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Chern class of a vector bundle and the associated projective space bundle

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2

$begingroup$

I have a very basic question regarding Chern classes. Let $X$ be a smooth projective variety and $mathcalE$ a vector bundle on it. Let $pi:mathbbP(mathcalE)to X$ denote the projective space bundle over $X$ associated to $mathcalE$.

How are the Chern classes $c_i$ of the vector bundle $mathcalE$ on $X$ related to the Chern classes $c_i'$ of the projective variety $mathbbP(mathcalE)$, i.e. the Chern classes of the tangent bundle of $mathbbP(mathcalE)$? My naive hope would be that we simply have $c_i'=pi^* c_i$. Is that true? Is there a good reference?

algebraic-geometry vector-bundles

share|cite|improve this question

asked Apr 13 at 14:31

HansHans

1,687616

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The answer is correct, but one can say more. There is a formula for the tangent bundle of a projective bundle that relates it to the pullback of the tangent bundle from below AND the relative tangent bundle. So there is a formula, but the formula makes it clear that what you are asking will almost never hold.
  $endgroup$
  – aginensky
  Apr 13 at 20:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Hi @aginensky ! That sounds interesting, can you maybe state this formula in a new answer or point me to a reference?
  $endgroup$
  – Hans
  Apr 15 at 7:55
  

  
  
  
  

add a comment | 

2

$begingroup$

I have a very basic question regarding Chern classes. Let $X$ be a smooth projective variety and $mathcalE$ a vector bundle on it. Let $pi:mathbbP(mathcalE)to X$ denote the projective space bundle over $X$ associated to $mathcalE$.

How are the Chern classes $c_i$ of the vector bundle $mathcalE$ on $X$ related to the Chern classes $c_i'$ of the projective variety $mathbbP(mathcalE)$, i.e. the Chern classes of the tangent bundle of $mathbbP(mathcalE)$? My naive hope would be that we simply have $c_i'=pi^* c_i$. Is that true? Is there a good reference?

algebraic-geometry vector-bundles

share|cite|improve this question

asked Apr 13 at 14:31

HansHans

1,687616

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The answer is correct, but one can say more. There is a formula for the tangent bundle of a projective bundle that relates it to the pullback of the tangent bundle from below AND the relative tangent bundle. So there is a formula, but the formula makes it clear that what you are asking will almost never hold.
  $endgroup$
  – aginensky
  Apr 13 at 20:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Hi @aginensky ! That sounds interesting, can you maybe state this formula in a new answer or point me to a reference?
  $endgroup$
  – Hans
  Apr 15 at 7:55
  

  
  
  
  

add a comment | 

$begingroup$

I have a very basic question regarding Chern classes. Let $X$ be a smooth projective variety and $mathcalE$ a vector bundle on it. Let $pi:mathbbP(mathcalE)to X$ denote the projective space bundle over $X$ associated to $mathcalE$.

How are the Chern classes $c_i$ of the vector bundle $mathcalE$ on $X$ related to the Chern classes $c_i'$ of the projective variety $mathbbP(mathcalE)$, i.e. the Chern classes of the tangent bundle of $mathbbP(mathcalE)$? My naive hope would be that we simply have $c_i'=pi^* c_i$. Is that true? Is there a good reference?

algebraic-geometry vector-bundles

share|cite|improve this question

asked Apr 13 at 14:31

HansHans

1,687616

$endgroup$

share|cite|improve this question

asked Apr 13 at 14:31

HansHans

1,687616

share|cite|improve this question

asked Apr 13 at 14:31

HansHans

1,687616

asked Apr 13 at 14:31

HansHans

1,687616

asked Apr 13 at 14:31

HansHans

1,687616

1 Answer
1

active

oldest

votes

6

$begingroup$

The nice relation that you hope for is not true unfortunately. Here is a counter-example: consider the trivial rank 2 vector bundle $E= mathcalO oplus mathcalO$ over $mathbbCP^1$. Then $$mathbbP(E) cong mathbbCP^1 times mathbbCP^1.$$

Now, note that $c_2(E)=0$ since $E$ is a trivial bundle, but $int_mathbbP(E) c_2(TmathbbP(E)) = 4$, since this is equal to the topological Euler characteristic of $mathbbP(E)$ (it is a standard result that the integral of the top Chern class of a complex manifold is equal to the topological Euler characteristic).

In general, I don't think that there will be a nice relation (although I could be wrong) since when $E$ has different Chern numbers, the topological type of $mathbbP(E)$ will be different in general, so we are not even talking about Chern classes on the same complex manifold. However, you can actually compute some topological invariants of $mathbbP(E)$ from the chern classes of $E$: see Proposition 15 of Okonek and Van de Ven's "Cubic forms and complex 3-folds".

share|cite|improve this answer

edited Apr 13 at 14:57

answered Apr 13 at 14:44

Nick LNick L

1,422210

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add a comment | 

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6

$begingroup$

The nice relation that you hope for is not true unfortunately. Here is a counter-example: consider the trivial rank 2 vector bundle $E= mathcalO oplus mathcalO$ over $mathbbCP^1$. Then $$mathbbP(E) cong mathbbCP^1 times mathbbCP^1.$$

Now, note that $c_2(E)=0$ since $E$ is a trivial bundle, but $int_mathbbP(E) c_2(TmathbbP(E)) = 4$, since this is equal to the topological Euler characteristic of $mathbbP(E)$ (it is a standard result that the integral of the top Chern class of a complex manifold is equal to the topological Euler characteristic).

In general, I don't think that there will be a nice relation (although I could be wrong) since when $E$ has different Chern numbers, the topological type of $mathbbP(E)$ will be different in general, so we are not even talking about Chern classes on the same complex manifold. However, you can actually compute some topological invariants of $mathbbP(E)$ from the chern classes of $E$: see Proposition 15 of Okonek and Van de Ven's "Cubic forms and complex 3-folds".

share|cite|improve this answer

edited Apr 13 at 14:57

answered Apr 13 at 14:44

Nick LNick L

1,422210

$endgroup$

add a comment | 

6

$begingroup$

The nice relation that you hope for is not true unfortunately. Here is a counter-example: consider the trivial rank 2 vector bundle $E= mathcalO oplus mathcalO$ over $mathbbCP^1$. Then $$mathbbP(E) cong mathbbCP^1 times mathbbCP^1.$$

Now, note that $c_2(E)=0$ since $E$ is a trivial bundle, but $int_mathbbP(E) c_2(TmathbbP(E)) = 4$, since this is equal to the topological Euler characteristic of $mathbbP(E)$ (it is a standard result that the integral of the top Chern class of a complex manifold is equal to the topological Euler characteristic).

In general, I don't think that there will be a nice relation (although I could be wrong) since when $E$ has different Chern numbers, the topological type of $mathbbP(E)$ will be different in general, so we are not even talking about Chern classes on the same complex manifold. However, you can actually compute some topological invariants of $mathbbP(E)$ from the chern classes of $E$: see Proposition 15 of Okonek and Van de Ven's "Cubic forms and complex 3-folds".

share|cite|improve this answer

edited Apr 13 at 14:57

answered Apr 13 at 14:44

Nick LNick L

1,422210

$endgroup$

add a comment | 

$begingroup$

The nice relation that you hope for is not true unfortunately. Here is a counter-example: consider the trivial rank 2 vector bundle $E= mathcalO oplus mathcalO$ over $mathbbCP^1$. Then $$mathbbP(E) cong mathbbCP^1 times mathbbCP^1.$$

Now, note that $c_2(E)=0$ since $E$ is a trivial bundle, but $int_mathbbP(E) c_2(TmathbbP(E)) = 4$, since this is equal to the topological Euler characteristic of $mathbbP(E)$ (it is a standard result that the integral of the top Chern class of a complex manifold is equal to the topological Euler characteristic).

In general, I don't think that there will be a nice relation (although I could be wrong) since when $E$ has different Chern numbers, the topological type of $mathbbP(E)$ will be different in general, so we are not even talking about Chern classes on the same complex manifold. However, you can actually compute some topological invariants of $mathbbP(E)$ from the chern classes of $E$: see Proposition 15 of Okonek and Van de Ven's "Cubic forms and complex 3-folds".

share|cite|improve this answer

edited Apr 13 at 14:57

answered Apr 13 at 14:44

Nick LNick L

1,422210

$endgroup$

share|cite|improve this answer

edited Apr 13 at 14:57

answered Apr 13 at 14:44

Nick LNick L

1,422210

answered Apr 13 at 14:44

Nick LNick L

1,422210

answered Apr 13 at 14:44

Nick LNick L

1,422210