All Questions
Tagged with line-bundles riemann-surfaces
9 questions
1
vote
1
answer
373
views
Flat connection of a degree zero line bundle on curve
The question is clear from the title. Suppose we have a line bundle on a compact smooth complex curve $X$, and a line bundle $\mathcal{L}=\mathcal{O}_X(p-q)$, where $p$ and $q$ are divisors, then what ...
- 93
1
vote
0
answers
469
views
Dimension of global holomorphic sections of a line bundle
Let $K$ be the canonical line bundle of a compact Riemann surface $M$ of genus $g$. Consider the pull back of $K$ on $M \times M$ via projection on the first factor. What is the dimension of the space ...
- 35
3
votes
0
answers
195
views
Is Wronskian a line bundle for Riemann surfaces?
Suppose $f_1,\dots,f_g$ are holomorphic functons on a domain $U\subset\mathbb{C}$. By the Wronskian determinant $f_1,\dots,f_g$ one means the determinant of the matrix of derivatives $f_k^{(m)},$ ...
- 954
4
votes
2
answers
307
views
Existence of a connection $A$ on a holomorphic line bundle $L$, s.t $F(A)=(\deg L)\omega$
I'm reading this paper and at page 67, he states that for any line bundle $L$ over a Rieman surface there is a connection $A$ whose curvature is
$$
F(A)=(\deg L)\omega,
$$
where $\omega$ is a positive ...
- 311
17
votes
2
answers
2k
views
Square root of the determinant line
Let $\Sigma$ be a compact Riemann surface equipped with a spin structure (a square root of $\Omega^1_\Sigma$, denoted $\Omega^{1/2}_\Sigma$).
Let $\Gamma(\Omega^1_\Sigma)$ be the space of holomorphic ...
- 43.2k
0
votes
1
answer
1k
views
Is there a toplogically trivial line bundle over a compact Riemann surfaces that isn't holomorphically trivial? [closed]
Is a complex line bundle over a compact Riemann surface topologically trivial iff it is holomorphically trivial? If so, how does one demonstrate that, and if not, what is a counterexample?
- 127
0
votes
1
answer
265
views
Trivial Line Bundle-Riemann surfaces
What are the Hermitian metrics in a trivial line bundle on a Riemann surface X?
I read that a Hermitian metric in the trivial line bundle is equivalent to a $\mathcal{C}^{\infty}$ weight function $\...
- 59
17
votes
3
answers
3k
views
Is there an algebraic construction of the Quillen (determinant) Line Bundle?
Let's consider the moduli space of representations of $\pi=\pi_1(\Sigma)$ (a surface group) into $G$ (a lie group). Call this $X=\operatorname{Hom}(\pi,G)$, and let $Y=\operatorname{Hom}(\pi,G)/\\!/G$...
- 18.7k
4
votes
2
answers
825
views
finite-dimensionality of cohomology groups on compact riemann surfaces
does the finite dimensionlity of the first cohomology group ($ H^1 $) of the sheaf of meromorphic sections of a holomorphic line bundle on a compact riemann surface follow easily from the finite ...
- 73