 14:12
Student7Student7
1839
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add a comment |
$begingroup$
In Blochs "Cycles on arithmetic schemes and Euler characteristics of curves" he defines the bivariant chern class for bundles on a scheme.
Now I wanted to calculate a bivariant chern class with $n=1$ for what I need to calculate the first chern class $c_1(xi)$, where $$xi=pr_0^*xi_0-pr_1^*xi_1=Gtimes_{G_0} E_0-Gtimes_G xi_1=Gtimes_Y E_0-xi_1$$
for the canonical rank-$e_i$-subbundles $xi_0=E_0$ and $xi_1$.
Question: How can I find a more explicit (as explicit as possible) representation of $c_1(xi$)? I want to calculate $deg(c_1(xi))$.
Any hint can help!
I read parts of Fultons "Intersection theory", where Fulton defines the bivariant chern class in a slightly more general setting, but it did not help that much.
(Notation: He considers a scheme Y of finite type over a regular noetherian base, a closed subscheme $Xsubset Y$, a two-term-complex $E_1rightarrow E_0$ of vector bundles and $e_i=rk(E_i)$. $G=Grass_{e_1}(E_1oplus E_0)$ is the Grassmannian of rank-$e_1$-subbundles of $E_1oplus E_0$, $pr_0:Grightarrow G_0=Grass_{e_0}(E_0)=Y$ the structure map and $pr_1=id:Grightarrow G$.)
characteristic-classes algebraic-vector-bundles
asked Nov 13 '18 at 14:12
Student7Student7
1839
$endgroup$
add a comment |
$begingroup$
In Blochs "Cycles on arithmetic schemes and Euler characteristics of curves" he defines the bivariant chern class for bundles on a scheme.
Now I wanted to calculate a bivariant chern class with $n=1$ for what I need to calculate the first chern class $c_1(xi)$, where $$xi=pr_0^*xi_0-pr_1^*xi_1=Gtimes_{G_0} E_0-Gtimes_G xi_1=Gtimes_Y E_0-xi_1$$
for the canonical rank-$e_i$-subbundles $xi_0=E_0$ and $xi_1$.
Question: How can I find a more explicit (as explicit as possible) representation of $c_1(xi$)? I want to calculate $deg(c_1(xi))$.
Any hint can help!
I read parts of Fultons "Intersection theory", where Fulton defines the bivariant chern class in a slightly more general setting, but it did not help that much.
(Notation: He considers a scheme Y of finite type over a regular noetherian base, a closed subscheme $Xsubset Y$, a two-term-complex $E_1rightarrow E_0$ of vector bundles and $e_i=rk(E_i)$. $G=Grass_{e_1}(E_1oplus E_0)$ is the Grassmannian of rank-$e_1$-subbundles of $E_1oplus E_0$, $pr_0:Grightarrow G_0=Grass_{e_0}(E_0)=Y$ the structure map and $pr_1=id:Grightarrow G$.)
characteristic-classes algebraic-vector-bundles
asked Nov 13 '18 at 14:12
Student7Student7
1839
$endgroup$
In Blochs "Cycles on arithmetic schemes and Euler characteristics of curves" he defines the bivariant chern class for bundles on a scheme.
Now I wanted to calculate a bivariant chern class with $n=1$ for what I need to calculate the first chern class $c_1(xi)$, where $$xi=pr_0^*xi_0-pr_1^*xi_1=Gtimes_{G_0} E_0-Gtimes_G xi_1=Gtimes_Y E_0-xi_1$$
for the canonical rank-$e_i$-subbundles $xi_0=E_0$ and $xi_1$.
Question: How can I find a more explicit (as explicit as possible) representation of $c_1(xi$)? I want to calculate $deg(c_1(xi))$.
Any hint can help!
I read parts of Fultons "Intersection theory", where Fulton defines the bivariant chern class in a slightly more general setting, but it did not help that much.
(Notation: He considers a scheme Y of finite type over a regular noetherian base, a closed subscheme $Xsubset Y$, a two-term-complex $E_1rightarrow E_0$ of vector bundles and $e_i=rk(E_i)$. $G=Grass_{e_1}(E_1oplus E_0)$ is the Grassmannian of rank-$e_1$-subbundles of $E_1oplus E_0$, $pr_0:Grightarrow G_0=Grass_{e_0}(E_0)=Y$ the structure map and $pr_1=id:Grightarrow G$.)
characteristic-classes algebraic-vector-bundles
characteristic-classes algebraic-vector-bundles
asked Nov 13 '18 at 14:12
Student7Student7
1839
asked Nov 13 '18 at 14:12
Student7Student7
1839
edited Dec 13 '18 at 10:02
Student7
asked Nov 13 '18 at 14:12
Student7Student7
1839
asked Nov 13 '18 at 14:12
Student7Student7
1839
asked Nov 13 '18 at 14:12
Student7Student7
1839
1839
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