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Tagged with complex-geometry vector-bundles
160
questions
6
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0
answers
200
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Holomorphic fibre bundles over noncompact Riemann surfaces
Some days ago I came across the paper "Holomorphic fiber bundles over Riemann surfaces", by H. Rohrl.
At the beginning of Section 1, the following theorem is quoted:
Theorem. Every fiber ...
- 271
2
votes
1
answer
183
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Identifying the circle bundle of the canonical line bundle $\mathcal{O}(-n-1)$ over a projective space $\mathbb{CP}^n$
It's not hard to see the following fact: the circle bundle of the tautological line bundle $\mathcal{O}(-1)\rightarrow \mathbb{CP}^n$ is $S^{2n+1}$, the unit sphere inside $C^{n+1}.$ I want to see ...
- 923
2
votes
1
answer
176
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Proving that $H^1(X,\mathcal{Hom}(\mathcal{G},\mathcal{E})) \cong \text{Ext}^1(\mathcal{G},\mathcal{E})$ holds for locally free sheaves
The following passage is from a thesis I'm reading:
Suppose we have a short exact sequence of vector bundles $$0 \to \mathcal{E} \to \mathcal{F}\to \mathcal{G} \to 0.$$ Since these sheaves are ...
- 63
1
vote
1
answer
443
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Vector bundles on $\mathbb{P}^1$
I am considering an alternative proof of Grothendieck's classification of vector bundles on $\mathbb{P}^1$. Given a vector bundle $E$ on $\mathbb{P}^1$ one can associate a graded module $\Gamma(E)$ ...
- 21
3
votes
0
answers
78
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Let $f: X \to D$ be a proper holomorphic submersion. Does a holomorphic form on $X$, closed on each fiber of $f$, have holomorphic coefficients?
Let $D$ be the unit disc in $\mathbb{C}^n$ and let $f: X \to D$ be a proper surjective holomorphic submersion, which is trivial as a smooth fiber bundle, with connected fiber $F$. We get an induced ...
- 538
0
votes
1
answer
307
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Self-intersection of zero section of line bundle over elliptic base curve
Let $C$ be an elliptic curve over $k=\mathbb{C}$ and $\mathcal{L}$ a line bundle of degree $d$. It induces naturally a $\mathbb{A}^1$-fibration $L \to C$ where $L=\underline{\operatorname{Spec}} (\...
- 5,780
4
votes
2
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385
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Is a positive degree self map on a Riemann surface homotopic to a holomorphic self map?
Let $S$ be a compact Riemann surface and $f:S\to S$ be a continuous self map of positive degree. Is $f$ homotopic to a holomorphic map on $S$?
Motivation: I had intention to consider this question ...
- 452
0
votes
0
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124
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Rank of a tangent map related to holomorphic line bundles
Let $L,\,\,J$ be two holomorphic line bundles over a compact Riemann surface $X$ of genus $g_X>0$ such that
(1) $d_1:=\dim H^0\big(\operatorname{Hom}(L,J)\big)>0$ and $d_2:=\dim H^0\big(\...
- 371
13
votes
1
answer
498
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Local systems arising from higher rational homotopy groups
I should mention I have very little background in algebraic topology and don't really know much about homotopy groups besides the definition.
I am aware that for a topological space $X$ and a point $x ...
- 233
1
vote
1
answer
163
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Vector subbundles of a given one in $\mathbb{CP}^1$
I apologize if this question is not suited for MathOverflow. This has been crossposted in MathStackExchange here and it is related to some open questions on that site that remain unsolved.
I would ...
- 13
5
votes
1
answer
176
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Does it make sense to define a holomorphic structure on $\mathbb{C}P^\infty$ and vector bundles over it?
Let $E \to \mathbb{C}P^\infty$ be any topological complex vector bundle over the infinite complex projective space. I'm wondering if it makes sense to possibly define a "holomorphic structure&...
- 1,755
3
votes
0
answers
288
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Ampleness of the normal bundle to the Albanese image
Let $X$ be a projective surface of general type over $\mathbb{C}$, and assume that $\Omega_X$ is globally generated. Then the Albanese map $a \colon X \to \operatorname{Alb}(X)$ is a local embedding ...
- 65.7k
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1
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241
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Stability of sheaves of non-constant rank
Let $E\to X$ be a coherent sheaf over a compact (projective) Kahler manifold. The definition of stability of sheaves as stated in Huybrechts-Lehn (Definition 1.2.12) says that $E$ is stable if for all ...
- 767
1
vote
0
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88
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A diagonal Chern curvature
Let $E\to X$ be a rank $r$ holomorphic Hermitian vector bundle over a (compact) Kahler manifold.
Are there conditions on $E$ and/or the metric $h$ such that there exists everywhere a local frame in ...
- 767
0
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0
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169
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Is there a meaning to the equation $c_1(E,h)=\lambda \omega$?
Let $(X,\omega)$ be a Kahler manifold and $(E,h)\to X$ a Hermitian holomorphic vector bundle on $X$. Denote by $c_1(E,h)\in \Omega^{1,1}(X)$ the first Chern form of $E$ with respect to the metric $h$. ...
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