Skip to main content
MathOverflow

Return to Question

edited body; edited title
Source Link
user40184
user40184

Bundle over $\mathbb{C}^{n}\setminus{O0}$

Let $\pi:\mathbb{C}^{n}\setminus{O}\rightarrow\mathbb{CP}^{n-1}, n\geq 3$$\pi:\mathbb{C}^{n}\setminus{0}\rightarrow\mathbb{CP}^{n-1}, n\geq 3$ be the projection from affine space without the origin to the projective space. If we pull back the tangent bundle of $\mathbb{CP}^{n-1}$ we would get a nontrivial bundle over $\mathbb{C}^{n}\setminus{O}$$\mathbb{C}^{n}\setminus{0}$. Now my question would be: what is $H^{1}(\mathbb{C}^{n}\setminus{O},\pi^{*}\mathcal{T}_{\mathbb{CP}^{n-1}})$$H^{1}(\mathbb{C}^{n}\setminus{0},\pi^{*}\mathcal{T}_{\mathbb{CP}^{n-1}})$?

Bundle over $\mathbb{C}^{n}\setminus{O}$

Let $\pi:\mathbb{C}^{n}\setminus{O}\rightarrow\mathbb{CP}^{n-1}, n\geq 3$ be the projection from affine space without the origin to the projective space. If we pull back the tangent bundle of $\mathbb{CP}^{n-1}$ we would get a nontrivial bundle over $\mathbb{C}^{n}\setminus{O}$. Now my question would be: what is $H^{1}(\mathbb{C}^{n}\setminus{O},\pi^{*}\mathcal{T}_{\mathbb{CP}^{n-1}})$?

Bundle over $\mathbb{C}^{n}\setminus{0}$

Let $\pi:\mathbb{C}^{n}\setminus{0}\rightarrow\mathbb{CP}^{n-1}, n\geq 3$ be the projection from affine space without the origin to the projective space. If we pull back the tangent bundle of $\mathbb{CP}^{n-1}$ we would get a nontrivial bundle over $\mathbb{C}^{n}\setminus{0}$. Now my question would be: what is $H^{1}(\mathbb{C}^{n}\setminus{0},\pi^{*}\mathcal{T}_{\mathbb{CP}^{n-1}})$?

Replaced diff. geometry tag by alg. geometry since the problem is purely algebraic(holomorphic)
Link
edit approved Sep 8, 2016 at 11:34
SashaP
edit approved Sep 8, 2016 at 11:34
SashaP
  • 7.4k
  • 1
  • 31
  • 46
Fixed a typo in the dimension of porjective space and clarified the definition of
Source Link
edit approved Sep 8, 2016 at 10:02
SashaP
edit approved Sep 8, 2016 at 10:02
SashaP
  • 7.4k
  • 1
  • 31
  • 46

Let $\pi:\mathbb{C}^{n}\setminus{O}\rightarrow\mathbb{CP}^{n-1}, n\geq 3$, if be the projection from affine space without the origin to the projective space. If we pull back the tangent bundle of $\mathbb{CP}^{n-1}$ we would get a nontrivial bundle over $\mathbb{C}^{n}\setminus{O}$. Now my question would be: what is $H^{1}(\mathbb{C}^{n}\setminus{O},\pi^{*}\mathcal{T}_{\mathbb{CP}^{n}})$$H^{1}(\mathbb{C}^{n}\setminus{O},\pi^{*}\mathcal{T}_{\mathbb{CP}^{n-1}})$?

Let $\pi:\mathbb{C}^{n}\setminus{O}\rightarrow\mathbb{CP}^{n-1}, n\geq 3$, if we pull back the tangent bundle of $\mathbb{CP}^{n-1}$ we would get a nontrivial bundle over $\mathbb{C}^{n}\setminus{O}$. Now my question would be: what is $H^{1}(\mathbb{C}^{n}\setminus{O},\pi^{*}\mathcal{T}_{\mathbb{CP}^{n}})$?

Let $\pi:\mathbb{C}^{n}\setminus{O}\rightarrow\mathbb{CP}^{n-1}, n\geq 3$ be the projection from affine space without the origin to the projective space. If we pull back the tangent bundle of $\mathbb{CP}^{n-1}$ we would get a nontrivial bundle over $\mathbb{C}^{n}\setminus{O}$. Now my question would be: what is $H^{1}(\mathbb{C}^{n}\setminus{O},\pi^{*}\mathcal{T}_{\mathbb{CP}^{n-1}})$?

Source Link
user40184
user40184